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<h1 class="Header__Title">支持向量机</h1>
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<article id="https://www.notion.so/3c0db3a790ba4de2a2685e166b11141f" class="PageRoot"><ul id="https://www.notion.so/b7069dfb62744775bf34caba9ec8dac0" class="ColorfulBlock ColorfulBlock--ColorGray TableOfContents"><li class="TableOfContents__Item"><a href="#https://www.notion.so/1eefe3390c294617bd83e93e04a18202"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">基本概念——线性可分类问题</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/68df4417e7d443bab1b4a61a809fe01a"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">线性可分</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/84d68a214237479eb5e6f4c53288eb3c"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">定义</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/0a2ec726bb0f40b2a7bcbe370da69014"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">优化问题</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/c9ac5810e51e40a19293af9e40fd4bc8"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">小结</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/72cd14da756f413ba75faec9f86bdadc"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">非线性问题</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/37ee5cf7e9f74344889e09bb8db00bc6"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">软间隔</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/d004565952574259b5ef1b7b238fb8b1"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">非线性可分</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/4ff2014eb303462683b7fe5734a9d1ed"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">核函数</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/1dfc3708698642ee9f75defa9cc514a4"><div style="margin-left:0px"><span class="SemanticStringArray"><span class="SemanticString">对偶问题</span></span></div></a></li><li class="TableOfContents__Item"><a href="#https://www.notion.so/88d77d2c5e81400d850ae23d57995701"><div style="margin-left:24px"><span class="SemanticStringArray"><span class="SemanticString">定义</span></span></div></a></li></ul><h2 id="https://www.notion.so/1eefe3390c294617bd83e93e04a18202" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/1eefe3390c294617bd83e93e04a18202"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">基本概念——线性可分类问题</span></span></h2><div id="https://www.notion.so/6ea5b41b2e784702ab2da129e21414e0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">支持向量机是基于统计学习理论的一种实用的机器学习方法。</span></span></p></div><div id="https://www.notion.so/07b0847ce5914f8ebd8177b842634df5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">SVM在解决</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">小样本</mark></span><span class="SemanticString">、非线性及高维模式识别问题中表现出许多特有的优势，并能够推广应用到函数拟合等其他机器学习问题中。</span></span></p></div><div id="https://www.notion.so/1d17b0d8c5f34d2393993c27f4f0777a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">机器学习算法的时间线：</span></span></p></div><div id="https://www.notion.so/6d6b8825ca8342d1b6fdcfe314579174" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fb2cff37f-ab82-425d-bcb6-a1d95054f3bb%2FUntitled.png?width=1793&table=block&id=6d6b8825-ca83-42d1-b6fd-cfe314579174"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fb2cff37f-ab82-425d-bcb6-a1d95054f3bb%2FUntitled.png?width=1793&table=block&id=6d6b8825-ca83-42d1-b6fd-cfe314579174" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><h3 id="https://www.notion.so/68df4417e7d443bab1b4a61a809fe01a" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/68df4417e7d443bab1b4a61a809fe01a"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">线性可分</span></span></h3><div id="https://www.notion.so/91e020e7360e4c99b6b827e350be5229" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F8c88b442-c556-4089-8127-e6963fea038f%2FUntitled.png?width=713&table=block&id=91e020e7-360e-4c99-b6b8-27e350be5229"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F8c88b442-c556-4089-8127-e6963fea038f%2FUntitled.png?width=713&table=block&id=91e020e7-360e-4c99-b6b8-27e350be5229" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/0f2d670b7f4b4c0aac1d7e8273877e10" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">问题：无数条线，哪个直线是最好的？</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/7bd32baaed4e483d96f82710c938fa7b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">如何定义这条线？</span></span></li><li id="https://www.notion.so/b719f93fc6db417690a7be2715122ef3" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">定义一个衡量每一条线的标准，每条线都能算出来一个性能指标。</span></span></li><li id="https://www.notion.so/5419b687716e4e4397dfcd7c3d275a64" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">哪条线能让这个性能指标最大？</span></span></li></ul><div id="https://www.notion.so/c6d5d919f6fa4214a865823966cc74fa" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">好的决策边界：间隔大</strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/71ed1d38c3994baa8e94487f8d6ef981" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">决策边界离两类数据应尽可能远</span></span></li></ul><div id="https://www.notion.so/765abf2cf7c149f2a157bb0740f6f0e5" class="ColumnList"><div id="https://www.notion.so/fadd282b2a20414c8ac3053b52a16ae6" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.4375)"><div id="https://www.notion.so/5cd513c6be6c4e6ba8d38860ebec94ed" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fbb07bbdd-b82e-4e88-84e2-8fb76350fc36%2FUntitled.png?width=298&table=block&id=5cd513c6-be6c-4e6b-a8d3-8860ebec94ed"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fbb07bbdd-b82e-4e88-84e2-8fb76350fc36%2FUntitled.png?width=298&table=block&id=5cd513c6-be6c-4e6b-a8d3-8860ebec94ed" style="width:298px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div><div id="https://www.notion.so/0eae4802b551426ea02fe2cf5e295f5f" class="Column" style="width:calc((100% - var(--column-spacing) * 1) * 0.5625)"><div id="https://www.notion.so/a9539c46b68048faa0be4ff0435effbc" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fddd0134b-9016-4a28-9bf0-0d6114c4b534%2FUntitled.png?width=238&table=block&id=a9539c46-b680-48fa-a0be-4ff0435effbc"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fddd0134b-9016-4a28-9bf0-0d6114c4b534%2FUntitled.png?width=238&table=block&id=a9539c46-b680-48fa-a0be-4ff0435effbc" style="width:238px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div></div></div><div id="https://www.notion.so/ab940cb0cba848f6813ff831712a3c54" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">将平行线插到的向量称为支持向量</span></span></p></div><div id="https://www.notion.so/82fc91d3818a4f9685ef58dbe2f011f0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">让这条线平行移动，直到能够插到某一个或几个圆圈为止</span></span></p></div><div id="https://www.notion.so/5bcd0a20d8dd45398fb07204baeb706f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">这个距离作为性能指标，绿色线是这个距离最大的一条线</span></span></p></div><h3 id="https://www.notion.so/84d68a214237479eb5e6f4c53288eb3c" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/84d68a214237479eb5e6f4c53288eb3c"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">定义</span></span></h3><ul class="BulletedListWrapper"><li id="https://www.notion.so/c18fed9bc5024a679d927ae2939e74c8" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">训练数据和标签：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></span></li></ul><div id="https://www.notion.so/813a9696d7c04b9ba352eb4a9946613b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">其中</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_i=\begin{bmatrix}
