From 9c682e64d18c9cddfa2575adc68c57693862b0f5 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Thu, 4 Dec 2014 17:17:04 -0500
Subject: yaron poster

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+\documentclass[final]{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage[scale=1.41]{beamerposter} % Use the beamerposter package for laying out the poster
+
+\usetheme{confposter} % Use the confposter theme supplied with this template
+
+\usepackage{color, bbm}
+\setbeamercolor{block title}{fg=dblue,bg=white} % Colors of the block titles
+\setbeamercolor{block body}{fg=black,bg=white} % Colors of the body of blocks
+\setbeamercolor{block alerted title}{fg=white,bg=dblue!70} % Colors of the highlighted block titles
+\setbeamercolor{block alerted body}{fg=black,bg=dblue!10} % Colors of the body of highlighted blocks
+% Many more colors are available for use in beamerthemeconfposter.sty
+
+%-----------------------------------------------------------
+% Define the column widths and overall poster size
+% To set effective sepwid, onecolwid and twocolwid values, first choose how many columns you want and how much separation you want between columns
+% In this template, the separation width chosen is 0.024 of the paper width and a 4-column layout
+% onecolwid should therefore be (1-(# of columns+1)*sepwid)/# of columns e.g. (1-(4+1)*0.024)/4 = 0.22
+% Set twocolwid to be (2*onecolwid)+sepwid = 0.464
+% Set threecolwid to be (3*onecolwid)+2*sepwid = 0.708
+
+\newlength{\sepwid}
+\newlength{\onecolwid}
+\newlength{\twocolwid}
+\newlength{\threecolwid}
+\setlength{\paperwidth}{48in} % A0 width: 46.8in
+\setlength{\paperheight}{40in} % A0 height: 33.1in
+\setlength{\sepwid}{0.024\paperwidth} % Separation width (white space) between columns
+\setlength{\onecolwid}{0.22\paperwidth} % Width of one column
+\setlength{\twocolwid}{0.464\paperwidth} % Width of two columns
+\setlength{\threecolwid}{0.708\paperwidth} % Width of three columns
+\setlength{\topmargin}{-1in} % Reduce the top margin size
+%-----------------------------------------------------------
+
+\usepackage{graphicx}  % Required for including images
+
+\usepackage{booktabs} % Top and bottom rules for tables
+
+
+
+%----------------------------------------------------------------------------------------
+%	TITLE SECTION 
+%----------------------------------------------------------------------------------------
+
+\title{Sparse Recovery for Graph Reconstruction } % Poster title
+
+\author{Eric Balkanski, Jean Pouget-Abadie} % Author(s)
+
+\institute{Harvard University} % Institution(s)
+%----------------------------------------------------------------------------------------
+\begin{document}
+\addtobeamertemplate{block end}{}{\vspace*{2ex}} % White space under blocks
+\addtobeamertemplate{block alerted end}{}{\vspace*{2ex}} % White space under highlighted (alert) blocks
+
+\setlength{\belowcaptionskip}{2ex} % White space under figures
+\setlength\belowdisplayshortskip{2ex} % White space under equations
+
+\begin{frame}[t] % The whole poster is enclosed in one beamer frame
+
+\begin{columns}[t] % The whole poster consists of three major columns, the second of which is split into two columns twice - the [t] option aligns each column's content to the top
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The first column
+
+%----------------------------------------------------------------------------------------
+%	INTODUCTION
+%----------------------------------------------------------------------------------------
+
+
+%\vspace{- 15.2 cm}
+%\begin{center}
+%{\includegraphics[height=7em]{logo.png}} % First university/lab logo on the left
+%\end{center}
+
+%\vspace{4.6 cm}
+
+\begin{block}{Introduction}
+
+{\bf Graph Reconstruction}:
+
+\begin{itemize}
+\item \{${\cal G}, \vec p$\}: directed graph, edge probabilities
+\item $F$: cascade generating model
+\item ${\cal M} := F\{{\cal G}, \vec p\}$: cascade
+\end{itemize}
+
+{\bf Objective}:
+\begin{itemize}
+\item Find algorithm which computes $F^{-1}({\cal M}) = \{{\cal G}, \vec p\}$ w.h.p., i.e. recovers graph from cascades.
