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notes/04_density-transform/1_density-transform.tex

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+
+\section{The ICA problem:}
+
+Let $\vec s = (s_1, s_2,...,s_N)^\top$ denote the concatenation of independent sources 
+and $\vec x \in \R^N$ describe our observations. $\vec x$ relates to $\vec s$ through a 
+\emph{linear transformation} $\vec A$:
+
+\begin{equation}
+\label{eq:ica}
+\vec x = \vec A \, \vec s.
+\end{equation}
+
+We refer to $\vec A$ as the \emph{mixing matrix} and Eq.\ref{eq:ica} as the \emph{ICA problem}, 
+which is recovering $\vec s$ from only observing $\vec x$.
+
+\underline{Example scenario:}
+
+Two speakers are placed in a room and emit signals $s_1$ and $s_2$. 
+The speakers operate indepdendent of one another.  
+Two microphones are placed in the room and start recording. 
+The first microphone is placed slightly closer to speaker 2, while 
+the second microphone is placed slightly closer to speaker 1.
+$x_1$ and $x_2$ denote the recordings of the first and second microphone respectively. 
+When we listen to the recordings we expect to hear a mix of $s_1$ and $s_2$. 
+Since microphone 1 was placed closer to speaker 2, when we only listen to $x_1$ we hear more of $s_2$ than $s_1$. 
+The opposite can be said when we listen only to $x_2$.
+
+Acoustic systems are linear. This means that $x_1$ is a superposition of \emph{both} sources $s_1$ and $s_2$. 
+We will assume here that the contribution of a source $s_i$ 
+to an observation $x_j$ is inversely proportional to the distance between the source and the microphone. 
+The distance-contribution relationship is \emph{linear}. We don't need this to be any more realistic. 
+
+If we had a measurement of the distance between each microphone and each speaker, 
+we would tell exactly what the contribution of each of $s_1$ and $s_2$ is to each recorded observation. 
+If we know the exact contribution of a source to an observation, we can look at both observations and recover each source in full.
+
+This is what ICA tries to solve, except that it does not have any knowledge about the spatial setting. It is blind.
+
+\underline{Outline:}
+
+Before we tackle ICA itself, we first look at the more basic principle of \emph{density transformation} and 
+the \emph{convservation of probability}.\\
+We start with more specific cases of applying density transformations, 
+namely \emph{pseudo random number generators} and what the inverse of \emph{cumulative distribution functions (cdf)} can be used for.\\
+Finally we discuss how to generalize this in order to transform one probability density function (pdf) into another.
+
+\subsection{PRNG:}
+
+How can we sample from the uniform distribution $\in \lbrack0, 1)$?
+
+\begin{itemize}
+\item Create a sequence, preferrably with a wrong period.
+\item minimal pattern and sub-subsequences
+\item determinism has an advantage:
+\begin{enumerate}
+	\item \emph{reproducible} sequneces
+	\item efficiency; the starting element or ``seed'' and length of the sequence is suffcient is representative of the entire sequnece.
+\end{enumerate}
+\end{itemize}
+
+\underline{Linear congruential generator (LCG):}
+
+Start with a seed $y_0 \in \overbrace{\left\{0,\ldots,m-1\right\}}^{=:\;\mathcal{M}}$ with $m \in \N$ ($m$ controls the granularity). 
+The next sample $y_t$ is computed as:
+
+\begin{equation}
+y_t = \left( \, a \; y_{t-1} \; + \; b \, \right) \, \text{mod} \; m,
+\end{equation}
+where\\[-0.7cm]
+\begin{align*}
+a \in \mathcal{M}&\; \text{is the multiplier,} \\
+b \in \mathcal{M}&\; \text{is the increment.}
+\end{align*}
+
+Then $u_i = \frac{y_i}{m} \approx \,\mathcal{U} \in \lbrack0, 1)$.
+
+Although finicky and requiring careful parameterization, LCG gives us something for drawing from a uniform distribution. 
+Next, we look at how to draw samples of a random variable $X$ with a desired pdf $p_X(x)$. 
+using uniformly sampled values $\tilde z \in [0,1]$.
