fixes and fixed point iteration

kashefy · Jun 1, 2020 · ab8bcc0 · ab8bcc0
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diff --git a/notes/06_fastica/0_recap_ica_whitening.tex b/notes/06_fastica/0_recap_ica_whitening.tex
@@ -24,28 +24,36 @@ \section{The ICA Problem}
 \end{itemize}
 \end{frame}
 
-\begin{frame}
-\underline{Outline:}
-\begin{itemize}
-    \item ICA on whitened data
-    \begin{itemize}
-        \item Whitening/sphering
-        \item Amibguities in ICA
-        \item PCA is \emph{half} the ICA Problem
-    \end{itemize}
-    \item the problem with gaussians
-    \item maximizing nongaussianity
-    \begin{itemize}
-        \item Kurtosis-based
-        \item negentropy
-    \end{itemize}
-\end{itemize}
-\end{frame}
-
-\newpage
+%\begin{frame}
+%\underline{Outline:}
+%\begin{itemize}
+    %\item ICA on whitened data
+    %\begin{itemize}
+        %\item Whitening/sphering
+        %\item Amibguities in ICA
+        %\item PCA is \emph{half} the ICA Problem
+    %\end{itemize}
+    %\item the problem with gaussians
+    %\item maximizing nongaussianity
+    %\begin{itemize}
+        %\item Kurtosis-based
+        %\item negentropy
+    %\end{itemize}
+%\end{itemize}
+%\end{frame}
+
+%\newpage
 
 \section{Whitening revisited}
 
+\mode<presentation>{
+\begin{frame} 
+    \begin{center} \huge
+        \secname
+    \end{center}
+\end{frame}
+}
+
 \begin{frame}{\secname}
 
 \notesonly{
@@ -87,6 +95,9 @@ \section{Whitening revisited}
 
 \pause
 
+
+\svspace{-5mm}
+
 \begin{equation}
 \label{eq:covw}
 \vec \Sigma_v = \mathrm{Cov}(\vec v) = \E \lbrack \, \vec v \, \vec v^\top \rbrack = \vec I_N

diff --git a/notes/06_fastica/2_pcaica.tex b/notes/06_fastica/2_pcaica.tex
@@ -46,10 +46,11 @@ \subsection{Whitening in the context of ICA}
 \widetilde{\vec A} := \vec Q \, \vec A 
 \end{equation}
 
-Which yields\only<3>{ a new ICA problem}:
+Which yields\only<3>{ a new ICA problem:
 \begin{equation}
 \vec u = \vec Q\, \vec x = \vec Q\,\vec A \, \vec s = \widetilde{\vec A} \, \vec s
 \end{equation}
+}
 
 \end{frame}
 

diff --git a/notes/06_fastica/4_kurt.tex b/notes/06_fastica/4_kurt.tex
@@ -466,6 +466,37 @@ \subsubsection{Kurtosis-based ICA: the gradient algorithm}
 }
 \end{frame}
 
+\subsubsection{Fixed-point iteration}
+
+\begin{frame}{\subsubsecname}
+
+\begin{itemize}
+\item A fixed point: $x=g(x)=g(g(\ldots g(x)\ldots))$\\
+
+Example:
+\begin{align}
+    g(x) &= x^{2} - 3x + 4\\
+    g(2) &= 2^{2} - 3\cdot 2 + 4 = 4 - 6 + 4 = 2
+\end{align}
+
+\item Fixed point iteration:
+\begin{equation}
+x^{(t+1)} = g(x^{(t)}), t=0,1,2,\ldots
+\end{equation}
+\end{itemize}
+
+\begin{center}
+    \includegraphics[height=3.5cm]{img/fixed_point_iter_cos} 
+    \mode<article>{
+    \captionof{figure}{
+    Fixed point iteration example
+    }
+    \label{fig:fixedpointcos}
+    } 
+\end{center}
+
+\end{frame}
+
 \begin{frame}
 
 \notesonly{

diff --git a/notes/06_fastica/5_fastica.tex b/notes/06_fastica/5_fastica.tex
@@ -10,12 +10,19 @@ \subsection{Negentropy}
     \end{center}
 \end{frame}
 }
+\notesonly{
 
