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16_lapsvm.tex
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\section{Laplacian Support Vector Machines}
\subsection{Motivation}
\begin{frame}
\frametitle{Motivation}
\structure{Example: two class ``clock'' data set}
\begin{figure}
\centering
\subfloat[Large set of unlabeled samples (black squares) and only one labeled sample per class (blue circle, green diamond)]{
\resizebox{.28\linewidth}{!}{
\input{\texfigdir/clock1.pstex_t}
}
}
\quad
\subfloat[Result of a maximum margin supervised classification]{
\resizebox{.28\linewidth}{!}{
\input{\texfigdir/clock2.pstex_t}
}
}
\quad
\subfloat[Result of a semi-supervised classification with intrinsic norm from manifold regularization]{
\resizebox{.28\linewidth}{!}{
\input{\texfigdir/clock3.pstex_t}
}
}
\end{figure}
\end{frame}
\subsection{Learning from Labeled and Unlabeled Data}
\begin{frame}
\frametitle{Learning from labeled and unlabeled data}
\structure{Training data: $\mathcal{S} = \mathcal{L} \cup \mathcal{U}$}
\begin{itemize}
\item labeled data: $\mathcal{L} = \{(\vec{x}_i, y_i), \quad i=1,\ldots,l\}$
\item unlabeled data: $\mathcal{U} = \{\vec{x}_i, \quad i = \underbrace{l+1,\ldots,m}_u\}$
\end{itemize}
\pspread
\structure{Graph Laplacian $\mat{L}$ associated with $\mathcal{S}$:}
\begin{itemize}
\item $\mat{L} = \mat{D} - \mat{W}$
\item adjacency matrix $\mat{W}$
\item diagonal matrix $\mat{D}$ with the degree of each node: $d_{ii} = \sum_{j=1}^m w_{ij}$
\end{itemize}
\pspread
\structure{Kernel matrix $\mat{K}$:} $k_{ij} = k(\vec{x}_i, \vec{x}_j)$
\pspread
\structure{Decision boundary $f(\vec{x})$:} $\vec{f} = [f(\vec{x}_i),~i=1,\ldots,m]^T$
\end{frame}
\begin{frame}
\frametitle{Learning from labeled and unlabeled data \cont}
\structure{Regularization framework for function learning:}
\begin{center}
\tikz[baseline]{
\node[fill=bl1!100, anchor=base, rounded corners=3pt, inner xsep=3mm] (d1) {
\color{bl3}
$\displaystyle
f^* = \argmin_{f \in \mathcal{H}_k}
\sum_{i=1}^l V(\vec{x}_i, y_i, f) +
\gamma_\text{A} \, \|f\|^2_\text{A} +
\gamma_\text{I} \, \|f\|^2_\text{I}
$
};
}
\end{center}
\vspace{.25cm}
\structure{Loss function $V(\vec{x}_i, y_i, f)$}
\begin{itemize}
\item Squared loss function \structure{$(y_i - f(\vec{x}_i))^2$} for Regularized Least Squares (RLS)
\item Hinge loss function \structure{$\max [0, 1-y_i f(\vec{x}_i)]$} for SVM
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Learning from labeled and unlabeled data \cont}
\structure{Regularization framework for function learning:}
\begin{center}
\tikz[baseline]{
\node[fill=bl1!100, anchor=base, rounded corners=3pt, inner xsep=3mm] (d1) {
\color{bl3}
$\displaystyle
f^* = \argmin_{f \in \mathcal{H}_k}
\sum_{i=1}^l V(\vec{x}_i, y_i, f) +
\gamma_\text{A} \, \|f\|^2_\text{A} +
\gamma_\text{I} \, \|f\|^2_\text{I}
$
};
}
\end{center}
\vspace{.15cm}
\structure{Regularization terms}
\begin{itemize}
\item \emph{Ambient} norm $\|\cdot\|_\text{A}$:
\begin{itemize}
\item norm of the function $f$ in the Reproducing Kernel Hilbert Space (RKHS)
\item enforces a smoothness condition on the possible solutions \\[.25cm]
\end{itemize}
\item \emph{Intrinsic} norm $\|\cdot\|_\text{I}$:
\begin{itemize}
\item norm of the function $f$ in the low dimensional manifold
\item enforces a smoothness along the sampled $\mathcal{M}$
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Reproducing Kernel Hilbert Spaces (RKHS)}
\structure{Hilbert space}
\begin{itemize}
\item \structure{abstract vector space} with any finite or \structure{infinite} number of dimensions
\item possesses the structure of the \structure{inner product}
\item allows the measurement of \structure{angles} and \structure{lengths}