+
+\subsection{Inverse CDF:}
+
+If $F_{X}(x)$ is the cumulative
+  distribution function (cdf) of a random variable $X$, then the
+random variable $Z = F_{X}(X)$ is uniformly distributed on the
+interval $[0,1]$. This result provides 
+% (after inverting the relationship) 
+a general recipe to generate
+samples $\tilde x$ of a random variable $X$ with a desired pdf $p_X(x)$
+from uniformly distributed random numbers $\tilde z \in [0,1]$:
+\begin{enumerate}
+\item Compute the cdf $F_X(x)$ of the desired pdf $p_X(x)$
+
+\begin{equation}
+F_X(x) = P(X \leq x) = \int_{-\infty}^{x} p(y)\,dy
+\end{equation}
+
+The cdf is a one-to-one mapping of the domain of the cdf to the interval $[0,1]$.
+ If $Z$ is a uniform random variable, then $X=F_X^{-1}(Z)$ has the distribution $F$.
+
+\item Determine the inverse transformation $F^{-1}$.
+
+\item Sample uniformly distributed numbers (in $[0,1]$), $\tilde z$.
+\item Get the samples $\tilde x=F^{-1}(\tilde z)$ from $X$. 
+\end{enumerate}
+
+At this point we can use a PRNG to sample from the uniform distribution and 
+by plugging those samples into the inverse cdf we can obtain samples from a desired pdf.
+
+\newpage
+
+\section{Density Transformation:}
+
+Let $X_1$ and $X_2$ be jointly continuous random variables with 
+density function $f_{X_1, X_2}$:
+
+$$
+f_{X_1, X_2}(x_1, x_2) = f(\vec x) \qquad \vec x \in \Omega \subset \R^2
+$$
+
+and let $\vec u = \vec u(\vec x) = ( u_1(\vec x), u_2(\vec x)) = ( u_1(x_1, x_2), u_2(x_1, x_2)) $ be a one-to-one mapping/transformation.
+
+%\begin{figure}
+%\centering
+\includegraphics[width=0.7\textwidth]{img/u.pdf}
+%\caption*{$\vec{\psi} \in \mathcal{F} = \overline{\operatorname{span} \vec{\phi}(\mathbb{R}^N)}$}
+%\end{figure}
+
+The area of the small rectangle in $\Omega$ is $A = dx_1\, dx_2$.\\
+
+The goal is to show that in order for probability to be conserved, the areas on both spaces have to be equal. 
+We will therefore demonstrate that:
+$$
+\int_{\Omega} f(\vec{x}) \mathbf{d}\vec{x}
+=\int_{u(\Omega)} f({\vec x(\vec u)}) \frac{1}{\left|\det \frac{\partial \vec{u}(\vec{x})}{\partial \vec{x}} \right|} \mathbf{d}\vec{u}.
+$$
+
+\begin{itemize}
+\item Because $dx_1$ and $dx_2$ are \emph{infinitesimally small} we can consider the mapping 
+$\vec u$ to act as a linear transformation, resulting in a different shape in $u(\Omega)$. 
+\item The shape of the shaded area in $u(\Omega)$ is \emph{approximately} a parallelogram.
+\item We will compute the ratio of the areas between the two transforms.
+\item If we want to go back from $u(\Omega)$ to $\Omega$ we only need $\frac{1}{\text{ratio}}$.
+\end{itemize}
+
+\underline{Important:}
+Approximating the above as a linear transformation (i.e. a small rectangle turns into a parallelogram) only holds because for very small $d\vec x$.
+
+To get the area of the parallelogram in $u(\Omega)$ we need to find out what $\vec u(dx_1, dx_2))$ is.
+The area of the parallelogram then becomes the magnitude of the cross product between the components of $\vec u(\vec x)$.
