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main.tex
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% FOR MATRIX NOTATION
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% ---------------------------------------------------------
% CUSTOM ENVIRONMENTS
% ---------------------------------------------------------
% \theoremstyle{plain}
% \newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}
%
% \theoremstyle{definition}
% \newtheorem{definition}{Definition}[section]
% \newtheorem{corollary}{Corollary}[theorem]
%
% \theoremstyle{remark}
% \newtheorem{remark}{Remark}[section]
%----------------------------------------------------------------------------------------
% TITLE PAGE
%----------------------------------------------------------------------------------------
\title{Memory-Efficient Separable Simplex-Structured Matrix Factorization via the Frank-Wolfe Method } % The short title appears at the bottom of every slide, the full title is only on the title page
\author{ Tri Nguyen} % Your name
\institute[OSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
{Qualifying Exam \\
Oregon State University \\ % Your institution for the title page
% \medskip
% \textit{[email protected] \endgraf } % Your email address
% }
}
\date{\today} % Date, can be changed to a custom date
\makeatletter
\makeatother
\begin{document}
%------------------------------------------------
\begin{frame}
\titlepage % Print the title page as the first slide
\note{
Hello everyone,
My name is Tri Nguyen. Thank you for serving as my committee. Today I'm glad to present my work for my qualifying exam.
}
\end{frame}
%------------------------------------------------
%------------------------------------------------
%------------------------------------------------
%\begin{frame}
%\frametitle{Contents} % Table of contents slide, comment this block out to remove it
%\tableofcontents % Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation
%\end{frame}
%------------------------------------------------
%------------------------------------------------
%----------------------------------------------------------------------------------------
% PRESENTATION SLIDES
%----------------------------------------------------------------------------------------
% \includeonlyframes{current}
\section{Outline}
\begin{frame}
\tableofcontents
\note{
\begin{itemize}
\item A brief outline of the talk. We'll go through 4 parts. In the first section, I'll introduce the problem, elaborate the setting, and show some examples why we are interested in it.
\item Then a quick look at how this problem has been solved. Particularly, there are 2 related different approaches, and we'll see what are their drawback and what could we offer to improve.
\item In the third part, we advocate using an optimization method named Frank-Wolfe.
We'll try to show our intuition and rationale behind the proposal.
\item Lastly, we showcase performance of the proposal via \textbf{both} synthetic experiment and real data experiments.
\end{itemize}
}
\end{frame}
\section{Problem of Interest}%
\subsection{Problem Setting}%
\label{sub:separable_nmf}
\begin{frame}
\frametitle{Simplex Structured Matrix Factorization}
{
\setbeamercolor{block title}{bg=BurntOrange,fg=black}
\begin{block}{Simplex Structured Matrix Factorization (SSMF)}
Data matrix $\bm{X} \in \mathbb{R}^{N \times M}$ is assumed to be generated by $\bm{W} \in \mathbb{R}^{N \times K}, \bm{H} \in \mathbb{R}^{K \times M}, K \ll \min(M, N)$ such that
\[
\bm{X} = \bm{W} \bm{H} + \bm{V} \quad \text{\rm subject to } \bm{H} \geq 0 , \bm{1}^{\T}\bm{H} = \bm{1}^{\T}
\]
\end{block}}
\begin{block}
Given $\bm{X}$, how do we find the latent factors $\bm{W}, \bm{H}$?
\end{block}
% In addition, $\bm{W}$ is assumed to be full rank.
\begin{itemize}
\item Closely related to nonnegative matrix factorization.
% Since $\bm{W} \geq 0$, the constraint $ \bm{1}^{\T}\bm{H}= \bm{1}^{\T}$ can always be enforced by normalizing $\bm{X}$.
\item Has received significant attention across many domains \citep{keshava2002spectral,CANMS,arora2012practical,arora2012learning,recht2012factoring,mao2017mixed,panov2018consistent,huang2019detecting,fu2016robust}
\end{itemize}
\note{
\begin{itemize}
\item We are interesting a branch of matrix factorization, namely simplex structure matrix factorization.
\item In particular, the model assumes that the data matrix $\bm{X}$ is generated as a production of 2 low-rank matrix $\bm{W}, \bm{H}$, as we will refer as latent factor. The inner dimension $K$ is assumed to be relatively small compared to $M, N$.
\item In addition, it is assumed that columns of $\bm{H}$ reside in a probability simplex. Note that we do not require nonnegativity on $\bm{W}$ as in NMF.
\item This model are closely related to NMF in a sense that we can always convert a NMF model into SSMF model by performing a normalization on columns of $\bm{X}$.
\item This model has witnessed a large interest from various domain, including machine learning, signal processing.
\item What? So the problem is: Given $\bm{X}$, how do we find the ground truth $\bm{W}, \bm{H}$.