 Negentropy $J(\widehat{s})$ of the reconstructed sources $\widehat{\vec s}$ is defined as:
 
 \begin{equation}
-\label{eq:negentropy}
-	J(\widehat{s}) \coloneqq H(\widehat{s})_\normal - H(\widehat{s})
+	J(\widehat{s}) \coloneqq \underbrace{ H(\widehat{s})_\normal}_{
+			\substack{ 	\text{entropy of a Gaussian} \\
+					\text{distribution with} \\
+					\text{same variance}}}
+		- \underbrace{ H(\widehat{s}) }_{
+			\substack{	\text{entropy of the true} \\
+					\text{distribution} \\
+					\text{(variance } \sigma^2 \text{)} }}
 \end{equation}
 
 where
@@ -24,6 +31,7 @@ \subsection{Negentropy}
 \label{eq:diffentropyshat}
 H(\widehat{s}) := - \int p(\widehat{s}) \log p(\widehat{s}) d\widehat{s}
 \end{equation}
+}
 
 \begin{frame} 
 \frametitle{Negentropy}
@@ -33,7 +41,7 @@ \subsection{Negentropy}
 $$
 
 \begin{block}{Definition of negentropy}
-\begin{equation*}
+\begin{equation}
 	J(\widehat{s}) \coloneqq \underbrace{ H(\widehat{s})_\normal}_{
 			\substack{ 	\text{entropy of a Gaussian} \\
 					\text{distribution with} \\
@@ -42,7 +50,7 @@ \subsection{Negentropy}
 			\substack{	\text{entropy of the true} \\
 					\text{distribution} \\
 					\text{(variance } \sigma^2 \text{)} }}
-\end{equation*}
+\end{equation}
 \end{block}
 }
 
@@ -57,8 +65,32 @@ \subsection{Negentropy}
   \itR \textbf{Problem:} requires estimation of density $p(\widehat{s})$
 \end{itemize}
 
+\end{frame}
+
+\begin{frame}
+\frametitle{Optimizing negentropy}
+
+\slidesonly{
+\begin{equation}
+	J(\widehat{s}) \coloneqq \underbrace{ H(\widehat{s})_\normal}_{
+			\substack{ 	\text{entropy of a Gaussian} \\
+					\text{distribution with} \\
+					\text{same variance}}}
+		- \underbrace{ H(\widehat{s}) }_{
+			\substack{	\text{entropy of the true} \\
+					\text{distribution} \\
+					\text{(variance } \sigma^2 \text{)} }}
+\end{equation}
+}
+
 \question{Should we minimize or maximize negentropy?}
 
+\slidesonly{
+\begin{center}
+	\includegraphics[width=0.25\textwidth]{img/meme_maxormin}%
+\end{center}
+}
+
 \end{frame}
 
 \subsubsection{Approximations of negentropy}
@@ -164,7 +196,10 @@ \subsubsection{Contrast functions}
 \end{frame}
 
 \begin{frame}
-cf. lecture slides for optimization of negentropy using contrast functions.
+\begin{itemize}
+\item[] cf. lecture slides for optimization of negentropy using contrast functions.
+\item[] Hyv\"arinen, Karhunen \& Oja, Independent Component Analysis, Wiley, 2001
+\end{itemize}
 \end{frame}
 
 \begin{frame}

diff --git a/notes/06_fastica/img/fixed_point_iter_cos.pdf b/notes/06_fastica/img/fixed_point_iter_cos.pdf
diff --git a/notes/06_fastica/img/meme_maxormin.jpg b/notes/06_fastica/img/meme_maxormin.jpg
diff --git a/notes/06_fastica/tutorial.tex b/notes/06_fastica/tutorial.tex
@@ -56,7 +56,12 @@
 \end{frame}
 
 \begin{frame}
+\mode<presentation>{
+\tableofcontents[hideallsubsections]
+}
+\mode<article>{
 \tableofcontents
+}
 \end{frame}
 
 \newpage