+
+\newpage
+
+We first only consider the vector due to $dx_1$ in blue:
+
+\includegraphics[width=0.75\textwidth]{img/x1.pdf}
+
+The difference vector between the two correspsonding points in $u(\Omega)$ becomes:
+
+\begin{equation*}
+\begin{array}{r}
+\rmat{
+u_1 (x_1 + dx_1, x_2)\\
+u_2 (x_1 + dx_2, x_2)
+} - 
+\rmat{
+u_1 (x_1, x_2)\\
+u_2 (x_1, x_2)
+} \\[0.7cm]
+=
+\rmat{
+u_1 (x_1 + dx_1, x_2) - u_1 (x_1, x_2)\\
+u_2 (x_1 + dx_2, x_2) - u_2 (x_1, x_2)
+}
+\end{array}
+\end{equation*}
+
+Because $dx_1$ is so small, we can approximate the transformed vector by the derivative, 
+which is essentially taking the limit. The difference vector in $u(\Omega)$ becomes:
+
+$$
+u: \vec e_1 \mapsto 
+\rmat{
+\frac{\partial u_1}{\partial x_1} \Delta x_1\\[0.2cm]
+\frac{\partial u_2}{\partial x_1} \Delta x_1
+}
+$$
+
+\question{What about $dx_2$?}
+
+- We use the same procedure (one the red vector $\vec e_2$) and get:
+
+$$
+u: \vec e_2 \mapsto 
+\rmat{
+\frac{\partial u_1}{\partial x_2} \Delta x_2\\[0.2cm]
+\frac{\partial u_2}{\partial x_2} \Delta x_2
+}
+$$
+
+The area of the parallelogram spanned by the two difference vectors in $u(\Omega)$ 
+is the magnitude of their cross product:
+\begin{align*}
+&\left| \; 
+\rmat{
+\frac{\partial u_1}{\partial x_1} \Delta x_1\\[0.2cm]
+\frac{\partial u_2}{\partial x_1} \Delta x_1
+}
+\times
+\rmat{
+\frac{\partial u_1}{\partial x_2} \Delta x_2\\[0.2cm]
+\frac{\partial u_2}{\partial x_2} \Delta x_2
+}
+\; \right|\\ 
+&=
+\left| \; 
+\underbrace{
+\frac{\partial u_1}{\partial x_1} \frac{\partial u_2}{\partial x_2}
+\; - \;
+\frac{\partial u_1}{\partial x_2} \frac{\partial u_2}{\partial x_1}
+}_{\text{the Jacobian determinant}}
+\; \right| \Delta x_1 \Delta x_2 \\
+&= 
+\left| \; \det \quad
+\underbrace{
+\left(
+\frac{\partial (u_1, u_2)}{\partial (x_1,x_2)}
+\right)
+}_{\text{the Jacobian}}
+\; \right| \Delta x_1 \Delta x_2
+\end{align*}
+
+This matrix of partial derivatives is called the \emph{Jacobian}:
+$$
+\frac{\partial (u_1, u_2)}{\partial (x_1,x_2)} = 
+\frac{\partial (u_1(\vec x), u_2(\vec x))}{\partial (x_1,x_2)} = 
+\frac{\partial (\vec u(\vec x))}{\partial (\vec x)} =
+\underbrace{
+\rmat{
+{\partial u_1}/{\partial x_1} & {\partial u_1}/{\partial x_2}\\[0.2cm]
+{\partial u_2}/{\partial x_1} & {\partial u_2}/{\partial x_2}
+}%rmat
+}_{
+\substack{
+\text{matrix of}\\
+\text{partial derivatives}
+}%substack
+}
+$$
+
+If we are given an area spanned by too small vectors (e.g. $\vec e_1$, $\vec e_2$) 
+we can compute the area of the corresponding parallelogram in $u(\Omega)$ 
+by multiplying the original area by the Jacobian determinant.
+
+\question{What if we want to transform from $u(\Omega)$ back to $\Omega$?}
+
+\includegraphics[width=0.7\textwidth]{img/reverse.pdf}
+
+We apply the inverse mapping. A very small rectangle in $u(\Omega)$ transforms into a parallelogram in $\Omega$. 
+This reverse transformation would require the inverse of the above matrix of partial derivatives.
+The matrix of partial derivatives tells us how to transform infinitesimally small vectors back and forth.
