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Diffstat (limited to 'poster/CS284r_poster')
| -rw-r--r-- | poster/CS284r_poster/beamerthemeconfposter.sty | 184 | ||||
| -rw-r--r-- | poster/CS284r_poster/cracking_cascades_classposter.tex | 461 |
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diff --git a/poster/CS284r_poster/beamerthemeconfposter.sty b/poster/CS284r_poster/beamerthemeconfposter.sty new file mode 100644 index 0000000..bcba6a7 --- /dev/null +++ b/poster/CS284r_poster/beamerthemeconfposter.sty @@ -0,0 +1,184 @@ +%============================================================================== +% Beamer style for the poster template posted at +% www.nathanieljohnston.com/index.php/2009/08/latex-poster-template +% +% Created by the Computational Physics and Biophysics Group at Jacobs University +% https://teamwork.jacobs-university.de:8443/confluence/display/CoPandBiG/LaTeX+Poster +% Modified by Nathaniel Johnston (nathaniel@nathanieljohnston.com) in August 2009 +% ============================================================================= + +\ProvidesPackage{beamerthemeconfposter} +\RequirePackage{tikz} % for drawing the nice rounded boxes +\usetikzlibrary{arrows,backgrounds} +\RequirePackage[T1]{fontenc} +\RequirePackage{lmodern} +\RequirePackage{textcomp} +\RequirePackage{amsmath,amssymb} +\usefonttheme{professionalfonts} +\newcommand{\makeruleinbox}{{\usebeamercolor[bg]{block alerted title}\centering\hspace*{-0.7cm}\rule{\inboxrule}{0.5cm}}} +\usepackage{ragged2e} +\usepackage{wrapfig} +%----------------------------------------------------------- +% Define a whole bunch of custom colours and fonts +%----------------------------------------------------------- + +\definecolor{lgreen} {RGB}{180,210,100} +\definecolor{dblue} {RGB}{20,66,129} +\definecolor{ddblue} {RGB}{11,36,69} +\definecolor{lred} {RGB}{220,0,0} +\definecolor{nred} {RGB}{224,0,0} +\definecolor{norange}{RGB}{244,127,36} +\definecolor{nyellow}{RGB}{255,221,0} +\definecolor{ngreen} {RGB}{98,158,31} +\definecolor{dgreen} {RGB}{78,138,21} +\definecolor{nblue} {RGB}{28,130,185} +\definecolor{jblue} {RGB}{20,50,100} +\definecolor{jalpha} {RGB}{0,0,0} + +% set the basic colors +\setbeamercolor{palette primary} {fg=black,bg=white} +\setbeamercolor{palette secondary} {fg=black,bg=white} +\setbeamercolor{palette tertiary} {bg=jblue,fg=white} +\setbeamercolor{palette quaternary}{fg=black,bg=white} +\setbeamercolor{structure}{fg=jblue} +\setbeamercolor{titlelike} {bg=jblue,fg=white} +\setbeamercolor{frametitle} {bg=jblue!10,fg=jblue} +\setbeamercolor{cboxb}{fg=black,bg=jblue} +\setbeamercolor{cboxr}{fg=black,bg=red} + +% set colors for itemize/enumerate +\setbeamercolor{item}{fg=ngreen} +\setbeamercolor{item projected}{fg=white,bg=ngreen} + +% set colors for blocks +%\setbeamercolor{block title}{fg=ngreen,bg=white} +%\setbeamercolor{block body}{fg=black,bg=white} + +%set colors for alerted blocks (blocks with frame) +\setbeamercolor{block alerted title}{fg=white,bg=jblue} +\setbeamercolor{block alerted body}{fg=black,bg=jblue!10} + +% set the fonts +\setbeamerfont{section in head/foot}{series=\bfseries} +\setbeamerfont{block title}{series=\bfseries} +\setbeamerfont{block alerted title}{series=\bfseries} +\setbeamerfont{frametitle}{series=\bfseries} +\setbeamerfont{frametitle}{size=\Large} + +% set some beamer theme options +\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true] +\setbeamertemplate{sections/subsections in toc}[square] +\setbeamertemplate{items}[circle] +\setbeamertemplate{blocks}[width=0.0] +\beamertemplatenavigationsymbolsempty + +% set bibliography style +\setbeamertemplate{bibliography item}[text] +\setbeamercolor{bibliography item}{fg=black,bg=white} +\setbeamercolor{bibliography entry author}{fg=black,bg=white} +\setbeamercolor{bibliography item}{fg=black,bg=white} + +% define some length variables that are used by the template +\newlength{\inboxwd} +\newlength{\iinboxwd} +\newlength{\inboxrule} +\makeatletter +\makeatother + +%============================================================================== +% build the poster title +%============================================================================== +\setbeamertemplate{headline}{ + \leavevmode + \begin{columns} + \begin{column}{.2\linewidth} + \vskip1cm + \centering + %\includegraphics[height=4in]{UIUC-logo} + \end{column} + \vspace{1cm} + \begin{column}{.6\linewidth} + \vskip1cm + \centering + \usebeamercolor{title in headline}{\color{jblue}\Huge{\textbf{\inserttitle}}\\[0.5ex]} + \usebeamercolor{author in headline}{\color{fg}\Large{\insertauthor}\\[1ex]} + \usebeamercolor{institute in headline}{\color{fg}\large{\insertinstitute}\\[1ex]} + \vskip1cm + \end{column} + \vspace{1cm} + \begin{column}{.2\linewidth} + \vskip1cm + \centering + %\includegraphics[height=4in]{graph500-logo} + \vskip1cm + \end{column} + \vspace{1cm} + \end{columns} + \vspace{0.5in} + \hspace{0.5in}\begin{beamercolorbox}[wd=47in,colsep=0.15cm]{cboxb}\end{beamercolorbox} + \vspace{0.1in} +} + +% Block definition +\setbeamertemplate{block begin} +{ + \par\vskip\medskipamount + \begin{beamercolorbox}[colsep*=0ex,dp={2ex},center]{block title} + \vskip-0.25cm + \usebeamerfont{block title}\large\insertblocktitle + \begin{flushleft} + \vskip-1cm + \begin{tikzpicture}[remember picture,overlay] + \shade [inner color=gray,outer color=white] + (0,0) rectangle (\textwidth,0.3cm); + \end{tikzpicture} + \end{flushleft} + \end{beamercolorbox} + {\parskip0pt\par} + \ifbeamercolorempty[bg]{block title} + {} + {\ifbeamercolorempty[bg]{block body}{}{\nointerlineskip\vskip-0.5pt}} + \usebeamerfont{block body} + \vskip-0.5cm + \begin{beamercolorbox}[colsep*=0ex,vmode]{block body} + \justifying +} + +\setbeamertemplate{block end} +{ + \end{beamercolorbox} + \vskip\smallskipamount +} + +% Alert block definition (with frame) +\setbeamertemplate{block alerted begin} +{ + \par\vskip\medskipamount + \begin{beamercolorbox}[sep=0ex,rounded=true,center,dp={2ex}]{block alerted title} + \vskip0.01cm + \usebeamerfont{block title}\large\insertblocktitle + \end{beamercolorbox} + {\parskip0pt\par} + \usebeamerfont{block body} + \vskip-0.8cm + \begin{beamercolorbox}[sep=0.5cm, rounded=true,center]{block alerted title} + \setlength{\inboxwd}{\linewidth} + \addtolength{\inboxwd}{-1cm} + \begin{beamercolorbox}[rounded=true,wd={\inboxwd},center]{block alerted body} + \setlength{\iinboxwd}{\inboxwd} + \setlength{\inboxrule}{\inboxwd} + \addtolength{\iinboxwd}{-0.5cm} + \addtolength{\inboxrule}{0.5cm} + \begin{center} + \begin{minipage}{\iinboxwd} + \justifying +} + +\setbeamertemplate{block alerted end} +{ + \end{minipage} + \end{center} + \end{beamercolorbox} + \end{beamercolorbox} + \vskip\smallskipamount +}
\ No newline at end of file diff --git a/poster/CS284r_poster/cracking_cascades_classposter.tex b/poster/CS284r_poster/cracking_cascades_classposter.tex new file mode 100644 index 0000000..3f8204e --- /dev/null +++ b/poster/CS284r_poster/cracking_cascades_classposter.tex @@ -0,0 +1,461 @@ +\documentclass[final]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[scale=1.6]{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}{Problem Statement} +\begin{center} +\bf{How can we reconstruct a graph on which observed cascades spread?} +\end{center} +\end{block} + + +%{\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. + + +%--------------------------------------------------------------------------------- +%--------------------------------------------------------------------------------- + +\begin{block}{Voter Model} + +\begin{figure} +\centering +\includegraphics[width=0.6\textwidth]{images/voter.png} +\end{figure} + + + +\vspace{0.5 cm} +{\bf Description} + +\vspace{0.5 cm} + +\begin{itemize} +\item $\mathbb{P}$({\color{blue} blue} at $t=0) = p_{\text{init}}$ +\item $\mathbb{P}$({\color{blue} blue} at $t+1) = \frac{\text{Number of {\color{blue}blue} neighbors}}{\text{Total number of neighbors}}$ +\end{itemize} + +\vspace{0.5 cm} + +{\bf Sparse Recovery Formulation} + +\vspace{0.5 cm} + + +To recover the neighbors of $v_1$, observe which nodes are {\color{red} red} (1) or {\color{blue} blue} (0) at time step $t$: +\begin{align*} +&v_2 \hspace{0.2 cm} v_3 \hspace{0.2 cm} v_4 \hspace{0.2 cm} v_5 &\\ +\vspace{1 cm} +M = & \left( \begin{array}{cccc} +0 & 0 & 1 & 1 \\ +1 & 1 & 0 & 0 \\ +\end{array} \right) & \begin{array}{l} \hspace{ - 5cm} +\text{time step 0} \\ + \hspace{ - 5cm} \text{time step 1} \\ +\end{array} +\end{align*} + +and which color $v_1$ is at time step $t+1$ due to $M$: + +\begin{align*} +b_1 = & \left( \begin{array}{c} +1 \\ +1 \\ +\end{array} \right) & \begin{array}{l} \hspace{ - 5cm} +\text{time step 1} \\ + \hspace{ - 5cm} \text{time step 2} \\ + \end{array} +\end{align*} + +Then , + +\begin{equation} +\boxed{M \vec x_1 = \vec b_1 + \epsilon \nonumber} +\end{equation} + +where $(\vec x_{1})_j := \frac{\text{1}}{\text{deg}(i)} \cdot \left[\text{j parent of 1 in }{\cal G}\right] $ + + +\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}{Independent Cascades Model} +\begin{figure} +\centering +\includegraphics[width=0.6\textwidth]{images/icc.png} +\end{figure} + +\vspace{0.5 cm} + +{\bf Description} + +\vspace{0.5 cm} + +\begin{itemize} +\item Three possible states: {\color{blue} susceptible}, {\color{red} infected}, {\color{yellow} inactive } +\item $\mathbb{P}$(infected at t=0)$=p_{\text{init}}$ +\item Infected node $j$ infects its susceptible neighbors $i$ with probability $p_{j,i}$ independently +\end{itemize} + +\vspace{0.5 cm} + +{\bf Sparse Recovery Formulation} + +\vspace{0.5 cm} + +To recover the neighbors of $v_5$,observe which nodes are {\color{red} red} (1), {\color{blue} blue} (0), or {\color{yellow} yellow} (0) at time step $t$: +\begin{align*} +&v_1 \hspace{0.2 cm} v_2 \hspace{0.2 cm} v_3 \hspace{0.2 cm} v_4 \\ +\vspace{1 cm} +M = & \left( \begin{array}{cccc} +1 & 0 & 0 & 0 \\ +0 & 1 & 1 & 0 \\ +\end{array} \right) \begin{array}{l} \hspace{ 1cm} +\text{time step 0} \\ + \hspace{ 1cm} \text{time step 1} \\ + \end{array} +\end{align*} + +and if $M$ caused $v_5$ to be infected at time step $t+1$: + +\begin{align*} +b_5 = & \left( \begin{array}{c} +0 \\ +1 \\ +\end{array} \right) \begin{array}{l} \hspace{ 1cm} +\text{time step 1} \\ + \hspace{ 1cm} \text{time step 2} \\ + \end{array} +\end{align*} + + +Then, + +\begin{equation} +\boxed{e^{M \vec \theta_5} = (1 - \vec b_5) + \epsilon} \nonumber +\end{equation} + +where $(\vec \theta_5)_j := \log ( 1 - p_{j,5}) $ + +\vspace{1 cm} + + +This problem is a {\bf Noisy Sparse Recovery} problem, which has been studied extensively. Here, the vectors $\vec x_i$ are deg(i)-sparse. + + +\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}{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} - (1 - \vec b_i) \|_2 \nonumber +\end{equation} +\end{itemize} + +\end{block} + +\begin{block}{Restricted Isometry Property (RIP)} +{\bf Definition} +\begin{itemize} +\item Characterizes a quasi-orthonormality of M on sparse vectors. + +\item The RIP constant $\delta_k$ is 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 The smaller $\delta_k$ is, the better we can recover $k$-sparse vectors $x$. +\end{itemize} + + + + +\end{block} + +\begin{block}{Theoretical Guarantees} + +With small RIP constants $(\delta \leq 0.25)$ for $M$ and some assumption on the noise $\epsilon$: + +{\bf Theorem \cite{candes}} + +If node $i$ has degree $\Delta$ and $n_{\text{rows}}(M) \geq C_1 \mu \Delta \log n$, then, w.h.p., + +$$\| \hat x - x^* \|_2 \leq C (1 + \log^{3/2}(n))\sqrt{\frac{\Delta \log n}{n_{\text{rows}}(M) }}$$ + + +\end{block} + + + + +%---------------------------------------------------------------------------------------- +% RESULTS +%---------------------------------------------------------------------------------------- + +\begin{block}{RIP Experiments} + +\begin{center} +\begin{table} +\begin{tabular}{c | c | c | c | c } +& $c$ = 100, &$c$ = 1000,& $c$ = 100, &$c$ = 1000,\\ +& $i$ = 0.1& $i$ = 0.1& $i$ = 0.05& $i$ = 0.05\\ + \hline + $\delta_4$ & 0.54 & 0.37 &0.43&0.23 \\ + \end{tabular} + \caption{RIP constant for a small graph. Here, $c$ is the number of cascades and $i$ is $p_{\text{init}}$.} +\end{table} +\end{center} + + +\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} + +\begin{center} + + +{\bf Graph reconstruction can naturally be expressed as Sparse Recovery. Understanding properties of $M$, for example RIP, leads to theoretical guarantees on the reconstruction.} + +\end{center} + +\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} |