+\end{itemize}
+
+{\bf Approach}
+\begin{itemize}
+\item Frame graph reconstruction as a {\it Sparse Recovery} problem for two cascade generating models.
+\end{itemize}
+
+%Given a set of observed cascades, the \textbf{graph reconstruction problem} consists of finding the underlying graph on which these cascades spread. We assume that these cascades come from the classical \textbf{Independent Cascade Model} where at each time step, newly infected nodes infect each of their neighbor with some probability.
+
+%In previous work, this problem has been formulated in different ways, including a convex optimization and a maximum likelihood problem. However, there is no known algorithm for graph reconstruction with theoretical guarantees and with a reasonable required sample size.
+
+%We formulate a novel approach to this problem in which we use \textbf{Sparse Recovery} to find the edges in the unknown underlying network. Sparse Recovery is the problem of finding the sparsest vector $x$ such that $\mathbf{M x =b}$. In our case, for each node $i$, we wish to recover the vector $x = p_i$ where $p_{i_j}$ is the probability that node $j$ infects node $i$ if $j$ is active. To recover this vector, we are given $M$, where row $M_{t,k}$ indicates which nodes are infected at time $t$ in observed cascade $k$,  and $b$, where $b_{t+1,k}$ indicates if node $i$ is infected at time $t+1$ in cascade $k$. Since most nodes have a small number of neighbors in large networks, we can assume that these vectors are sparse. Sparse Recovery is a well studied problem which can be solved efficiently and with small error if $M$ satisfies certain properties. In this project, we empirically study to what extent $M$ satisfies the Restricted Isometry Property.
+
+\end{block}
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+\begin{block}{Voter Model}
+{\bf Description}
+
+\begin{itemize}
+\item Two types of nodes, {\color{red} red} and {\color{blue} blue}.
+\item $\mathbb{P}$(node is {\color{blue} blue} at $t=0) = p_{\text{init}}$
+\item $\mathbb{P}$(node is  {\color{blue} blue} at t+1) =  $\frac{\text{Number of {\color{blue}blue} neighbors}}{\text{Total number of neighbors}}$
+\item Observe until horizon T
+\end{itemize}
+
+{\bf Sparse Recovery Formulation}
+\begin{itemize}
+\item Definitions:
+\end{itemize}
+\begin{align}
+(\vec m^t_k)_j &:= \left[\text{node j is  {\color{blue} blue} at t in cascade k}\right] \nonumber \\
+(\vec x_{i})_j &:= \frac{\text{1}}{\text{deg}(i)} \cdot \left[\text{j parent of i in }{\cal G}\right] \nonumber \\
+b^t_{k,i} &:= \text{node i is {\color{blue} blue} at t+1 in cascade k} \nonumber 
+\end{align}
+
+\begin{itemize}
+\item Observation:
+\end{itemize}
+\begin{align}
+\langle \vec m^t_k, \vec x_i \rangle &= \mathbb{P}(\text{node i is {\color{blue} blue} at t+1 in cascade k}) \nonumber \\
+&=: w^{t+1}_{k,i} \nonumber
+\end{align}
+
+\begin{itemize}
+\item We observe $M := \text{vstack}(\vec m^t_k)$
+\item We observe $b_i := \text{vstack}(b^t_{k,i})$ where $$b^t_{k,i} \sim {\cal B}(w^t_{k,i}) = w^t_{k,i} - \epsilon$$
+\item For each node i, solve for $\vec x_i$:
+\begin{equation}
+\boxed{M \vec x_i = \vec b_i + \epsilon \nonumber}
+\end{equation}
+\end{itemize}
+\end{block}
+
+
+
+
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+\begin{block}{Independent Cascade Model}
+
+{\bf Description}
+\begin{itemize}
+\item Three possible states: susceptible, infected, inactive
