From a34494de1917c2e5ab41307b084c9c1fadcf74bb Mon Sep 17 00:00:00 2001 From: Youssef Kashef Date: Wed, 10 Jun 2020 23:51:54 +0200 Subject: [PATCH] soft kmeans --- notes/08_clustering/1_kmeans.tex | 26 ++-- notes/08_clustering/4_pairwise.tex | 12 +- notes/08_clustering/5_pairwise_data.tex | 63 ++++++--- notes/08_clustering/6_pairwsie_meanfield.tex | 121 +++++++++++++++--- notes/08_clustering/7_soft_kmeans_eucl.tex | 67 +++++++++- notes/08_clustering/img/m3_circ_data.pdf | Bin 0 -> 17558 bytes .../08_clustering/img/m3_circ_data_polar.pdf | Bin 0 -> 20967 bytes notes/08_clustering/img/m3_circ_pdist.pdf | Bin 0 -> 262664 bytes .../08_clustering/img/m3_circ_pdist_polar.pdf | Bin 0 -> 240685 bytes notes/08_clustering/img/section4_fig6a.pdf | Bin 0 -> 11855 bytes notes/08_clustering/tutorial.tex | 10 +- 11 files changed, 242 insertions(+), 57 deletions(-) create mode 100644 notes/08_clustering/img/m3_circ_data.pdf create mode 100644 notes/08_clustering/img/m3_circ_data_polar.pdf create mode 100644 notes/08_clustering/img/m3_circ_pdist.pdf create mode 100644 notes/08_clustering/img/m3_circ_pdist_polar.pdf create mode 100644 notes/08_clustering/img/section4_fig6a.pdf diff --git a/notes/08_clustering/1_kmeans.tex b/notes/08_clustering/1_kmeans.tex index 7293c69..b4d6406 100644 --- a/notes/08_clustering/1_kmeans.tex +++ b/notes/08_clustering/1_kmeans.tex @@ -11,19 +11,19 @@ \section{K-means} \end{frame} } -\begin{frame}{Finding structure in the data} - +\begin{frame} K-means is simple. +\end{frame} -\pause +\begin{frame}{Finding structure in the data} \begin{center} \slidesonly{ - \includegraphics<2>[width=6cm]{img/clustering_no-color_no-labels_2p} - \includegraphics<3>[width=6cm]{img/clustering_no-color_no-labels_4p} - \includegraphics<4>[width=6cm]{img/clustering_no-color_no-labels} + \includegraphics<1>[width=6cm]{img/clustering_no-color_no-labels_2p} + \includegraphics<2>[width=6cm]{img/clustering_no-color_no-labels_4p} + \includegraphics<3>[width=6cm]{img/clustering_no-color_no-labels} } - \includegraphics<5->[width=6cm]{img/clustering_color} + \includegraphics<4->[width=6cm]{img/clustering_color} \notesonly{ \captionof{figure}{Example clustering of 2D points into M=2 clusters} } @@ -31,7 +31,7 @@ \section{K-means} \slidesonly{ -\visible<6>{ +\visible<5>{ \begin{itemize} \item proximity must count for something \item Points that fall within a region form a cluster @@ -58,8 +58,14 @@ \subsubsection{In plain English} \question{What does clustering give us?} -- Eventually, instead of describing each point by its absolute location, we will be able to describe it by the cluster it is assigned to. -We will also be able to describe the entire dataset by the partitions we've found that separate the clusters. +\begin{itemize} +\item[-] Instead of describing each point by its absolute location, we will be able to describe it by the cluster it is assigned to. +\begin{equation} +\text{point}~\vec x \in \R^N \quad \longrightarrow \quad~\text{cluster index}~q \in \N +\end{equation} +\item We will be able to describe the entire dataset by the partitions we've found that separate the clusters. +\item We'll draw relations between simple clustering and other algorithms (embedding algorithms, density estimation) +\end{itemize} \end{frame} diff --git a/notes/08_clustering/4_pairwise.tex b/notes/08_clustering/4_pairwise.tex index 94fedc7..05f3968 100644 --- a/notes/08_clustering/4_pairwise.tex +++ b/notes/08_clustering/4_pairwise.tex @@ -5,14 +5,16 @@ \section{Pairwise clustering} \begin{center} \huge \secname \end{center} - \begin{center} - Proximity to $\underset{\text{other points}}{\cancel{\text{prototype}}}$ - \end{center} + \begin{itemize} + \item Proximity to $\underset{\text{other points}}{\cancel{\text{prototype}}}$ + \item A proper application of mean-field annealing + \item A building block for ``soft'' K-means + \end{itemize} \end{frame} } -\begin{frame}{Notions of distance} +\begin{frame}{Notions of proximity} \textbf{Recall} that K-means clusters points based on their proximity to some prototype.