x_{i1} \\
x_{i2} \\
\dots \\
x_{im}
\end{bmatrix}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mrow><mi>i</mi><mi>m</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">x_i=\begin{bmatrix}
x_{i1} \\
x_{i2} \\
\dots \\
x_{im}
\end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:4.80303em;vertical-align:-2.15003em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">为向量，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y_i=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString">或</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="-1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString">为标签。</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/f3e529c4c69343038b33bdc99d260657" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">线性模型：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(w,b)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(w,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> </span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w^\text{T}x+b=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>w</mi><mtext>T</mtext></msup><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w^\text{T}x+b=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.924661em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></mark></span><span class="SemanticString">（超平面 Hyperplane）</span></span></li></ul><div id="https://www.notion.so/1cac1c9c026c48469e951fa50f83418c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span></span></span><span class="SemanticString">：向量，维度与</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span></span></span><span class="SemanticString">一样 </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w=\begin{bmatrix}
w_{1} \\
w_{2} \\
\dots \\
w_{m}
\end{bmatrix}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>w</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>w</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>w</mi><mi>m</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">w=\begin{bmatrix}
w_{1} \\
w_{2} \\
\dots \\
w_{m}
\end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:4.80303em;vertical-align:-2.15003em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString"> </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span><span class="SemanticString">：常数</span></span></p></div><div id="https://www.notion.so/7a11b9075dfb43d7ab042a635bcf6b7c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">要找到描述这个超平面的线性方程</span></span></p></div><div id="https://www.notion.so/902d2a1aa1e34173a60bc789ca1fe323" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">通过所有</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span></span></span></span><span class="SemanticString">的取值，来算出</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/2e462a5c3e81401098f958f0e029d004" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">一个训练集线性可分是指：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="{}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex">{}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;vertical-align:0em;"></span><span class="mord"></span></span></span></span></span></span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{(x_i,y_i)\},i=1∼N"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>∼</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\{(x_i,y_i)\},i=1∼N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span></span></span></span></span></span></li></ul><div id="https://www.notion.so/125eda4fcf8b4cd78ec12ac5224ed54e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\exists (w,b)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi><mo stretchy="false">(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exists (w,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∃</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，使对</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\forall i=1∼N"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>i</mi><mo>=</mo><mn>1</mn><mo>∼</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\forall i=1∼N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">∀</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span></span></span></span></span><span class="SemanticString">，有：</span></span></p></div><ol class="NumberedListWrapper"><li id="https://www.notion.so/c9a079f3da40433694e4968ba3155d6f" class="NumberedList" value="1"><span class="SemanticStringArray"><span class="SemanticString">若</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i=+1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y_i=+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">+</span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString">，则</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w^\text{T}x_i+b\ge0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w^\text{T}x_i+b\ge0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.991331em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">。</span></span></li><li id="https://www.notion.so/ec4e85082f8746bc8e8da6b12c0501dd" class="NumberedList" value="2"><span class="SemanticStringArray"><span class="SemanticString">若</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i=-1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y_i=-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span></span><span class="SemanticString">，则</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w^\text{T}x_i+b\le0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w^\text{T}x_i+b\le0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.991331em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">。