+
+%Transformation between the spaces for infinitesimally small vectors:
+
+%$$
+%\rmat{
+%u_1(x_1) & u_1(x_2)\\
+%u_2(x_1) & u_2(x_2)
+%}
+%\rmat{
+%a\\
+%b
+%}
+%=
+%\rmat{
+%u_1(x_1)\\
+%u_2(x_1)
+%}
+%a
+%+
+%\rmat{
+%u_1(x_2)\\
+%u_2(x_2)
+%}
+%b
+%$$
+
+\question{Do we really have to compute the inverse of the matrix?}
+
+- No, the determinant of the inverse matrix is 1 / det of the original matrix:
+$$
+\frac{1}{\left| \det \left( \frac{\partial \vec u}{\partial \vec x} \right) \right|} =
+{\left| \det \left( \frac{\partial \vec x}{\partial \vec u} \right) \right|}
+$$
+
+\underline{Transformation between probability densities:}
+
+Conservation of probability: The area represents the probability of the event, 
+transforming it into another space should nt cause any increase or decrease in the probability of the event.
+
+Therefore, if we multiply the area of the rectangle in $\Omega$ by the pdf $f(\vec x)$ at $\vec x$, we get the probability of the parallelogram in $u(\Omega)$:
+
+$$
+\int_{\Omega} f(\vec{x}) \mathbf{d}\vec{x}
+=\int_{u(\Omega)} f({\vec x(\vec u)}) \left|\det \frac{\partial \vec{x}(\vec{u})}{\partial \vec{u}} \right| \mathbf{d}\vec{u}
+$$
+$$
+=\int_{u(\Omega)} f({\vec x(\vec u)}) \frac{1}{\left|\det \frac{\partial \vec{u}(\vec{x})}{\partial \vec{x}} \right|} \mathbf{d}\vec{u}.
+$$
+

notes/04_density-transform/Makefile

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+all: slides notes clean
+#all:  handout
+
+projname = tutorial
+targetname = $(projname)_$(shell basename $(CURDIR))
+compile = pdflatex
+projnameS = $(projname).slides
+projnameH = $(projname).handout
+projnameA = $(projname).notes
+
+slides: $(projname).slides.tex $(projname).tex
+	$(compile) $(projname).slides.tex
+#	bibtex $(projname).slides
+#	$(compile) --interaction=batchmode  $(projname).slides.tex
+#	$(compile) --interaction=batchmode  $(projname).slides.tex
+	mv $(projname).slides.pdf $(targetname).slides.pdf 
+
+handout: $(projname).handout.tex $(projname).tex
+	$(compile) $(projname).handout.tex
+	mv $(projname).handout.pdf $(targetname).handout.pdf 
+
+# Repeat compilation for the references to show up correctly
+notes: $(projname).notes.tex $(projname).tex
+	$(compile) $(projname).notes.tex
+#	bibtex $(projname).notes
+#	$(compile) --interaction=batchmode  $(projname).notes.tex
+	$(compile) --interaction=batchmode  $(projname).notes.tex
+	mv $(projname).notes.pdf $(targetname).notes.pdf 
+
+clean: cleans cleanh cleana
+
+cleans:
+	rm -f $(projnameS).aux $(projnameS).bbl $(projnameS).log $(projnameS).out $(projnameS).toc $(projnameS).lof $(projnameS).glo $(projnameS).glsdefs $(projnameS).idx $(projnameS).ilg $(projnameS).ind $(projnameS).loa $(projnameS).lot $(projnameS).loe $(projnameS).snm $(projnameS).nav
+
+cleanh:
+	rm -f $(projnameH).aux $(projnameH).bbl $(projnameH).log $(projnameH).out $(projnameH).toc $(projnameH).lof $(projnameH).glo $(projnameH).glsdefs $(projnameH).idx $(projnameH).ilg $(projnameH).ind $(projnameH).loa $(projnameH).lot $(projnameH).loe $(projnameH).snm $(projnameH).nav
+
+cleana:	
+	rm -f $(projnameA).aux $(projnameA).bbl $(projnameA).log $(projnameA).out $(projnameA).toc $(projnameA).lof $(projnameA).glo $(projnameA).glsdefs $(projnameA).idx $(projnameA).ilg $(projnameA).ind $(projnameA).loa $(projnameA).lot $(projnameA).loe $(projnameA).snm $(projnameA).nav
+