\item Why? Finding $\bm{W}, \bm{H}$ is meaningful as they carries physical meaning depending on particular applications.
\item {\blue \cite{CANMS} is using pure pixel model in addition to SSMF}
% \item {\blue It has been widely adopted
% \begin{itemize}
% \item Hyperspectral unmixing \citep{keshava2002spectral}
% \item Bind source separation \citep{CANMS}
% \item Topic discovery \citep{arora2012practical,gillis2013robustness,arora2012learning}
% \item Community detection \cite{mao2017mixed}
% \item NMF in general \citep{recht2012factoring}
% \end{itemize}
% }
\end{itemize}
}
\end{frame}
\subsection{Applications}%
\label{sub:applications}
\begin{frame}
\frametitle{Application: Topic Modeling}
\begin{figure}[ht]
\centering
{
\fontsize{7pt}{9pt}\selectfont% or whatever fontsize you like
\def\svgwidth{\columnwidth}
\import{./figures/}{topic_modeling_demo.pdf_tex}
}
\caption*{A demonstration of $\bm{x}_\ell \approx \bm{W} \bm{h}_\ell$}
\end{figure}
\begin{itemize}
\item
$\bm{X}$ is a vocab-document matrix, then $\bm{X} = \bm{W}\bm{H}$ where
\begin{itemize}
\item $\bm{H} \geq 0,\bm{1}^{\T}\bm{H} = \bm{1}^{\T} $
\item $K$ is number of topics
\end{itemize}
\item This model has been used in \citep{arora2012practical,arora2012learning,recht2012factoring,huang2016anchor,arora2016computing}
\end{itemize}
\note{
\begin{itemize}
\item For example, in topic modeling, if a document is presented using bag-of-word, then a document is assumed to be a convex combination of some small set of topics, in this case presented as 3 vectors:\ldots
\item The task of topic discovery is to find the latent representation of topic, in this case the $\bm{W}$ matrix.
Note that coefficients are sum-up-to 1 in this case.
\end{itemize}
}
\end{frame}
\begin{frame}
\frametitle{Application: Community Detection}
\begin{columns}
\begin{column}{0.7\textwidth}
\begin{itemize}
\item The mixed membership stochastic blockmodels \citep{airoldi2008mixed}
\begin{align*}
&P_{i,j} = \bm{h}_i^{\T} \bm{B} \bm{h}_j \\
& \bm{A}(i,j) = \bm{A}(j, i) \sim \text{Bernoulli}(\bm{P}(i, j))
\end{align*}
where $\bm{h}_i= [h_{1,i}, \ldots , h_{K, i}]^{\T}$ represents membership of node $i$, $\bm{B}$ represents community-community connection.
\end{itemize}
\end{column}
\begin{column}{0.3\textwidth}
\begin{figure}
\centering
\resizebox{\textwidth}{!}
{
\incfig{community-detection-demo} }
\caption*{Demonstration of a graph with $K=2$ communities}
\end{figure}
\end{column}
\end{columns}
\begin{columns}
\begin{column}{1.\textwidth}
\begin{itemize}
\item By physical interpretation, $\bm{H} \geq 0, \bm{1}^{\T}\bm{H}=\bm{1}^{\T}$.
\item Range space of $\bm{H}$ can be estimated from
% a matrix $\bm{X}$ constructing from
$K$ leading eigenvectors of $\bm{A}$ (denoted as matrix $\bm{X}$).
\citep{mao2017mixed,mao2017estimating,panov2018consistent,lei2015consistency}
\[
\bm{X} = \bm{W} \bm{H} + \bm{N}
\]
%todo think about this sentence
\end{itemize}
\end{column}
\end{columns}
\note{
\begin{itemize}
\item Another application is community detection, where given a graph, we wish to discovery a small number of community constituted by set of nodes that are considered to be close to each other.
\item As a well-known model, the mixed membership stochastic blockmodels models . Under this model, the task amounts to finding the membership matrix $\bm{H}$ from an observed adjacency matrix.
\item As a membership vector, it is natural that $\bm{h}$ is in a probability simplex.
\item This is nothing but the SSMF as we saw earlier.
\item {\blue Despite having a broad range of applications, solving a SSMF models is a difficult task.}
\end{itemize}
}
\end{frame}
\begin{frame}
\frametitle{Identifiability}
\begin{itemize}
% \item SSMF model with $\bm{X}, \bm{W}^{\star}, \bm{H}^{\star}$
% \item NMF in general is NP-hard \citep{vavasis2009complexity}.