</span></span></li></ol><div id="https://www.notion.so/3847c5e8b0944ff2824358301e1617ec" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">统一表示为：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i[w^\text{T}x_i+b]\ge 0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo stretchy="false">]</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y_i[w^\text{T}x_i+b]\ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f7a77e02a3714c0b9ea3d60b6fd3a22c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">线性可分数据的超平面</strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/efa0246fc14a40d6aecae081018ee287" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">当训练数据集线性可分时，存在无穷多个分离超平面可将两类数据正确分开。</span></span></li><li id="https://www.notion.so/258b0f04fcf54e66b1126a11858724a9" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">感知机</mark></span><span class="SemanticString">利用误分类最小的策略，求得分离超平面，但有无穷多个解。</span></span></li><li id="https://www.notion.so/0bce9efe109a495c98caf986ffbbadfa" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">支持向量机</mark></span><span class="SemanticString">利用间隔最大化求最优分离超平面，解唯一。</span></span></li></ul><h3 id="https://www.notion.so/0a2ec726bb0f40b2a7bcbe370da69014" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/0a2ec726bb0f40b2a7bcbe370da69014"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">优化问题</span></span></h3><ul class="BulletedListWrapper"><li id="https://www.notion.so/7af8b7c98a274716a78fb81816eacd9d" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">如何得到间隔最大的超平面</span></span></li></ul><div id="https://www.notion.so/2e2bf81aed76437d9a1d6336e8a9c3ae" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">支持向量机做了一个优化问题：</mark></span></span></p></div><div id="https://www.notion.so/d59477fcac7b45b7b5e12c3c00c4669a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">最小化（Minimize）</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\lVert w\rVert"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∥</mo><mi>w</mi><mo stretchy="false">∥</mo></mrow><annotation encoding="application/x-tex">\lVert w\rVert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mclose">∥</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/c325c27b833746a3931cbf088d9da5ba" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">约束条件（Subject to）</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i[w^\text{T}x_i+b]\ge 1\;(i=1∼N)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo stretchy="false">]</mo><mo>≥</mo><mn>1</mn><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>∼</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y_i[w^\text{T}x_i+b]\ge 1\;(i=1∼N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> </span></span></p></div><div id="https://www.notion.so/cd46d3ed7c5a40c78fb4cb790777d1dd" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w^\text{T}+b=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>w</mi><mtext>T</mtext></msup><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w^\text{T}+b=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.924661em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">与</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="aw^\text{T}+ab=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><msup><mi>w</mi><mtext>T</mtext></msup><mo>+</mo><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">aw^\text{T}+ab=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.924661em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">a</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">a</span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">是同一个平面，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a\in \R^+"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">a\in \R^+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.771331em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathbb">R</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.771331em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span></span></span></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/d077f45eae3b4d38b84655f0b5f196b2" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">点</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(x_0,y_0)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0,y_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">到平面</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w_1x+w_2x+b=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>w</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w_1x+w_2x+b=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">的距离：</span></span><div id="https://www.notion.so/890dd5581ef14fbd9279eedc9cc2cc7f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="d=\cfrac{|w_1x+w_2x+b|}{\sqrt{w_1^2+w_2^2}}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi mathvariant="normal">∣</mi><msub><mi>w</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub><mi>x</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">∣</mi></mrow><msqrt><mrow><msubsup><mi>w</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>w</mi><mn>2</mn><mn>2</mn></msubsup></mrow></msqrt></mfrac></mstyle></mrow><annotation encoding="application/x-tex">d=\cfrac{|w_1x+w_2x+b|}{\sqrt{w_1^2+w_2^2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.7199999999999998em;vertical-align:-1.13em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.1602em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9498em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em;"><span style="top:-2.433692em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em;"><span style="top:-2.433692em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.9098em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg width='400em' height='1.28em' viewBox='0 0 400000 1296' preserveAspectRatio='xMinYMin slice'><path d='M263,681c0.7,0,18,39.7,52,119
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class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.13em;"><span></span></span></span></span></span><span></span></span></span></span></span></span></span></span></p></div></li><li id="https://www.notion.so/cd23093aa4a64dd88584bd98625b589f" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">向量</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">到平面</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w^\text{T}x+b=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>w</mi><mtext>T</mtext></msup><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w^\text{T}x+b=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.924661em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">的距离：</span></span><div id="https://www.notion.so/8f336660a239467db77ba013b6f57263" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="d=\cfrac{|w^\text{T}x_0+b|}{\lVert w\rVert}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi mathvariant="normal">∣</mi><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>b</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mo