\\ @@ -33,7 +35,7 @@ \section{Pairwise clustering} \end{frame} -\begin{frame}{Notions of distance: pairwise distance} +\begin{frame}{Notions of proximity: pairwise distance} \begin{center} \begin{minipage}{0.32\textwidth} diff --git a/notes/08_clustering/5_pairwise_data.tex b/notes/08_clustering/5_pairwise_data.tex index dbab745..8a78565 100644 --- a/notes/08_clustering/5_pairwise_data.tex +++ b/notes/08_clustering/5_pairwise_data.tex @@ -1,4 +1,4 @@ -\subsection{The data} +\subsection{Data representation} \begin{frame}{\secname:~\subsecname} @@ -67,12 +67,43 @@ \subsection{The data} \begin{frame} \slidesonly{\frametitle{Other sources for pairwise distances}} -A further use-case is one where pairwise distance is not measured explicitly. The data representation can already be in a pairwise fashion as a result of an algorithm. Example: Sequence alignment procedures and graph-similarity measures produce pairwise representations.\\ +A further use-case is one where pairwise distance is not measured explicitly. The data representation can already be in a pairwise fashion as a result of an algorithm.\\ +Example: Sequence alignment procedures and graph-similarity measures produce pairwise representations.\\ We will opt for the \emph{squared Euclidean distance} for our pairwise clustering algorithm. \end{frame} +\begin{frame}{Possibilities: transformed distances} + +\begin{center} +\begin{minipage}{0.99\textwidth} + \begin{center} + \begin{minipage}{0.35\textwidth} + \includegraphics[width=0.9\textwidth]{img/m3_circ_data} + \end{minipage} + \begin{minipage}{0.35\textwidth} + \includegraphics[width=0.9\textwidth]{img/m3_circ_pdist} + \end{minipage} + \end{center} +\end{minipage} +\end{center} + +\begin{center} +\begin{minipage}{0.99\textwidth} + \begin{center} + \begin{minipage}{0.35\textwidth} + \includegraphics[width=0.9\textwidth]{img/m3_circ_data_polar} + \end{minipage} + \begin{minipage}{0.35\textwidth} + \includegraphics[width=0.9\textwidth]{img/m3_circ_pdist_polar} + \end{minipage} + \end{center} +\end{minipage} +\end{center} + +\end{frame} + \subsection{Cost} \begin{frame} @@ -119,7 +150,18 @@ \subsection{Relation of pairwise clustering to K-means} \begin{frame}\frametitle{\subsecname} +\notesonly{ When we choose squared Euclidean distance as the distance measure for pairwise clustering, we can show that this choice let's pairwise clustering effectively find the same solution as K-means clustering. +} + +\begin{center} +\resizebox{.7\textwidth}{!}{% +\begin{tabular}{cllll} +Pairwise Clustering & \multicolumn{1}{c}{$\Longrightarrow$} & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}Some\\ clustering solution\end{tabular}} & \multicolumn{1}{c}{$\Longleftarrow$} & K-means \\ +\begin{tabular}[c]{@{}c@{}}using\\ $d_{\alpha\alpha'}$ := sq. Eucl. distance\end{tabular} & & & & +\end{tabular}% +} +\end{center} \end{frame} @@ -327,20 +369,3 @@ \subsection{Relation of pairwise clustering to K-means} \end{frame} -\subsection{The mean-field approximation for pairwise clustering} - -\begin{frame} - Nomenclature ($\otimes$ $\rightarrow$ \emph{set-product, Cartesian product})\\ - - \begin{tabular}{r l p{9cm}} -$\big\{ \vec{m}^{(\alpha)} \big\}$: & & set of all $M$-dimensional binary vectors $\big( m_1^{(\alpha)}, m_2^{(\alpha)}, \ldots, - m_M^{(\alpha)} \big)^\top$ which fulfill the normalization condition (exactly one element equals 1). \\\\ -$\mathscr{M}$: & & $\big\{ \vec{m}^{(1)} \big\} \otimes \big\{ \vec{m}^{(2)} \big\} \otimes \ldots \otimes \big\{ \vec{m}^{(p)} \big\}$\\ -& & set-product (Cartesian product) between all possible binary assignment variables i.e.