\item Given a SSMF model with $\bm{X} = \bm{W}^{\star} \bm{H}^{\star}$, finding $\bm{W}^{\star}, \bm{H}^{\star}$ is a difficult problem.
\begin{subequations}
\label{problem:criteria}
\begin{alignat}{2}
& \text{find} && \quad \bm{W}, \bm{H} \\
& \text{subject to } && \bm{X} = \bm{W}\bm{H} \\
&&& \bm{H} \geq 0, \bm{1}^{\T}\bm{H} = \bm{1}^{\T}
\end{alignat}
\end{subequations}
\item The solution is not unique. There exists non-singular $\bm{Q}$ such that
\[
\bm{X} = \bm{W}^{\star} \bm{H}^{\star} = (\underbrace{\bm{W}^{\star}\bm{Q}^{-1}}_{\bm{W}'}) (\underbrace{\bm{Q} \bm{H}^{\star}}_{\bm{H}'}), \text{ and } \bm{H}' \geq 0, \bm{1}^{\T}\bm{H}' = \bm{1}^{\T}
\]
% \item In topic modeling, each column in $\bm{W}^{\star}$ is believed to represent topics. However, the sought $\bm{W}'$ is some mixture of topics
\begin{definition}[Identifiability \citepb{fu2018nonnegative}]
A SSMF model where $\bm{X}= \bm{W}^{\star} \bm{H}^{\star}$ is called identifiable respect to criterion \eqref{problem:criteria} if
for all $\bm{W}, \bm{H}$ satisfying criterion \eqref{problem:criteria}, it holds that
$\bm{W}=\bm{W}^{\star} \boldsymbol \Pi , \bm{H} = \boldsymbol \Pi^{\T}\bm{H}^{\star} $, where $\boldsymbol \Pi$ is a permutation matrix.
\end{definition}
\end{itemize}
\note{
\begin{itemize}
% \item Note that we do not just want to explain $\bm{X}$ with some matrices $\bm{W}, \bm{H}$. What we really want is finding the ground truth $\bm{W}^{\star}, \bm{H}^{\star}$. However, NMF in general is an NP-hard problem.
\item A natural attempt is to consider finding \ldots , and hope that the found solution could reveal $\bm{W}^{\star}, \bm{H}^{\star}$
\item Unfortunately, the solution of Problem 1 is not unique. We can see why. It is trivial to construct $\bm{Q}$ \ldots
\item $\bm{W}', \bm{H}'$ in this case would not bring much useful information. For example, in topic modeling , $\bm{W}^{\star}$ represent the topics, while the sought $\bm{W}'$ represent a mixed of topics. Hence we haven't really got a good representation of topics by using $\bm{W}'$.
\item By that reason, we focus our interest to those models whose can be identified. In particular, by saying SSMF model is identifiable we mean that if criterion (1) has solution $\bm{W}, \bm{H}$ , then they are just some permutation of the ground truth.
\item This kind of ambiguity is unharmed.
\item The definition is borrowed from this work and modified to our specific SSMF problem.
\item {\blue the point is: problem is hard}
\end{itemize}}
\end{frame}
\begin{frame}
\frametitle{Separability Condition}
\begin{block}{Separability condition \citepb{donoho2003does}}
There exists set $\mathcal{K}$ so that $\bm{H}^{\star}(:, \mathcal{K}) = \bm{I}$.
\end{block}
\begin{itemize}
\item Have been adapted in many works \citep{chan2011simplex,nascimento2005vertex,arora2016computing,gillis2014fast}
% {\red not very concise}
\item Finding $\mathcal{K}$ is the key to estimate ground truth $\bm{W}^{\star}, \bm{H}^{\star}$.
\begin{itemize}
\item In noiseless case, $\bm{X}(:, \mathcal{K}) = \bm{W}^{\star} \bm{H}^{\star}(:, \mathcal{K}) = \bm{W}^{\star}$.
\end{itemize}
\item Physical interpretation
\begin{itemize}
\item Anchor word \citep{arora2012learning} in topic modeling
\item Pure node \citep{mao2017mixed} in community detection
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.55\textwidth}
\centering
\resizebox{\textwidth}{!}{
\fontsize{4pt}{6pt}\selectfont% or whatever fontsize you like
\def\svgwidth{\columnwidth}
\import{./figures/}{anchor-word-demo.pdf_tex}
}
\caption*{Demonstration of anchor word}
\end{subfigure}
\begin{subfigure}[b]{0.30\textwidth}
\centering
{ \fontsize{7pt}{9pt}\selectfont% or whatever fontsize you like
\def\svgwidth{\columnwidth}
\import{./figures/}{pure-node-demo.pdf_tex} }
\caption*{Demonstration of pure node}
\end{subfigure}
\end{figure}
\item Expert annotator in crowd-sourcing \citep{ibrahim2019crowdsourcing}
\item Pure pixels in hyperspectral unmixing \citep{ma2014asignal}
\end{itemize}
\end{itemize}
\note{
\begin{itemize}
\item There are many works that investigate in finding conditions that can guarantee identifiability.