stretchy="false">∥</mo><mi>w</mi><mo stretchy="false">∥</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">d=\cfrac{|w^\text{T}x_0+b|}{\lVert w\rVert}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.526em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mclose">∥</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span></span></span></span></span></span></span></span><span class="SemanticString">，其中</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\lVert w\rVert=\sqrt{w_1^2+w_2^2+\cdots+w_m^2}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∥</mo><mi>w</mi><mo stretchy="false">∥</mo><mo>=</mo><msqrt><mrow><msubsup><mi>w</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>w</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>w</mi><mi>m</mi><mn>2</mn></msubsup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\lVert w\rVert=\sqrt{w_1^2+w_2^2+\cdots+w_m^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mclose">∥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.24em;vertical-align:-0.2902em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9498em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em;"><span style="top:-2.433692em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em;"><span style="top:-2.433692em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.9098em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg width='400em' height='1.28em' viewBox='0 0 400000 1296' preserveAspectRatio='xMinYMin slice'><path d='M263,681c0.7,0,18,39.7,52,119
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M1001 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2902em;"><span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">。</span></span></p></div></li></ul><div id="https://www.notion.so/3569fb3d69dc448184719ff1c2643a1f" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">SVM是最大化间隔的分类算法</mark></span></span></p></div><div id="https://www.notion.so/3af446f2508e4919a9b9b891bfb54a19" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">用</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span><span class="SemanticString">缩放</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(w,b)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(w,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">→</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="(aw,ab)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>w</mi><mo separator="true">,</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(aw,ab)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，最终使其在支持向量</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">上有：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="|w^\text{T}x_0+b|=1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|w^\text{T}x_0+b|=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/05a07648d78943cb98b533dcb2a00504" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">此时支持向量与平面的距离：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="d=\cfrac{1}{\lVert w\rVert}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mo stretchy="false">∥</mo><mi>w</mi><mo stretchy="false">∥</mo></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">d=\cfrac{1}{\lVert w\rVert}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.526em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5899999999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mclose">∥</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span></span></span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/295d8d8f35a54ef5909cf8fb8d69e6d5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">最小化</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\lVert w\rVert"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∥</mo><mi>w</mi><mo stretchy="false">∥</mo></mrow><annotation encoding="application/x-tex">\lVert w\rVert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mclose">∥</span></span></span></span></span></mark></span><span class="SemanticString">，其他点大于</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="d"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/0433458c072540ed8ea3df2a9e840354" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">y_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">取正负1，可以协调两个类，也可以改成任意整数，差距就是</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="a"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/a177938a749247e7a03aaa4a8c64bcda" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">数据集如果线性可分</strong></span><span class="SemanticString">，就能求得</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span></span></span><span class="SemanticString">和</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="b"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span></span></span></span></span></span><span class="SemanticString">，满足条件：</span></span></p></div><div id="https://www.notion.so/624e80351eea45bea9a0b6370cf314bb" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\frac{1}{2}\lVert w\rVert^2=\frac{1}{2}(w_1^2+w_2^2+\cdots+w_m^2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">∥</mo><mi>w</mi><msup><mo stretchy="false">∥</mo><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msubsup><mi>w</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>w</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>w</mi><mi>m</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}\lVert w\rVert^2=\frac{1}{2}(w_1^2+w_2^2+\cdots+w_m^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mclose"><span class="mclose">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0622159999999998em;vertical-align:-0.24810799999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString"> </span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\frac{1}{2}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span><span class="SemanticString">是为了求导方便</span></span></p></div><div id="https://www.notion.so/5df21cb58e3f4368a1abaf0f38edbe0c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i[w^\text{T}x_i+b]\ge 1\;(i=1∼N)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo stretchy="false">]</mo><mo>≥</mo><mn>1</mn><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>∼</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y_i[w^\text{T}x_i+b]\ge 1\;(i=1∼N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/66a38c48f1b54936b5f48d66ba4f878b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">优化问题</strong></span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/38ae7c7a939a4f4c97da56dc95181ff7" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">目标优化问题是一个凸优化问题中的一种二次规划问题</span></span></li><li