\ all possible valid assignments for the full dataset\\\\ -$\mathscr{M}_{\gamma}$:& & $\big\{ \vec{m}^{(1)} \big\} \otimes \ldots \otimes \big\{ \vec{m}^{(\gamma - 1)} \big\} \otimes - \big\{ \vec{m}^{(\gamma + 1)} \big\} \otimes \ldots \otimes - \big\{ \vec{m}^{(p)} \big\}$\\ -& &\ set of all possible assignments for all data points \\& & \hspace{0.03cm} except $\gamma$ -\end{tabular} - -\end{frame} diff --git a/notes/08_clustering/6_pairwsie_meanfield.tex b/notes/08_clustering/6_pairwsie_meanfield.tex index 58e72f3..7b12cd5 100644 --- a/notes/08_clustering/6_pairwsie_meanfield.tex +++ b/notes/08_clustering/6_pairwsie_meanfield.tex @@ -1,43 +1,46 @@ +\subsection{The mean-field approximation for pairwise clustering} \begin{frame}[t] -\slidesonly{\frametitle{The mean-field approximation for pairwise clustering}} +\slidesonly{\frametitle{\subsecname}} \svspace{-3mm} \begin{block}{assignment noise $\rightarrow$ Gibbs distribution} -$$ +\begin{equation} P_{ \big( \big\{ m_q^{(\alpha)} \big\} \big) } = \frac{1}{Z_p} \exp \Big\{ -\beta \overbrace{ E_{\big[ \big\{ m_q^{(\alpha)} \big\} \big]} }^{= \, E_p} \Big\} -$$ +\end{equation} where -$$ +\begin{equation} Z_p = \sum\limits_{\mathscr{M}} \exp \Big\{ -\beta E_p \Big\} -$$ +\end{equation} \end{block} \notesonly{ This is approximated by the mean-fields: } \begin{block}{factorizing distribution} -$$ +\begin{equation} Q_{ \big[ \big\{ m_q^{(\alpha)} \big\} \big] } = \frac{1}{Z_Q} \exp \Big\{ -\beta \sum\limits_{q, \gamma} m_q^{(\gamma)} \underbrace{ e_q^{(\gamma)} }_{ \text{{\tiny mean-fields}} } \Big\} -$$ +\end{equation} where: -$$ +\begin{equation} Z_Q = \sum\limits_{\mathscr{M}} \exp \Big\{ -\beta \sum\limits_{q, \gamma} m_q^{(\gamma)} e_q^{(\gamma)} \Big\} -$$ +\end{equation} \end{block} \end{frame} +\subsubsection{Calculation of the moments} + \begin{frame}\frametitle{Recap calculation of the moments (general mean-field case)} The factorization of the distribution $Q$ simplifies the calculation of the moments. This is based on the individual state variables being \emph{uncorrelated}. @@ -49,6 +52,9 @@ \end{equation} \end{frame} + +\mode
{ + \begin{frame} \slidesonly{ \frametitle{Factorizing moments (general mean-field case)} @@ -82,22 +88,48 @@ \end{frame} \begin{frame} -\slidesonly{\frametitle{Calculation of moments}} -$$ +\notesonly{Some nomenclature, before we demonstrate the same factorization for the assignment variables in pairwise clustering.} +\slidesonly{ Nomenclature }($\otimes$ $\rightarrow$ \emph{set-product, Cartesian product})\\ + + \begin{tabular}{r l p{9cm}} +$\big\{ \vec{m}^{(\alpha)} \big\}$: & & set of all $M$-dimensional binary vectors $\big( m_1^{(\alpha)}, m_2^{(\alpha)}, \ldots, + m_M^{(\alpha)} \big)^\top$ which fulfill the normalization condition (exactly one element equals 1). \\\\ +$\mathscr{M}$: & & $\big\{ \vec{m}^{(1)} \big\} \otimes \big\{ \vec{m}^{(2)} \big\} \otimes \ldots \otimes \big\{ \vec{m}^{(p)} \big\}$\\ +& & set-product (Cartesian product) between all possible binary assignment variables i.e.\ all possible valid assignments for the full dataset\\\\ +$\mathscr{M}_{\gamma}$:& & $\big\{ \vec{m}^{(1)} \big\} \otimes \ldots \otimes \big\{ \vec{m}^{(\gamma - 1)} \big\} \otimes + \big\{ \vec{m}^{(\gamma + 1)} \big\} \otimes \ldots \otimes + \big\{ \vec{m}^{(p)} \big\}$\\ +& &\ set of all possible assignments for all data points \\& & \hspace{0.03cm} except $\gamma$ +\end{tabular} + +\end{frame} + +} + +\begin{frame} +\slidesonly{\frametitle{Calculation of moments for the assignment variables}} +\begin{equation} \begin{array}{lll} \big< m_q^{(\gamma)} \big>_Q & = \frac{1}{Z_Q} \sum\limits_{\mathscr{M}} m_q^{(\gamma)} \exp \Big\{ -\beta \sum\limits_{r, \delta} m_{r}^{(\delta)} e_{r}^{(\delta)} \Big\} \end{array} -$$ +\end{equation} +\slidesonly{ \begin{itemize} \itr The factorization of $Q$ simplifies the calculation of the moments. \end{itemize} +Here,\\ +$\mathscr{M}$ is the set-product (Cartesian product) between \textbf{all possible} binary assignment variables i.e.\ all possible valid assignments for the full dataset. +} + \end{frame} +\mode