+\item Infected node $j$ infects its susceptible neighbors $i$ with probability $p_{j,i}$ independently
+\item $\mathbb{P}$(inactive at t+1| infected at t) = 1
+\item $\mathbb{P}$(infected at t=0)$=p_{\text{init}}$
+\end{itemize}
+
+
+{\bf Sparse Recovery Formulation}
+\begin{itemize}
+
+\item Definitions:
+\begin{align}
+(\vec \theta_i)_j &:= \log ( 1 - p_{j,i}) \nonumber \\
+(\vec m^t_k)_j &= \left[\text{node j is infected at t in cascade k}\right] \nonumber
+\end{align}
+
+\item Observation:
+\begin{equation}
+\langle \vec m^t_k, \vec x_i \rangle = \log \mathbb{P}(\text{node i {\it not} infect. at t+1 in casc. k}) \nonumber
+\end{equation}
+\item With same notations, we solve for each node i:
+\begin{equation}
+\boxed{e^{M \vec \theta_i} = (1 - \vec b_i) + \epsilon} \nonumber
+\end{equation}
+where: $e^{M \vec \theta_i}$ is taken element-wise
+\end{itemize}
+\end{block}
+
+
+
+
+
+%---------------------------------------------------------------------------------
+%---------------------------------------------------------------------------------
+
+
+
+\end{column} % End of the first column
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The first column
+
+%----------------------------------------------------------------------------------------
+%	CONSTRAINT SATISFACTION - BACKTRACKING
+%----------------------------------------------------------------------------------------
+\begin{block}{Example}
+
+\begin{figure}
+\centering
+\includegraphics[width=0.6\textwidth]{cascades.png}
+\caption{Two cascades with two time steps each. Red nodes are the infected and active nodes.}
+\end{figure}
+
+Figure 1 illustrates two cascades. In this case, if we wish to recover $p_A$, then our problem would be is given by the formula
+\[ \left( \begin{array}{cccc}
+1 & 0 & 0 & 0 \\
+0 & 1 & 0 & 0 \\
+\end{array} \right) x_A =  \left( \begin{array}{c}
+0 \\
+1 \\\end{array} \right)\]
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+
+\begin{block}{Motivation}
+\begin{itemize}
+\item Vectors $\vec x_i$ are deg(i)-sparse.
+\item Matrix $M$ is well-conditioned (see experimental section)
+\item Voter model: quasi-textbook noisy sparse recovery
+\item Independent cascade: if $p_{j,i} < 1 - \eta$, exponential can be approximated by affine function
+\end{itemize}
+\end{block}
+
+\begin{block}{Algorithms}
+
+{\bf Voter Model}
+
+\begin{itemize}
+\item Solve for each node i:
+\begin{equation}
+\min_{\vec x_i} \|\vec x_i\|_1 + \lambda \|M \vec x_i - \vec b_i \|_2 \nonumber
+\end{equation}
+\end{itemize}
+
+{\bf Independent Cascade Model}
+
+\begin{itemize}
+\item Solve for each node i:
+\begin{equation}
+\min_{\vec \theta_i} \|\vec \theta_i\|_1 + \lambda \|e^{M \vec \theta_i} - \vec b_i \|_2 \nonumber
+\end{equation}
+where: $e^{M \vec \theta_i}$ is taken element-wise
+\end{itemize}
+
+\end{block}
+
+
+
+
+
+%----------------------------------------------------------------------------------------
+%	MIP
+%----------------------------------------------------------------------------------------
+
+% \begin{block}{RIP property}
+
+% %The Restricted Isometry Property (RIP) characterizes a quasi-orthonormality of the measurement matrix M on sparse vectors. 
+
+% For all k, we define $\delta_k$ as the best possible constant such that for all unit-normed ($l$2) and k-sparse vectors $x$
+
+% \begin{equation}
+% 1-\delta_k \leq \|Mx\|^2_2 \leq 1 + \delta_k
+% \end{equation}
+
+% In general, the smaller $\delta_k$ is, the better we can recover $k$-sparse vectors $x$!