One of the well-known conditions is separability condition.
\item It states that \ldots . The term was first coined in this work, and have been adapted in many works after that.
\item The condition has been exploited to when it comes to algorithm design. Particularly, the problem of finding $\bm{W}, \bm{H}$ boils down to finding the set $\mathcal{K}$. The rationale is like this.
This notation means a sub-matrix constructed by from columns of $\bm{X}$ with indices from $\mathcal{K}$. Therefore, knowing $\mathcal{K}$ already reveals $\bm{W}$. Then finding $\bm{H}$ becomes a trivial task.
\item In terms of application, the condition imposes interesting and reasonable physical interpretation.
\begin{itemize}
\item For example, in topic modeling, it asserts that for each topic, there exists a word that only belong to that topic.
\item In communities, there exists a node that only belongs to a single community.
\item Other similar interpretations are presented in other applications.
\end{itemize}
\end{itemize}
}
\end{frame}
\section{Related Works}%
\label{sec:what_have_people_done_}
\begin{frame}
\frametitle{A Self-Dictionary Perspective}
\begin{itemize}
% \item An interesting formulation arises from self-dictionary and sparse regression
\item Consider the self-dictionary and sparse regression formulation,
\citep{iordache2014collaborative,esser2012convex,recht2012factoring,Elhamifar2012}
\begin{alignat*}{2}
& \minimize_{\bm{C}} \quad && \norm{\bm{C}}_{\text{row-}0} \\
& \text{\rm subject to} && \bm{X} = \bm{X} \bm{C} \\
&&& \bm{C} \geq 0, \bm{1}^{\T}\bm{C} = \bm{1}^{\T}
\end{alignat*}
\item $\bm{C}_{\text{opt}}(\mathcal{K}, :) = \bm{H}, \bm{C}_{\text{opt}}(\mathcal{K}^{c}, :) = \bm{0}$ is an optimal solution point.
\begin{itemize}
\item $\norm{\bm{C}_\text{opt}}_{\text{row-}0} = K$.
\item For a full rank $\bm{W}$, one needs at least $K$ non-zero rows of $\bm{C}$ to construct $\bm{X}$.
\end{itemize}
\end{itemize}
% \begin{figure}
% \centering
% \includegraphics[width=0.7\textwidth]{figures/sdmmv_demo/demo.png}
% \caption*{Row-sparsity matrix $\bm{C}_{\text{opt}}$ }
% \end{figure}
\begin{figure}[ht]
\centering
\incfig[0.9]{demoooo}
\caption*{Row-sparsity matrix $\bm{C}_{\text{opt}}$ }
\label{fig:demoooo}
\end{figure}
\note{
\begin{itemize}
\item There have been many formulations developed for problem of finding $\mathcal{K}$ under separability condition. One interesting perspective is from self-diction and sparse regression, as shown in this formulation. The objective function is row-0 norm, which counts number of nonzero rows in $\bm{C}$.
\item The optimal solution of this problem has a particular structure, which is:
\begin{itemize}
\item A subset of rows of $\bm{C}$ with indices from $\mathcal{K}$ is $\bm{H}$
\item The other rows are $ \bm{0}$ rows.
\end{itemize}
First of all, [look at picture], we can see that this $\bm{C}$ satisfies all constraints. Objective value at $\bm{C}$ is $K$.
\item And for a full rank $\bm{W}$, one will need at least $K$ non-zero rows from $\bm{C}$.
\item With this structure of $\bm{C}_{\text{opt}}$, we can easily identify $\mathcal{K}$.
