added appendix section

2 files changed, 44 insertions, 23 deletions

diff --git a/paper/paper.tex b/paper/paper.tex
index ce8cde9..5311bec 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -12,7 +12,7 @@
 % For figures
 \usepackage{graphicx} % more modern
 %\usepackage{epsfig} % less modern
-\usepackage{subfigure} 
+\usepackage{subfigure}
 
 % For citations
 \usepackage{natbib}
@@ -34,7 +34,7 @@
 % Employ the following version of the ``usepackage'' statement for
 % submitting the draft version of the paper for review.  This will set
 % the note in the first column to ``Under review.  Do not distribute.''
-\usepackage{icml2015} 
+\usepackage{icml2015}
 
 % Employ this version of the ``usepackage'' statement after the paper has
 % been accepted, when creating the final version.  This will set the
@@ -72,7 +72,7 @@
 \newtheorem{proposition}{Proposition}
 \newtheorem{definition}{Definition}
 
-\begin{document} 
+\begin{document}
 
 \twocolumn[
 \icmltitle{Inferring Graphs from Cascades: A Sparse Recovery Framework}
@@ -88,17 +88,17 @@
 \icmladdress{Their Fantastic Institute,
             27182 Exp St., Toronto, ON M6H 2T1 CANADA}
 
-% You may provide any keywords that you 
-% find helpful for describing your paper; these are used to populate 
+% You may provide any keywords that you
+% find helpful for describing your paper; these are used to populate
 % the "keywords" metadata in the PDF but will not be shown in the document
 \icmlkeywords{boring formatting information, machine learning, ICML}
 
 \vskip 0.3in
 ]
 
-\begin{abstract} 
+\begin{abstract}
 \input{sections/abstract.tex}
-\end{abstract} 
+\end{abstract}
 
 \section{Introduction}
 \input{sections/intro.tex}
@@ -131,12 +131,14 @@
 \label{sec:linear_threshold}
 \input{sections/discussion.tex}
 
-%\section{Appendix}
-%\input{sections/appendix.tex}
+
 
 \newpage
 \bibliography{sparse}
 \bibliographystyle{icml2015}
 
-\end{document} 
+\newpage
+\section{Appendix}
+\input{sections/appendix.tex}
 
+\end{document}
diff --git a/paper/sections/appendix.tex b/paper/sections/appendix.tex
index 2f0162e..ff4c5dd 100644
--- a/paper/sections/appendix.tex
+++ b/paper/sections/appendix.tex
@@ -1,20 +1,39 @@
-\begin{comment}
-\begin{multline*}
-    \nabla^2 \mathcal{L}(\theta) =
-    \frac{1}{|\mathcal{T}|}\sum_{t\in \mathcal{T}}x^t(x^t)^T\bigg[
-        x_i^{t+1}\frac{f''f - f'^2}{f^2}(\inprod{\theta_i}{x^t})\\
-    - (1-x_i^{t+1})\frac{f''(1-f) + f'^2}{(1-f)^2}(\inprod{\theta_i}{x^t})\bigg]
-\end{multline*}
-\end{comment}
+\subsection{Proof for different lemmas}
 
+\subsubsection{Bounded gradient}
+\subsubsection{Approximate sparsity proof}
+\subsubsection{RE with high probability}
 
-\subsection{Proposition~\ref{prop:irrepresentability}}
-In the words and notation of Theorem 9.1 in \cite{vandegeer:2009}: 
+\subsection{Other continuous time processes binned to ours: prop. hazards model}
+
+\subsection{Irrepresentability vs. Restricted Eigenvalue Condition}
+In the words and notation of Theorem 9.1 in \cite{vandegeer:2009}:
 \begin{lemma}
 \label{lemm:irrepresentability_proof}
-Let $\phi^2_{\text{compatible}}(L,S) \defeq \min \{ \frac{s \|f_\beta\|^2_2}{\|\beta_S\|^2_1} \ : \ \beta \in {\cal R}(L, S) \}$, where $\|f_\beta\|^2_2 \defeq \{ \beta^T \Sigma \beta \}$ and ${\cal R}(L,S) \defeq \{\beta : \|\beta_{S^c}\|_1 \leq L \|\beta_S\|_1 \neq 0\}$. If $\nu_{\text{irrepresentable}(S,s)} < 1/L$, then $\phi^2_{\text{compatible}}(L,S) \geq (1 - L \nu_{\text{irrepresentable}(S,s)})^2 \lambda_{\min}^2$. 
+Let $\phi^2_{\text{compatible}}(L,S) \defeq \min \{ \frac{s
+\|f_\beta\|^2_2}{\|\beta_S\|^2_1} \ : \ \beta \in {\cal R}(L, S) \}$, where
+$\|f_\beta\|^2_2 \defeq \{ \beta^T \Sigma \beta \}$ and ${\cal R}(L,S) \defeq
+\{\beta : \|\beta_{S^c}\|_1 \leq L \|\beta_S\|_1 \neq 0\}$. If
+$\nu_{\text{irrepresentable}(S,s)} < 1/L$, then $\phi^2_{\text{compatible}}(L,S)
+\geq (1 - L \nu_{\text{irrepresentable}(S,s)})^2 \lambda_{\min}^2$.
 \end{lemma}
 
-Since ${\cal R}(3, S) = {\cal C}$, $\|\beta_S\|_1 \geq \|\beta_S\|_2$, and $\|\beta_S\|_1 \geq \frac{1}{3} \|\beta_{S^c}\|_1$ it is easy to see that $\|\beta_S\|_1 \geq \frac{1}{4} \|\beta\|_2$ and therefore that: $\gamma_n \geq \frac{n}{4s}\phi^2_{\text{compatible}}(3,S)$
+Since ${\cal R}(3, S) = {\cal C}$, $\|\beta_S\|_1 \geq \|\beta_S\|_2$, and
+$\|\beta_S\|_1 \geq \frac{1}{3} \|\beta_{S^c}\|_1$ it is easy to see that
+$\|\beta_S\|_1 \geq \frac{1}{4} \|\beta\|_2$ and therefore that: $\gamma_n \geq
+\frac{n}{4s}\phi^2_{\text{compatible}}(3,S)$
+
+Consequently, if $\epsilon > \frac{2}{3}$, then
+$\nu_{\text{irrepresentable}(S,s)} < 1/3$ and the conditions of
+Lemma~\ref{lemm:irrepresentability_proof} hold.
+
+
+
+\subsection{Lower bound for restricted eigenvalues (expected hessian) for
+different graphs}
+
+\subsection{Better asymptotic w.r.t expected hessian}
+
+\subsection{Confidence intervals?}
 
-Consequently, if $\epsilon > \frac{2}{3}$, then $\nu_{\text{irrepresentable}(S,s)} < 1/3$ and the conditions of Lemma~\ref{lemm:irrepresentability_proof} hold.
+\subsection{Active learning}