id="https://www.notion.so/2827959451b3431a9a63b7f69de512b2" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">二次规划</mark></span><span class="SemanticString">（quantity programming）</span></span><div id="https://www.notion.so/2bedf01638c04f4ba5daec79f149cbae" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">目标函数是二次项；约束条件是一次项</span></span></p></div><div id="https://www.notion.so/956a491260014f52851910c3dfbad176" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无解或者一个最小值</span></span></p></div><div id="https://www.notion.so/77bf5892f94a452586ce0bd625d750b9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">二次规划是计算机已解决的问题，其局部极值就是全局极值</span></span></p></div></li><li id="https://www.notion.so/a9e01b058dad4e2395d33b290b23d92d" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">用试探方法，求极值，例如梯度下降</span></span></li><li id="https://www.notion.so/fa24509df99543deac47607d5da5f92b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">多维情况下很困难，有时人眼无法看到所有情况</span></span></li></ul><h3 id="https://www.notion.so/c9ac5810e51e40a19293af9e40fd4bc8" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/c9ac5810e51e40a19293af9e40fd4bc8"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">小结</span></span></h3><ul class="BulletedListWrapper"><li id="https://www.notion.so/ee7ca93d9a0a42358ddc485de85e149b" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">SVM是最大化间隔（margin）的分类算法</strong></span></span></li><li id="https://www.notion.so/51b03ad27ac846b586af2e251dbb28ce" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">优化问题</span></span><div id="https://www.notion.so/58f4ae825b4b4a18b35bda81d4f1d3cc" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">训练样本</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\{(x_i,y_i)\}_{i=1∼N}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>∼</mo><mi>N</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{(x_i,y_i)\}_{i=1∼N}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span><span class="mrel mtight">∼</span><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></p></div></li><li id="https://www.notion.so/aa9095e5f29c47169103da3d53e6163a" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">优化目标</span></span><div id="https://www.notion.so/a1f3b00663434edc801e1de382040a77" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\min\frac{1}{2}\|w\|^2"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>min</mi><mo></mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">∥</mi><mi>w</mi><msup><mi mathvariant="normal">∥</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\min\frac{1}{2}\|w\|^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/8fe9b4af734f414ca905741e39643df3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="s.t.\;\; y_i[w^\text{T}x_i+b]\ge1\;\;(i=1∼N)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mtext> </mtext><mtext> </mtext><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo stretchy="false">]</mo><mo>≥</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>∼</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s.t.\;\; y_i[w^\text{T}x_i+b]\ge1\;\;(i=1∼N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord mathdefault">s</span><span class="mord">.</span><span class="mord mathdefault">t</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/e0b0c5e6a1504b6791212b6796fe8aa0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">凸优化问题中的二次规划问题</span></span></p></div></li></ul><h2 id="https://www.notion.so/72cd14da756f413ba75faec9f86bdadc" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/72cd14da756f413ba75faec9f86bdadc"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">非线性问题</span></span></h2><h3 id="https://www.notion.so/37ee5cf7e9f74344889e09bb8db00bc6" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/37ee5cf7e9f74344889e09bb8db00bc6"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">软间隔</span></span></h3><div id="https://www.notion.so/c42930fdbe634d0e9ddcb6cbbf67e374" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">训练数据中的特异点，去掉后，剩下大部分样本点组成的集合满足线性可分</span></span></p></div><div id="https://www.notion.so/ca8ae86b786c4f1fad4801e64354cbf5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">引入“软间隔”的概念，</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">允许支持向量机在一些样本上不满足约束</mark></span></span></p></div><div id="https://www.notion.so/a2e42880a5044f208828ec5a9db88779" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F0260d1d8-3b13-48b2-8317-88411b7ff3f5%2FUntitled.png?width=288&table=block&id=a2e42880-a504-4f20-8828-ec5a9db88779"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F0260d1d8-3b13-48b2-8317-88411b7ff3f5%2FUntitled.png?width=288&table=block&id=a2e42880-a504-4f20-8828-ec5a9db88779" style="width:288px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/0a7f8766e0314683b88f89bffd1434be" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">现实中，很难确定合适的核函数使得训练样本在特征空间中线性可分；一个线性可分的结果也很难断定是否是由过拟合造成的。</span></span></p></div><div id="https://www.notion.so/469058a672304b07a6f93cb4d80682c5" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">将线性不可分的学习问题转换为凸二次规划问题，即</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">软间隔最大化</mark></span></span></p></div><div id="https://www.notion.so/14cddfa0612c4bba8746bea8e82021b8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">线性支持向量机：</span></span></p></div><div id="https://www.notion.so/8d6c859775904c75b79bc10811e38322" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">最小化：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\frac{1}{2}\|w\|^2+\boxed{C\sum_{i=1}^{N}\xi_i}"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">∥</mi><mi>w</mi><msup><mi mathvariant="normal">∥</mi><mn>2</mn></msup><mo>+</mo><menclose notation="box"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>C</mi><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>ξ</mi><mi>i</mi></msub></mrow></mstyle></mstyle></mstyle></menclose></mrow><annotation encoding="application/x-tex">\frac{1}{2}\|w\|^2+\boxed{C\sum_{i=1}^{N}\xi_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.7860050000000003em;vertical-align:-1.6176689999999998em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1683360000000005em;"><span style="top:-5.786004999999999em;"><span class="pstrut" style="height:5.786005em;"></span><span class="boxpad"><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.168336em;"><span class="pstrut" style="height:5.786005em;"></span><span class="stretchy