+ 
+% \end{block}
+
+%----------------------------------------------------------------------------------------
+
+\end{column} 
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The first column within column 2 (column 2.1)
+
+
+%----------------------------------------------------------------------------------------
+
+
+\begin{block}{Theoretical Guarantees}
+
+{\bf Assumptions}
+\begin{itemize}
+\item Suppose that each row of $M$ is taken from a distribution $F$, such that
+\begin{itemize}
+	\item $\mathbb{E}_{a \in F}(a a^T) = I_n$
+	\item $\|a\|_{\infty} \leq \mu$
+\end{itemize}
+\item Suppose that w.h.p $\| M^T \epsilon \|_{\infty} \leq 2.5 \sqrt{\log n} $
+\end{itemize}
+
+{\bf Theorem \cite{candes}}
+
+If node $i$ has degree $\Delta$  and $n_{\text{rows}}(M) \geq C \mu \Delta \log n$, then, w.h.p.,
+
+$$\| \hat x - x^* \|_2 \leq C (1 + \log^{3/2}(n))\sqrt{\frac{\Delta \log n}{m}}$$
+
+{\bf Discussion}
+
+\begin{itemize}
+\item If we consider $M/p_{init}$, then $\mathbb{E}_{a \in F}(a a^T) \approx I_n$
+\item $\mathbb{E}(\epsilon) = 0$, hence $\| M^T \epsilon \|_{\infty} \leq 2.5 \sqrt{\log n}$ should hold w.h.p
+\end{itemize}
+
+\end{block}
+
+
+\begin{block}{Restriced Isometry Property (RIP)}
+{\bf Definition}
+\begin{itemize}
+\item The Restricted Isometry Property (RIP) characterizes a quasi-orthonormality of the measurement matrix M on sparse vectors. 
+
+\item For all k, we define $\delta_k$ as the best possible constant such that for all unit-normed ($l$2) and k-sparse vectors $x$:
+
+\begin{equation}
+1-\delta_k \leq \|Mx\|^2_2 \leq 1 + \delta_k \nonumber
+\end{equation}
+
+\item In general, the smaller $\delta_k$ is, the better we can recover $k$-sparse vectors $x$!
+\end{itemize}
+
+{\bf Relevance}
+\begin{itemize}
+\item Commonly studied in stable sparse recovery
+\item $\mathbb{E}_{a \in F}(a a^T) \approx I_n$ + RIP property (with $\delta = \frac{1}{4}$) should get similar conclusion as above
+\end{itemize}
+
+
+
+\end{block}
+
+
+
+%----------------------------------------------------------------------------------------
+%	RESULTS
+%----------------------------------------------------------------------------------------
+
+\begin{block}{Description of Experiments}
+
+
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+
+
+\end{column} % End of the second column
+
+\begin{column}{\sepwid}\end{column} % Empty spacer column
+
+\begin{column}{\onecolwid} % The third column
+
+%----------------------------------------------------------------------------------------
+%	IOVERALL COMPARISON
+%----------------------------------------------------------------------------------------
+
+%\vspace{- 14.2 cm}
+%\begin{center}
+%{\includegraphics[height=7em]{cmu_logo.png}} % First university/lab logo on the left
+%\end{center}
+
+%\vspace{4 cm}
+
+\begin{alertblock}{Experimental Results}
+
+
+
+
+\end{alertblock} 
+
+%----------------------------------------------------------------------------------------
+
+
+%----------------------------------------------------------------------------------------
+%	CONCLUSION
+%----------------------------------------------------------------------------------------
+
+\begin{block}{Conclusion}
+
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+%	REFERENCES
+%----------------------------------------------------------------------------------------
+
+\begin{block}{References}
+
+\begin{thebibliography}{42}
+
+\bibitem{candes}
+Candès, E., and Plan, Y.
+\newblock {\it A Probabilistic and RIPless Theory of Compressed Sensing}
+\newblock Information Theory, IEEE Transactions on, 57(11): 7235--7254,
+\newblock 2011.
+\end{thebibliography}
+
+\end{block}
+
+%----------------------------------------------------------------------------------------
+
+\end{column} % End of the third column
+
+\end{columns} % End of all the columns in the poster
+
+\end{frame} % End of the enclosing frame
+
+
+
+\end{document}
-- 
cgit v1.2.3-70-g09d2