\item is \structure{complete} \\[.3cm]
\end{itemize}
\pause
\structure{Reproducing Kernel Hilbert Space}
\begin{itemize}
\item Hilbert space of functions
\item can be defined by kernels
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Manifolds}
A \structure{manifold} is a topological space that on a small enough scale \\ resembles
the Euclidean space. \\[.3cm]
\begin{figure}
\centering
\subfloat{
\includegraphics[height=3.3cm]{\pngdir/globe.\png} \pause
}
\qquad \qquad
\subfloat{
\alt<4->{
\copyrightbox[b]{
\makebox[.4\linewidth]{
\href{https://commons.wikimedia.org/wiki/File:Waldseem%C3%BCller-Globus.jpg
}
{
\includegraphics[height=3.3cm]{\jpgdir/waldseemueller-globus.\jpg}
}
}
}{Martin Waldseemüller, Public domain, via Wikimedia Commons}
}{\alt<3>{
\includegraphics[height=3.3cm]{\pngdir/worlmap_notflat.\png}
}{
\includegraphics[height=3.3cm]{\pngdir/worldmap.\png}
}}
}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Manifold Learning}
\structure{The Swiss Roll Problem}
\begin{figure}
\subfloat{
\includegraphics[height=4cm]{\pngdir/swiss_roll.\png}
}
\hspace{2cm} \pause
\subfloat{
\copyrightbox[b]{
\makebox[.35\linewidth]{
\includegraphics[height=3.2cm]{\pngdir/swiss_roll_solution.\png}
}
}{Algorithm: S. Marsland, “Machine Learning: An Algorithmic Perspective”, Chapter 10, 2009.}
}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Learning from labeled and unlabeled data \cont}
\structure{Intrinsic norm $\|\cdot\|_\text{I}$:}
\begin{displaymath}
\|f\|_\text{I}^2 =
\sum_{i=1}^m \sum_{j=i}^m w_{ij} \big( f(\vec{x}_i) - f(\vec{x}_j) \big)^2 =
\vec{f}^T \mat{L} \, \vec{f}
\end{displaymath}
\end{frame}
\begin{frame}
\frametitle{Learning from labeled and unlabeled data \cont}
\structure{Representer Theorem} (Kimeldorf and Wahba, 1970): \\[.3cm]
The solution $f^*$ of this optimization problem has the form:
\begin{displaymath}
f^*(\vec{x}) = \sum_{i=1}^m \beta_i^* \cdot k(\vec{x}_i, \vec{x}) + \beta_0^*
\end{displaymath}
\end{frame}
\subsection{Laplacian Support Vector Machines}
\begin{frame}
\frametitle{Laplacian Support Vector Machines}
\structure{Constrained primal optimization problem based on the dual form:}
\begin{center}
\small
\tikz[baseline]{
\node[fill=bl1!100, anchor=base, rounded corners=3pt, inner xsep=3mm] (d1) {
\color{bl3}
$\displaystyle
\begin{aligned}
\min\limits_{\vec{\beta} \in \real^m, \vec{\xi}\in\real^l}
& \qquad \sum_{i=1}^l \xi_i +
\gamma_\text{A} \cdot \vec{\beta}^T \mat{K} \vec{\beta} +
\gamma_\text{I} \cdot \vec{\beta}^T \mat{K} \mat{L} \mat{K} \vec{\beta} \\[.3cm]
\mbox{subject to}
& \qquad y_i \left( \sum_{j=1}^m \vec{\beta}_j k(\vec{x}_j, \vec{x}_i) + \beta_0\right) \geq 1 - \xi_i, \quad i = 1,\ldots,l \\[.3cm]
& \qquad \xi_i \geq 0, \quad i = 1,\ldots,l
\end{aligned}
$
};
}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Laplacian Support Vector Machines \cont}
\structure{Lagrange function $L$:}
\begin{eqnarray*}
L(\vec\beta,\beta_0,\vec\xi,\vec\lambda,\vec\nu)
&=& \sum_{i=1}^l \xi_i +
\frac{1}{2} \vec{\beta}^T (2 \gamma_\text{A} \mat{K} + 2 \gamma_\text{I} \mat{K}\mat{L}\mat{K}) \vec{\beta} -{} \\
& & {}- \sum_{i=1}^l \lambda_i \left(y_i \Bigg(\sum_{j=1}^m \beta_i k(\vec{x}_i,\vec{x}_j) + \beta_0\Bigg) - 1 + \xi_i\right) -{} \\
& & {}- \sum_{i=1}^l \nu_i \xi_i
\end{eqnarray*}
\end{frame}
\begin{frame}
\frametitle{Laplacian Support Vector Machines \cont}
\structure{KKT condition:} the gradient w.\,r.\,t. the primal variables $\vec{\beta}, \beta_0, \vec{\xi}$ has to vanish
\pause
\begin{itemize}
\item Partial derivative w.\,r.\,t. $\beta_0$:
\begin{displaymath}
\frac{\partial L}{\partial \beta_0} \stackrel{!}{=} 0 \quad \Rightarrow \quad
\sum_{i=1}^l \lambda_i y_i = 0
\end{displaymath}
\pause
\item Partial derivative w.\,r.\,t. $\xi_i$:
\begin{displaymath}
\frac{\partial L}{\partial \xi_i} \stackrel{!}{=} 0 \quad \Rightarrow \quad
1 - \lambda_i - \nu_i = 0 \quad \Rightarrow \quad
0 \leq \lambda_i \leq 1
\end{displaymath}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Laplacian Support Vector Machines \cont}
\structure{Simplifying the Langrangian using the two identities above:}
{\small
\begin{eqnarray*}
L(\vec{\beta}, \vec{\lambda})
&=& \frac{1}{2} \vec{\beta}^T (2 \gamma_\text{A} \mat{K} + 2 \gamma_\text{I} \mat{K}\mat{L}\mat{K}) \vec{\beta} -{} \\
& & {}- \sum_{i=1}^l \lambda_i \left(y_i \Bigg(\sum_{j=1}^m \beta_i k(\vec{x}_i,\vec{x}_j) + \beta_0\Bigg) - 1 \right) \\
&=& \frac{1}{2} \vec{\beta}^T (2 \gamma_\text{A} \mat{K} + 2 \gamma_\text{I} \mat{K}\mat{L}\mat{K}) \vec{\beta} -
\vec{\beta}^T \mat{K} \mat{J}_{\mathcal{L}}^T \mat{Y} \vec{\lambda} +
\sum_{i=1}^l \lambda_i
\end{eqnarray*}
}
with $\mat{J}_{\mathcal{L}} = [\mat{I}~\mat{0}] \in \real^{l\times m}$:
\begin{itemize}
\item identity matrix $\mat{I} \in \real^{l \times l}$
\item rectangular matrix $\mat{0} \in \real^{l \times u}$ with all entries being 0 \\[.25cm]
\end{itemize}
and diagonal matrix $\mat{Y} \in \real^{l \times l}$ composed by the $l$ class labels $y_i$
\end{frame}
\begin{frame}
\frametitle{Laplacian Support Vector Machines \cont}
\structure{Partial derivative w.\,r.\,t. $\vec{\beta}$:}
\begin{eqnarray*}
\frac{\partial L}{\vec{\beta}} \stackrel{!}{=} \vec{0}
&\Rightarrow&
(2 \gamma_\text{A} \mat{K} + 2 \gamma_\text{I} \mat{K}\mat{L}\mat{K}) \vec{\beta} -
\mat{K} \mat{J}_{\mathcal{L}}^T \mat{Y} \vec{\lambda} = \vec{0} \\[.25cm]
&\Rightarrow&
\vec{\beta} = (2 \gamma_\text{A} \mat{I} + 2 \gamma_\text{I} \mat{K}\mat{L})^{-1}
\mat{J}_{\mathcal{L}}^T \mat{Y} \vec{\lambda}
\end{eqnarray*}
\pspread
\structure{Note:} direct relationship between parameters $\vec{\beta}$ and Lagrange multipliers $\vec{\lambda}$
\end{frame}
\begin{frame}
\frametitle{Laplacian Support Vector Machines \cont}
\structure{Substituting back in the Langrange expression leads to the dual problem:}
\begin{center}
\small
\tikz[baseline]{
\node[fill=bl1!100, anchor=base, rounded corners=3pt, inner xsep=3mm] (d1) {
\color{bl3}
$\displaystyle
\begin{aligned}
\max\limits_{\vec{\lambda} \in \real^l}
& \qquad \sum_{i=1}^l \lambda_i - \frac{1}{2} \vec{\lambda}^T \mat{Q} \vec{\lambda} \\[.3cm]
\mbox{subject to}
& \qquad 0 \leq \lambda_i \leq 1, \quad i = 1,\ldots,l \\
& \qquad \sum_{i=1}^l \lambda_i y_i = 0
\end{aligned}
$
};
}
\end{center}
where
{\small
\begin{displaymath}
\mat{Q} = \mat{Y} \mat{J}_{\mathcal{L}} \mat{K}
(2 \gamma_\text{A} \mat{I} + 2 \gamma_\text{I} \mat{K}\mat{L})^{-1}
\mat{J}_{\mathcal{L}}^T \mat{Y}
\end{displaymath}
}
\end{frame}
\subsection{Lessons Learned}
\begin{frame}
\frametitle{Lessons Learned}
\structure{Laplacian SVM:}
\begin{itemize}
\item Ongoing research topic \\[.25cm]
\item Extension of the Kernel SVM \\[.25cm]
\item Additional regularization term \\[.25cm]
\item Derivation of the dual problem
\end{itemize}
\end{frame}
\input{nextTime.tex}
\subsection{Further Readings}
\begin{frame}
\frametitle{Further Readings}
\begin{itemize}
\item Stefano Melacci, Mikhail Belkin: \\
\structure{Laplacian Support Vector Machines Trained in the Primal}, \\
Journal of Machine Learning Research, 12:1149-1184, 2011 \\[.25cm]
\item Mikhail Belkin, Partha Niyogi, Vikas Sindhwani: \\
\structure{Manifold Regularization: A Geometric Framework for Learning \\
from Labeled and Unlabeled Examples}, \\
Journal of Machine Learning Research, 7:2399-2434, 2006
\end{itemize}
\end{frame}
% \subsection{Comprehensive Questions}
%
%\begin{frame}
% \frametitle{Comprehensive Questions}
%
% \begin{itemize}
% \item
% \end{itemize}
%\end{frame}