fbox" style="height:3.7860050000000003em;border-style:solid;border-width:0.04em;"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.6176689999999998em;"><span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">（</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\xi_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ξ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\xi_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">为松弛变量，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="C\sum_{i=1}^{N}\xi_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><msub><mi>ξ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C\sum_{i=1}^{N}\xi_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">为正则项）</span></span></p></div><div id="https://www.notion.so/333970a4faf942eb9c6ff09ed6f2e436" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">约束条件：</span></span></p></div><div id="https://www.notion.so/e58a01cbb0ea416a87b22e7a9c1ff70e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（1）</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="y_i[w^\text{T}x_i+b]\ge1"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msup><mi>w</mi><mtext>T</mtext></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo stretchy="false">]</mo><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y_i[w^\text{T}x_i+b]\ge1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/6a585133513c4c64851e673e6a0d4d46" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（2）</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\xi_i\ge0\;,i=1∼N"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ξ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mtext> </mtext><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>∼</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\xi_i\ge0\;,i=1∼N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/0a0cfda6abb343fe9660d8d6c8054821" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">求解得到分离超平面</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="w^*x+b^*=0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>w</mi><mo>∗</mo></msup><mi>x</mi><mo>+</mo><msup><mi>b</mi><mo>∗</mo></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w^*x+b^*=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.772026em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span><span class="SemanticString">，以及决策函数</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="f(x)=sign(w^*x+b^*)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mi>i</mi><mi>g</mi><mi>n</mi><mo stretchy="false">(</mo><msup><mi>w</mi><mo>∗</mo></msup><mi>x</mi><mo>+</mo><msup><mi>b</mi><mo>∗</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)=sign(w^*x+b^*)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">s</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">n</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/0b95da8de9934a56ab06c7197b88cb3a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">软间隔参数</strong></span></span></p></div><div id="https://www.notion.so/cdc9b719dc7c4915ae79aa6402ab2993" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F816113b2-e89f-4eb5-832a-05783d2fde7a%2FUntitled.png?width=1440&table=block&id=cdc9b719-dc7c-4915-ae79-aa6402ab2993"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F816113b2-e89f-4eb5-832a-05783d2fde7a%2FUntitled.png?width=1440&table=block&id=cdc9b719-dc7c-4915-ae79-aa6402ab2993" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/54ac66981ea04025b86621dd8c8a8836" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">尽可能在保持间隔宽阔或限制间隔违例之间找到一个平衡</span></span></p></div><div id="https://www.notion.so/034d85a9b9754f1692ab6b91ec619d71" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">间隔或限制间隔违例之间的关系由超参数</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="C"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span></span></span></span></span><span class="SemanticString">控制，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="C"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span></span></span></span></span><span class="SemanticString">越大，间隔越窄，但是违例也更少，反之则间隔越宽，但是违例也越多。</span></span></p></div><div id="https://www.notion.so/a51bf87c5fc542168a142db196a70338" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F95c07802-f85b-466c-8290-c58a3e7b304d%2FUntitled.png?width=752&table=block&id=a51bf87c-5fc5-4216-8a14-2db196a70338"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F95c07802-f85b-466c-8290-c58a3e7b304d%2FUntitled.png?width=752&table=block&id=a51bf87c-5fc5-4216-8a14-2db196a70338" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><h3 id="https://www.notion.so/d004565952574259b5ef1b7b238fb8b1" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/d004565952574259b5ef1b7b238fb8b1"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">非线性可分</span></span></h3><div id="https://www.notion.so/4b5232f23ca748078936dbf292d768c1" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">仍然找直线，但是到一个高维空间中找直线</mark></span></span></p></div><div id="https://www.notion.so/f15cb4b8a7b24dd8a1ddf4fe001455fb" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1df58015-25ae-454a-8e6e-3f42888fe771%2FUntitled.png?width=997&table=block&id=f15cb4b8-a7b2-4dd8-a1dd-f4fe001455fb"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F1df58015-25ae-454a-8e6e-3f42888fe771%2FUntitled.png?width=997&table=block&id=f15cb4b8-a7b2-4dd8-a1dd-f4fe001455fb" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/0c290b133e25445f8da69b752a6d4700" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">定义一个高维映射</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">：低维</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span></span></span><span class="SemanticString"> → 高维</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f7327cb2fdc04e939e6391d80172a9bf" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">例如，最简单的非线性可分问题：异或问题</span></span></p></div><div id="https://www.notion.so/e29cd01f034342b98569b8f577f1526c" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div><div id="https://www.notion.so/c17c243491e549a8947227503dbbf75a" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">线性分类器变为一个超平面，把空间分为两个部分</span></span></p></div><div id="https://www.notion.so/ec006d03f30045f08bcedaad5044d9f8" class="Image Image--Normal"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fd3846308-341a-4c47-a0c2-ce9d40747ee7%2FUntitled.png?width=565&table=block&id=ec006d03-f300-45f0-8bce-daad5044d9f8"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2Fd3846308-341a-4c47-a0c2-ce9d40747ee7%2FUntitled.png?width=565&table=block&id=ec006d03-f300-45f0-8bce-daad5044d9f8" style="width:565px"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/f3aa6842157a4d08bf2b42883649a6e2" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">关键思想：</span></span></p></div><div id="https://www.notion.so/cc104ac0f7ea44a48cba105c2bfdcef7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">为了解决非线性分割问题，将</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">变换到一个高维空间</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/ba874c1ed0094c4f9170eded848d179a" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">输入空间：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x_i"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="SemanticString">所在的空间</span></span></li><li id="https://www.notion.so/4e3850eff28f4807a523392e7f5f4df4" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">特征空间：变换后</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_i)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">的空间</span></span></li></ul><div id="https://www.notion.so/d3b2b5993eb24357a88b817a005f86f2" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">如何变换？