\end{itemize}}
\end{frame}
% \begin{frame}
% \frametitle{demo}
% \begin{figure}
% \centering
% \includegraphics[width=\textwidth]{figures/sdmmv_demo/demonstration.eps}
% \end{figure}
%
% \begin{figure}[ht]
% \centering
% \incfig{demoooo}
% \caption{demoooo}
% \label{fig:demoooo}
% \end{figure}
%
% \end{frame}
% \begin{frame}
% \frametitle{A Self-Dictionary Perspective}
% \begin{figure}[t]
% \begin{subfigure}[b]{0.3\textwidth}
% \includegraphics[width=\textwidth]{figures/sdmmv_geometry.png}
% \caption*{$\bm{x}_\ell = \bm{W}\bm{h}_{\ell}$ \citep{fu2018nonnegative}}
%
% \end{subfigure}
% \quad \quad
% \begin{subfigure}[b]{0.55\textwidth}
% \centering
% \includegraphics[width=\textwidth]{figures/sdmmv_demo/demo.png}
% \caption*{Row-sparsity matrix $\bm{C}$ }
% \end{subfigure}
% \end{figure}
% \begin{itemize}
% \item Since $\bm{h}_\ell \geq 0, \bm{1}^{\T}\bm{h}_\ell=1$, then $\bm{x}_\ell \in \text{conv}(\bm{W})$
% \item By separability assumption, there exist data points $\bm{x}_\ell$'s at the vertices of $\text{conv}(\bm{W})$
% \item Hence, each column $\bm{x}_\ell$ is a convex combination of a small subset of columns in $\bm{X}$
% \item Self-dictionary with encouraging row-sparsity
% \end{itemize}
% \begin{alignat*}{2}
% & \minimize_{\bm{C}} \quad && \norm{\bm{C}}_{\text{row-}0} \\
% & \text{\rm subject to} && \bm{X} = \bm{X} \bm{C} \\
% &&& \bm{C} \geq 0, \bm{1}^{\T}\bm{C} = \bm{1}^{\T}
% \end{alignat*}
%
% % todo: change layout to | text | figure |
%
% \note{
% \begin{itemize}
% \item We are looking for a basis in $\bm{X}$ to represent all columns in $\bm{X}$
% \item To have a better look at this formulation, or how solving this formulation leads to $\mathcal{K}$? Next slide
% \end{itemize}}
% \end{frame}
% \begin{frame}
% \frametitle{A Self-Dictionary Perspective}
% \begin{itemize}
% \item Since columns of $\bm{H}$ belongs to a simplex, each column in $\bm{X}$ is a convex combination of $K$ columns of $\bm{W}$
% \item By separability assumption, and assume that $\mathcal{K} = \set{1, 2, \ldots, N}$
% \[
% \bm{X} = \bm{W} \bm{H} = \bm{W} [\bm{I}, \bm{H}'] = [\bm{W}, \bm{W}\bm{H}']
% \]
% \item Hence, each column of $\bm{X}$ is a convex combination of a small subset of columns in $\bm{X}$
% \item Self-dictionary with encouraging row-sparsity
% \end{itemize}
% \begin{columns}
% \begin{column}{0.3\textwidth}
% \begin{alignat*}{2}
% & \minimize_{\bm{C}} \quad && \norm{\bm{C}}_{\text{row-}0} \\
% & \text{subject to} && \bm{X} = \bm{X} \bm{C} \\
% &&& \bm{C} \geq 0, \bm{1}^{\T}\bm{C} = \bm{1}^{\T}
% \end{alignat*}
% \end{column}
% \begin{column}{0.7\textwidth}
% % \begin{figure}[t]
% % \centering
% % \def\svgwidth{\linewidth}
% % \import{./figures/sdmmv_demo}{demonstration.pdf_tex}
% % \end{figure}
% \end{column}
% \end{columns}
%
% \note{
% \begin{itemize}
% \item We are looking for a basis in $\bm{X}$ to represent all columns in $\bm{X}$
% \item To have a better look at this formulation, or how solving this formulation leads to $\mathcal{K}$? Next slide
% \end{itemize}}
% \end{frame}
\subsection{Greedy Approach}%
\label{sub:greedy_approach}
\begin{frame}
\frametitle{Greedy Approach}
\begin{alignat*}{2}
& \minimize_{\bm{C}} \quad && \norm{\bm{C}}_{\text{row-}0} \\
& \text{\rm subject to} && \bm{X} = \bm{X} \bm{C} \\
&&& \bm{C} \geq 0, \bm{1}^{\T}\bm{C} = \bm{1}^{\T}
\end{alignat*}
\begin{itemize}
\item The greedy approach identifies the set $\mathcal{K}$ by adding one index at a time \citep{fu2014self}.
\item Successive projection algorithm (SPA) \citep{MC01} is a representative.
\item Extracting $\mathcal{K}$ is guaranteed even in noisy case \citep{gillis2014fast}.
\item All greedy-based methods have a Gram-Schmidt structure which is prone to error propagation under noisy conditions.
\end{itemize}
\note{
\begin{itemize}
% \item Under this the formulation, there are 2 main approaches in the literature.
\item Solving this optimization problem is not a trivial task. It is firstly not a convex problem due to the row-0, and secondly, and it is a combinatorial problem in essence.
% So a native greedy search would be prohibited even if number of vertices $K$ is given.
\item One of the approaches has been largely studied is greedy approach. As the name suggested, methods in this approach construct set $\mathcal{K}$ by adding 1 index at a time.
\item A famous successive projection algorithm (SPA) is a representative of this approach.
\item And it has been shown that estimating exact $\mathcal{K}$ is guaranteed, even under noisy condition.