{ + \begin{frame} \slidesonly{\frametitle{Calculation of moments (derivation)} The factorization in \eqref{eq:factorizingMoments}) regarding valid assignments $\big\{ \vec{m}^{(\gamma)} \big\}$ for observation $\gamma$ and the rest of the variables excluding $\gamma$ (i.e. $\mathscr{M}_{\gamma}$) is valid for any functions $f,g$: @@ -153,6 +185,8 @@ \end{equation} \end{frame} +} + \begin{frame} \frametitle{Solution for calculating the moments} $$ @@ -174,7 +208,7 @@ \end{block} \end{frame} -\subsection{Soft clustering} +\subsubsection{Soft clustering} \begin{frame}{\subsecname} @@ -206,6 +240,8 @@ \subsection{Soft clustering} \end{figure} \end{frame} +\subsubsection{Determining the mean-field parameters} + \notesonly{ Going back to mean-field approximation: @@ -215,26 +251,71 @@ \subsection{Soft clustering} \begin{frame}[t] \slidesonly{\frametitle{Minimization of the KL-divergence}} \begin{block}{Mean Field equation (c.f. section on Stochastic Optimization for how to arrive at this result)} -$$ +\begin{equation} \fbox{$ \frac{\partial}{\partial e_l}\big_Q - \sum\limits_k e_k \frac{\partial}{\partial e_l} \big_Q = 0 $} -$$ +\end{equation} \end{block} -$$ \frac{\partial \big< E_p \big>_Q}{\partial e_q^{(\alpha)}} +\begin{equation} \frac{\partial \big< E_p \big>_Q}{\partial e_q^{(\alpha)}} - \sum\limits_{r, \gamma} \frac{ \overbrace{ \partial \big< m_r^{(\gamma)} \big>_Q }^{ \substack{ \text{depends only on} \\ \text{data point } \gamma }}}{ \partial e_q^{(\alpha)}} e_r^{(\gamma)} \eqexcl 0 -$$ -$$ +\end{equation} +\begin{equation} \frac{\partial \big< E_p \big>_Q}{\partial e_q^{(\alpha)}} - \sum\limits_r \frac{\partial \big< m_r^{(\alpha)} \big>_Q}{ \partial e_q^{(\alpha)}} e_r^{(\alpha)} \eqexcl 0 -$$ +\end{equation} + +\end{frame} + +%-------------------------------------------------------------------------- -switch to lecture slides.... +%-------------------------------------------------------------------------- +\begin{frame}[shrink=17] \frametitle{Mean-field annealing for pairwise clustering} +\begin{figure}[!th] +\footnotesize +\removelatexerror +\begin{algorithm}[H] + \DontPrintSemicolon + \textbf{Initialization:}\; + - (max.) number $M$ of partitions, initial ($\beta_0$) and final ($\beta_f$) values of the noise parameter, annealing factor $\eta$, convergence criterion $\theta$\; + - initialize mean-fields $e_q^{(\alpha)}$ with random numbers $\in [0, 1]$\; + +- $\beta \leftarrow \beta_0$\; +\While(annealing){$\beta < \beta_f$}{ +\Repeat ( EM $\text{(fixed point iteration)}$){ +$\big| \big( e_q^{(\alpha)} \big)_{\mathrm{new}} + - \big( e_q^{(\alpha)} \big)_{\mathrm{old}} \big| < \theta + \hspace{0.3cm}\forall q, \alpha$}{ +\hspace{-0.2cm}compute assignment probabilities: +$ \big< m_q^{(\alpha)} \big>_Q = \frac{ \exp \big\{ -\beta \big(e_q^{(\alpha)} + \big)_{\mathrm{old}}\big\} }{ \sum\limits_r \exp \big\{ -\beta \big( + e_r^{(\alpha)} \big)_{\mathrm{old}} \big\} } \hspace{0.3cm} \forall + q, \alpha +$\; +\hspace{-0.2cm}compute new mean-fields: +\hspace{-0.6cm} +$$\hspace{-0.2cm}\big( e_q^{(\alpha)} \big)_{\mathrm{new}} = \frac{2}{p} \frac{1}{ + \sum\limits_{\gamma} \big< m_q^{(\gamma)} \big>_Q } \sum\limits_{\delta} + \big_Q \cdot \bigg\{ d_{\delta \alpha} - \frac{1}{2} \frac{1}{ + \sum\limits_{\gamma} \big< m_q^{(\gamma)} \big>_Q } + \sum\limits_{\varepsilon} \big< m_q^{(\varepsilon)} + \big>_Q d_{\varepsilon \delta} \bigg\} \hspace{0.1cm}\forall q, \alpha$$ + +} +$\beta \leftarrow \eta \cdot \beta$\; +} +\label{alg:meanFieldClustering} +\caption{Mean-field annealing for pairwise clustering} +\end{algorithm} +\end{figure} \end{frame} + +%-------------------------------------------------------------------------- + diff --git a/notes/08_clustering/7_soft_kmeans_eucl.tex b/notes/08_clustering/7_soft_kmeans_eucl.tex index 0b98c87..ecd1a0d 100644 --- a/notes/08_clustering/7_soft_kmeans_eucl.tex +++ b/notes/08_clustering/7_soft_kmeans_eucl.tex @@ -1,4 +1,34 @@ \subsection{Special case: ``soft'' K-means with Euclidean distances} + +\begin{frame}{Turning Pairwise clustering into ``soft'' K-means} +Vectorial data: $\vec{x}^{(\alpha)}; \hspace{0.2cm} \alpha = 1, +\ldots, p; \hspace{0.2cm} \vec{x}^{(\alpha)} \in \mathbb{R}^N$, Euclidean squared distances, +$$ + d_{\alpha \alpha^{'}} + = \frac{1}{2} \big( \vec{x}^{(\alpha)} - \vec{x}^{(\alpha^{'})} + \big)^2 +$$ + +Given the definition of prototypes in K-means: +\begin{equation} + \vec{w}_q + = \frac{\sum\limits_{\gamma} \big< m_q^{(\gamma)} \big>_Q + \vec{x}^{(\gamma)}}{\sum\limits_{\gamma} + \big< m_q^{(\gamma)} \big>_Q} +\end{equation} + +The mean fields can be expressed in terms of these prototypes: +\begin{equation} + e_q^{(\alpha)} + = \frac{1}{2} \big( \vec{x}^{(\alpha)} - \vec{w}_q \big)^2 +\end{equation} + +\begin{itemize} +\itr center of mass of all objects weighted by their probability of assignment to cluster $q$ +\end{itemize} + +\end{frame} + % -------------------------------------------------------------------------- \begin{frame}[shrink=18]% \frametitle{Special case: ``soft'' K-means with Euclidean distances} \begin{figure}[!th] @@ -34,7 +64,7 @@ \subsection{Special case: ``soft'' K-means with Euclidean distances} $\beta \leftarrow \eta \beta$ } \label{alg:softKmeansEuclidean} -\caption{Soft K-means clustering for Euclidean distances} +\caption{Soft K-means clustering for Euclidean distances (batch version)} \end{algorithm} \end{figure} \begin{block}{Interpretation} @@ -45,3 +75,38 @@ \subsection{Special case: ``soft'' K-means with Euclidean distances} \end{block} \end{frame} % -------------------------------------------------------------------------- + +\subsection{Phase transitions in clustering} + +%-------------------------------------------------------------------------- +\begin{frame}[shrink=8] \frametitle{\subsecname} + +\question{What are the implications of ``annealing'' in clustering (pairwise clustering, soft K-means)?} + +Looking at average cost (Gibb's distribution of simulated annealing): +$$ + \big< E \big> = \frac{1}{Z} \sum\limits_{\big\{ m_p^{(\alpha)} \big\}} + E \exp \big\{ -\beta E \big\} +$$ + +The annealing (i.e. increase of $\beta$) implies: +\begin{itemize} + \item decrease of average cost + \item decrease of cluster size + \item increase in spatial resolution (hierarchical clustering) - coarse to granular clustering +\end{itemize} + +\end{frame} +% ----------------------------------------------------------------------- + +%-------------------------------------------------------------------------- +\begin{frame} \frametitle{Summarizing overview} +\begin{center}\includegraphics[scale=0.85]{img/section4_fig6a} +\end{center} +\begin{itemize} +%\item principled alternative to fuzzy clustering methods + \itr mean-field annealing is robust against convergence to local optima + \itr choice of terminal value of noise parameter $\beta \Rightarrow$ ``resolution'' + of cluster analysis +\end{itemize} +\end{frame} diff --git 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