</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/be948e5ac27045baaa9e33120bf759cd" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">利用一个适当的变换</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">ϕ</span></span></span></span></span></span><span class="SemanticString">，使分类变得容易</span></span></li><li id="https://www.notion.so/e25f9d2b04bf43589fdb3d976c1adfa7" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">特征空间中的线性算子等价于输入空间中的非线性算子</span></span></li></ul><div id="https://www.notion.so/2f924238a74e47efbf91780727c50d45" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">变换可能出现的问题：难以得到一个好的分类且计算开销大</span></span></p></div><div id="https://www.notion.so/c3dbaf6803624958a6bf643002cd76be" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">需要同时解决两个问题：</span></span></p></div><ul class="BulletedListWrapper"><li id="https://www.notion.so/8cf118cbe1514a97ab604ef338a522e9" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">最小化</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\|w\|"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi><mi>w</mi><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\|w\|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∥</span></span></span></span></span></span><span class="SemanticString">能得到好的分类</span></span></li><li id="https://www.notion.so/784d226b0c5041c385a4807dd1472794" class="BulletedList"><span class="SemanticStringArray"><span class="SemanticString">利用核函数技巧可以进行有效的计算</span></span></li></ul><div id="https://www.notion.so/36e9b47f7559468090d8067cec3c862f" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5c60af3a-8165-49f9-8a79-25f1499a4a7f%2FUntitled.png?width=712&table=block&id=36e9b47f-7559-4680-90d8-067cec3c862f"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5c60af3a-8165-49f9-8a79-25f1499a4a7f%2FUntitled.png?width=712&table=block&id=36e9b47f-7559-4680-90d8-067cec3c862f" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><h3 id="https://www.notion.so/4ff2014eb303462683b7fe5734a9d1ed" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/4ff2014eb303462683b7fe5734a9d1ed"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">核函数</span></span></h3><div id="https://www.notion.so/ece3ddcf26674c1a90d39cd9376789ca" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">高维映射</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="x\rightarrow\phi(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\rightarrow\phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><p id="https://www.notion.so/8eaa324137ab4914a2eec5f2e17748da" class="Equation" data-latex="\begin{cases}
y_i[w^{\text{T}}\phi(x_i)+b]\ge1-\xi_i \\
\xi_i\ge 0
\end{cases}"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">[</mo><msup><mi>w</mi><mtext>T</mtext></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo stretchy="false">]</mo><mo>≥</mo><mn>1</mn><mo>−</mo><msub><mi>ξ</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>ξ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}
y_i[w^{\text{T}}\phi(x_i)+b]\ge1-\xi_i \\
\xi_i\ge 0
\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.04601em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><div id="https://www.notion.so/49b93a0f48994167acfcc1e01f81f476" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">是无限维，无法求整体优化</span></span></p></div><div id="https://www.notion.so/fc6898923e5e421db2f40946a4ba5901" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_1)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">与</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">两个无限维向量</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">内积</mark></span><span class="SemanticString">：</span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">核函数（Kenerl Function）</mark></span></span></p></div><div id="https://www.notion.so/174d84243d93441eb0716ec0403c5e4e" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x_1,x_2)=\phi(x_1)^\text{T}\phi(x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><msup><mo stretchy="false">)</mo><mtext>T</mtext></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,x_2)=\phi(x_1)^\text{T}\phi(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></mark></span></span></p></div><div id="https://www.notion.so/e5c0ee75373043348bfb3f02206b21e7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">只要知道一个核函数，不知道无限维映射</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_i)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">的显式表达，仍然可以求解最优</span></span></p></div><div id="https://www.notion.so/cd925663a90f48c882f5371be44b06e7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">为什么可以通过求内积代替显式</mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">ϕ</span></span></span></span></span></mark></span><span class="SemanticString"><mark class="SemanticString__Fragment SemanticString__Fragment--HighlightedColor SemanticString__Fragment--ColorBlue">——对偶问题</mark></span></span></p></div><div id="https://www.notion.so/e687f6d1cc074893bce028c2f643d47d" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">假设知道了</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span></span><span class="SemanticString">，如何求解</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_i)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">，让优化问题可求解？