\item However, all methods following the greedy approach have a Gram-Schmidt structure in their algorithms. When noise is present, an error made in one iteration will be propagated to future iterations.
% This make methods under this approach sensitive to noise.
\end{itemize}
}
\end{frame}
\subsection{Convex Relaxation Approach}%
\label{sub:convex_relaxation_approach}
\begin{frame}
\frametitle{Convex Relaxation Approach}
\begin{itemize}
\item Relax the problem to a convex optimization problem \citep{gillis2018afast,gillis2014robust,gillis2013robustness,recht2012factoring,Elhamifar2012,Ammanouil2014blind}
\item An example of this approach is \citep{esser2012convex,fu2015robust,gillis2018afast}
\begin{alignat*}{2}
& \minimize_{\bm{C}} \quad && \dfrac{1}{2} \norm{\bm{X} - \bm{X}\bm{C}}_{\rm F}^2 + \lambda R(\bm{C}) \\
& \text{\rm subject to} && \bm{C} \geq 0, \bm{1}^{\T}\bm{C} = \bm{1}^{\T}
\end{alignat*}
where $R(\bm{C})$ is some regularization term to promote row-sparsity.
% , e.g.,
% \begin{itemize}
% \item \texttt{FastGradient} \citep{gillis2018afast}, $R(\bm{C}) = p^{\T} \text{diag}(\bm{C})$
% \end{itemize}
% $\norm{\bm{C}}_{\infty, 1} := \sum^{N}_{n=1} \norm{\bm{C}(n, :)}_{\infty}$.
\item $\mathcal{K}$ is identified in noisy conditions.
\item Often more robust than greedy approach.
\end{itemize}
\begin{columns}
\begin{column}{0.6\textwidth}
% However, this approach suffers from large memory consumption
\begin{alertblock}{Potential Memory Issue}
The variable $\bm{C}$ has size $N \times N$.
\end{alertblock}
A dense matrix $\bm{C}$ with $N=100000$ requires $74.5$GB.
\end{column}
\begin{column}{0.5\textwidth}
\begin{figure}
\begin{subfigure}{\textwidth}
\begin{tikzpicture}[scale=0.48]
\begin{axis}[
/pgf/number format/1000 sep={},
xlabel=$N$,
ylabel=Memory cost in RSS (GB),
legend pos=north west,
legend cell align={left},
xticklabel style={
/pgf/number format/fixed,
},
scaled x ticks=false,
% yticklabels={0, 1, 2, 3, 4, 5},
extra y ticks={1, 3, 5, 6},
% extra y tick labels={0.3, 5.0},
% extra tick style={tickwidth=\pgfkeysvalueof{/pgfplots/minor tick length}},
]
\addplot+[red,mark=*,only marks,mark size=3.5pt,mark options={fill=red!50!white}]
table [x=N, y=FG]{figures/synthetic_data/exp2/mem.dat};
\addplot+[dashed,blue,domain=1000:10000, samples=5,mark=none] {0.000000055*x^2 - 0.000024*x + 0.12};
\legend{\texttt{FastGradient}, curve $aN^2 + b N +c$}
\end{axis}
\end{tikzpicture}
\vspace{-0.25cm}
\caption*{Memory consumption of \texttt{FastGradient} {\footnotesize \citep{gillis2018afast}}}
\end{subfigure}
\end{figure}
\end{column}
\end{columns}
\note{
\begin{itemize}
\item The second approach is called convex relaxation.
\item As mentioned before, the original problem is non-convex, which makes it hard in terms of optimisation. So a natural thing to do is to use a convex opt problem as a surrogate. There has been many convex relaxation formulations proposed in the literature.
One example is this formulation where the objective function comprises of 2 terms: the fitting error and a regularization to promote row-sparsity of $\bm{C}$.
\item Under this formulation, identifiability is guaranteed.
\item Since the algorithm does not suffer from error propagation, it is often more robust then the previous approach. We'll see some evidence on this in our experiments.
\item However, memory is an obstacle for this approach. Size of variable $\bm{C}$ is $N$ by $N$. If it is a dense matrix, memory requirement will grow quadratically.
As an example, FastGradient is a method following this approach and is considered as state-of-the-art. We run FastGradient on synthetic data and measure its memory consumption. We can see that memory cost grows quadratically to $N$. This memory cost prohibits this approach's applicability to large scale problem when $N$ could reach to 100000.
\item The question is can we do any better?
\end{itemize}
}
\end{frame}
\section{Proposal: Frank-Wolfe}%
\label{sec:what_are_we_proposing_}
\begin{frame}
\frametitle{Proposal: Frank-Wolfe}
In order to gain noise robustness and memory efficiency while obtaining identifiability,
\begin{itemize}
\item We follow the convex relaxation approach.
\item We propose to use Frank-Wolfe as the optimization method to guarantee $O(KN)$ memory consumption.
\end{itemize}
\note{
Yes, in the followings, I'll present our proposal on the use of FW. Particularly,
\begin{itemize}
\item We follow the convex relaxation approach because of its noise robustness
\item We propose using Frank-Wolfe method as the optimization method that can guarantee a memory cost of $O(KN)$.
\end{itemize}
}
\end{frame}
\subsection{Warm-up: Noiseless Case}%
\begin{frame}
\frametitle{Warm-up with the Noiseless Case}
% We propose to use Frank-Wolfe as the method to work on convex relaxation formulation.
% For a noiseless case,
\begin{subequations}
\label{problem:noiseless_warmup}
\begin{alignat}{2}
& \minimize_{\bm{C}} \quad && \dfrac{1}{2} \norm{\bm{X} - \bm{X}\bm{C}}_{\rm F}^2 \\
& \text{\rm subject to} && \bm{C} \geq 0, \bm{1}^{\T}\bm{C} = \bm{1}^{\T}
\end{alignat}
\end{subequations}
Problem \eqref{problem:noiseless_warmup} can have several solutions
\begin{itemize}
\item A desired solution $\bm{C}^{\star}(\mathcal{K}, :) = \bm{H}, \bm{C}^{\star}(\mathcal{K}^{c}, :) = \bm{0}$
\item A trivial solution $\bm{I}_{N}$
\end{itemize}
\begin{figure}[!t]
\centering
\hspace{-0.5cm}
\begin{subfigure}[t]{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.4]
\begin{axis}[
/pgf/number format/.cd,
1000 sep={},
xlabel=$n$,
ylabel=$\norm{\bm{C}_{\text{opt}}(n, :)}_{\infty}$,
legend pos=south east,
legend cell align={left},
]
\addplot+[only marks, mark=o, mark size=3.5pt]
table [x=index, y=norm]{code/pgd_C.dat};
\addplot+[ycomb, mark=square*, mark size=4pt]
table [x=index, y=norm]{code/pgd_C_pure_pixel.dat};
\legend{$n \notin \mathcal{K}$, $n \in \mathcal{K}$}
\end{axis}
\end{tikzpicture}
\caption*{\texttt{APG - Objective value: $3.99{e-}5$}}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\centering
\begin{tikzpicture}[scale=0.4]
\begin{axis}[
/pgf/number format/.cd,
1000 sep={},
xlabel=$n$,
ylabel=$\norm{\bm{C}_{\text{opt}}(n, :)}_{\infty}$,
legend pos=south east,
legend cell align={left},
]
\addplot+[only marks, mark=o, mark size=3.5pt,]
table [x=index, y=norm]{code/fw_C.dat};
\addplot+[ ycomb, mark=square*, mark size=4pt]
table [x=index, y=norm]{code/fw_C_pure_pixel.dat};
\legend{$n \notin \mathcal{K}$, $n \in \mathcal{K}$}
\end{axis}
\end{tikzpicture}
\caption*{\texttt{FW} - Objective value: $2.86{e-}5$}
\end{subfigure}
\begin{subfigure}[t]{0.33\textwidth}
\centering
\begin{tikzpicture}[scale=0.4]
\begin{axis}[
/pgf/number format/.cd,
1000 sep={},
xlabel=iter,
ylabel=$\text{nnz}(\bm{C}$),
legend pos=north east,
legend cell align={left},
y label style={at={(axis description cs:-0.00,.5)},anchor=south},
]
\addplot[orange,style=very thick] table [x=iter, y=PGD]{code/nnz_fw_pgd.dat};
\addplot[blue,style=very thick] table [x=iter, y=FW]{code/nnz_fw_pgd.dat};
\legend{\texttt{APG}, \texttt{FW}}
\end{axis}
\end{tikzpicture}
\caption*{Number of nonzeros (nnz) of $\bm{C}$}
\end{subfigure}
\caption*{\texttt{Accelerated proximal gradient (APG)} vs \texttt{Frank-Wolfe (FW)}. Unlike \texttt{APG}, \texttt{FW} outputs exact $\bm{C}^{\star}$ and keeps $\bm{C}$ sparse during its procedure. $M=10, N=50, K=3$}
\end{figure}
\note{
\begin{itemize}
\item To see how FW can realize our goal, let's start with a noiseless case. Consider the following simple optimization problem. This problem is convex, but could have multiple solutions.
\item As example a solution $\bm{C}^{\star}$. It is a desired solution because by inspecting $\bm{C}^{\star}$, such as examining l1-norm of of the rows, we can expect that the l1-norm is $1$ if the corresponding index is from $\mathcal{K}$, 0 otherwise .