</span></span></p></div><div id="https://www.notion.so/155c361810394c798437983e5a6a8242" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">核函数也是有限制的，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span></span><span class="SemanticString">要满足某种特定条件，才能拆成内积，</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x_1,x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">能写成</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x_1)^\text{T}\phi(x_2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><msup><mo stretchy="false">)</mo><mtext>T</mtext></msup><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x_1)^\text{T}\phi(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">T</span></span></span></span></span></span></span></span></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span><span class="SemanticString">的充要条件：</span></span></p></div><div id="https://www.notion.so/bfaa12c447f343259bdfe81d1fd0f07b" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（1）交换性：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x_1,x_2)=K(x_2,x_1)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,x_2)=K(x_2,x_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/05a0dd7cc8bb4750b3cdbcfe657821e0" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">（2）半正定性：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\forall C_i,x_i\;,i=1,\dots,N"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><msub><mi>C</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>i</mi></msub><mtext> </mtext><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\forall C_i,x_i\;,i=1,\dots,N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∀</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span></span></span></span></span><span class="SemanticString">，有</span></span></p></div><p id="https://www.notion.so/723ab4750962433999f24daa33ef1ad7" class="Equation" data-latex="\sum_{i=1}^N\sum_{j=1}^NC_iC_jK(x_i,x_j)\ge0"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>C</mi><mi>i</mi></msub><msub><mi>C</mi><mi>j</mi></msub><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{i=1}^N\sum_{j=1}^NC_iC_jK(x_i,x_j)\ge0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2421130000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000006em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p><div id="https://www.notion.so/c5c6861db08f4f84a859020ef3377413" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F45235644-a656-4ea9-a3b5-1662ecdfd86e%2FUntitled.png?width=1290&table=block&id=c5c6861d-b08f-4f84-a859-020ef3377413"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F45235644-a656-4ea9-a3b5-1662ecdfd86e%2FUntitled.png?width=1290&table=block&id=c5c6861d-b08f-4f84-a859-020ef3377413" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/47bf4273f84c44a6b336f35c1eacae8d" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F540e3c75-0d5c-4ec2-be39-67adf52aefc0%2FUntitled.png?width=922&table=block&id=47bf4273-f84c-44a6-b336-f35c1eacae8d"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F540e3c75-0d5c-4ec2-be39-67adf52aefc0%2FUntitled.png?width=922&table=block&id=47bf4273-f84c-44a6-b336-f35c1eacae8d" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/6a735a9bbb004d9ea0aff45180635d46" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5ed5e7e9-248e-4b5f-88b3-5992f8b09b02%2FUntitled.png?width=1016&table=block&id=6a735a9b-bb00-4d9e-a0af-f45180635d46"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F5ed5e7e9-248e-4b5f-88b3-5992f8b09b02%2FUntitled.png?width=1016&table=block&id=6a735a9b-bb00-4d9e-a0af-f45180635d46" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/9e1c2b042a944398aa49f46075a1d1c5" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F88150cbf-00fe-4711-8329-a9eaede3b165%2FUntitled.png?width=960&table=block&id=9e1c2b04-2a94-4398-aa49-f46075a1d1c5"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F88150cbf-00fe-4711-8329-a9eaede3b165%2FUntitled.png?width=960&table=block&id=9e1c2b04-2a94-4398-aa49-f46075a1d1c5" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/c126b026e0f241fbbd26d408d5d140e3" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><strong class="SemanticString__Fragment SemanticString__Fragment--Bold">高斯核</strong></span></span></p></div><div id="https://www.notion.so/d8abfeb0836146fab461ecc8d93c69e7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">无限维的特征变换</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="\phi(x)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">ϕ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/019c9b0607074fa6817dcb56160701b8" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x,x')=\exp(-(x-x')^2)"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x,x')=\exp(-(x-x')^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f5db0103dc394c519e62963445f22c60" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F427fdce0-1971-414c-96d0-de0c1bd13d5f%2FUntitled.png?width=807&table=block&id=f5db0103-dc39-4c51-9e62-963445f22c60"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F427fdce0-1971-414c-96d0-de0c1bd13d5f%2FUntitled.png?width=807&table=block&id=f5db0103-dc39-4c51-9e62-963445f22c60" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><div id="https://www.notion.so/90b18f069ba546f4b3a03da2fc6e9be9" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"><span class="SemanticString">一般形式：</span><span class="SemanticString"><span class="SemanticString__Fragment SemanticString__Fragment--Math" data-latex="K(x,x')=\exp(-\lambda(x-x')^2)\;,\lambda>0"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">)</mo><mtext> </mtext><mo separator="true">,</mo><mi>λ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">K(x,x')=\exp(-\lambda(x-x')^2)\;,\lambda>0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathdefault">λ</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">λ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></span></span></p></div><div id="https://www.notion.so/f92367ee31fa4d32a0d1b42824e4c2ef" class="Image Image--PageWidth"><figure><a href="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F04e8056f-c842-4fcd-b0c3-51fb7a5a984f%2FUntitled.png?width=944&table=block&id=f92367ee-31fa-4d32-a0d1-b42824e4c2ef"><img src="https://www.notion.so/signed/https%3A%2F%2Fs3-us-west-2.amazonaws.com%2Fsecure.notion-static.com%2F04e8056f-c842-4fcd-b0c3-51fb7a5a984f%2FUntitled.png?width=944&table=block&id=f92367ee-31fa-4d32-a0d1-b42824e4c2ef" style="width:100%"/></a><figcaption><span class="SemanticStringArray"></span></figcaption></figure></div><h2 id="https://www.notion.so/1dfc3708698642ee9f75defa9cc514a4" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--2"><a class="Anchor" href="#https://www.notion.so/1dfc3708698642ee9f75defa9cc514a4"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">对偶问题</span></span></h2><h3 id="https://www.notion.so/88d77d2c5e81400d850ae23d57995701" class="ColorfulBlock ColorfulBlock--ColorDefault Heading Heading--3"><a class="Anchor" href="#https://www.notion.so/88d77d2c5e81400d850ae23d57995701"><svg width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a><span class="SemanticStringArray"><span class="SemanticString">定义</span></span></h3><div id="https://www.notion.so/8791d7263d8348d5b84902a13cc989c7" class="ColorfulBlock ColorfulBlock--ColorDefault Text"><p class="Text__Content"><span class="SemanticStringArray"></span></p></div></article>
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