% For $\bm{C}^{\star}$, if an $n \in \mathcal{K}$, then l1 norm of the $n$-th row of $\bm{C}^{\star}$ should be $1$. This perfectly holds in the optimal solution of FW.
\item There are other solutions as well, for example, an identity matrix, but it provides no information about $\mathcal{K}$.
\item To demonstrate FW's magic, we run Accelerated Proximal Gradient, a typical first order method for constrained optimization problem, and compare it with Frank-Wolfe.
\item Firstly, both methods converge and gives a good solutions in terms of objective value.
However, if we look at l1-norm in the optimal solution, only FW reveals perfect $\mathcal{K}$.
\item Secondly, and more interestingly, if we take a look at the density of $\bm{C}$ during the optimization produce, we can see that FW consistently keeps $\bm{C}$ being very sparse, compared to APG. This is the key for memory efficiency when using FW.
\end{itemize}
}
\end{frame}
% \begin{frame}[label=fine]
% \frametitle{Proposal: a Frank-Wolfe Approach}
% \begin{subequations}
% \label{problem:noiseless0}
% \begin{alignat}{2}
% & \minimize_{\bm{C}} \quad && \dfrac{1}{2} \norm{\bm{X} - \bm{X}\bm{C}}_{\rm F}^2 \\
% & \text{\rm subject to} && \bm{C} \geq 0, \bm{1}^{\T}\bm{C} = \bm{1}^{\T}
% \end{alignat}
% \end{subequations}
% \begin{theorem}[Noiseless Case, Memory Efficiency] \label{theorem:noiseless}
% %todo let the no repeated unit vectors a trap for questions
% Suppose that
% \begin{itemize}
% \item $\text{rank}(\bm{W})=K$, no repeated unit vectors exist in $\bm{H}$
% \item The noise is absent (i.e., $\bm{V}=\bm{0}$)
% \item Define $\bm{q}_\ell^t = \bm{W}^{\T}\bm{W}(\bm{H}\bm{c}_\ell^t - \bm{h}_\ell)$. Assume
% \begin{align}\label{eq:nodupq}
% q_{i,\ell}^t - \min_{j} q_{j,\ell}^t \neq 0,~\forall i \neq \argmin_{j} q_{j,\ell}^t,
% \end{align}
% \end{itemize}
% Then, using FW with initialization $\bm{C}^0 = \bm{0}$ to solve \eqref{problem:noiseless0} can reveal exact $\mathcal{K}$ using $O(KN)$ memory.
% \end{theorem}
%
% % \begin{block}{Fact}
% % Condition \eqref{eq:nodupq} is almost guaranteed if $\bm{W}$ is drawn from some joint continuous distribution.
% % \end{block}
%
% \note{
% \begin{itemize}
% \item Interesting result,
% \item Explain 4 assumptions, causally mention the triviality of the last assumption
% \item Emphasize: get the right solution, while using less memory
% \item {\blue Question: is using $\bm{C}=\bm{0}$ valid?}
% \item {\blue check word "almost guarantee", is it the same as $P=1$.}
% \end{itemize}
% }
% \end{frame}
\begin{frame}
\frametitle{Frank-Wolfe (FW) method \citep{frank1956algorithm}}
\begin{itemize}
\item Assume $f(\bm{x})$ is convex and $\mathcal{D}$ is a compact convex constraint
\begin{alignat*}{2}
& \minimize_{\bm{x}} \quad && f(\bm{x}) \\
& \text{\rm subject to} && \bm{x} \in \mathcal{D}
\end{alignat*}
\item FW's standard procedure: at iteration $t$,
\begin{align}
&\bm{s}^{t} \leftarrow \argmin_{\bm{s} \in \mathcal{D}} \; \nabla f(\bm{x}^{t})^{\T} \bm{s} \label{eq:fw_general_update}\\
&\bm{x}^{t+1} \leftarrow \bm{x}^{t} + \alpha^{t} (\bm{s}^{t} - \bm{x}^{t}) , \quad \alpha^{t}=2/(2+t) \nonumber
\end{align}
\item For our problem,
\begin{block}
When $\mathcal{D} = \set{\bm{x} \in \mathbb{R}^{n} \mid \bm{x} \geq 0, \bm{1}^{\T}\bm{x} = 1}$, solving $\eqref{eq:fw_general_update}$ only cost $O(n)$, i.e.,
\[
\bm{s} = \bm{e}_{n^{\star}}, \; n^{\star} = \argmin_{n} [\nabla f(\bm{x}^{t})]_n
\]
\end{block}
% \item FW only update $n^{\star}$-th element of $\bm{x}$ per iteration.
% \item With $\bm{x}^{0} = \bm{0}$, and $n^{\star} \in \mathcal{K}$, then $\text{supp}(\bm{x}^{t})=K$