From 6dcedc33f27f40c23fa7bf456fec09e5f9d0c056 Mon Sep 17 00:00:00 2001
From: Jean Pouget-Abadie <jean.pougetabadie@gmail.com>
Date: Thu, 30 Apr 2015 12:08:57 -0400
Subject: added appendix section

---
 paper/sections/appendix.tex | 49 +++++++++++++++++++++++++++++++--------------
 1 file changed, 34 insertions(+), 15 deletions(-)

(limited to 'paper/sections/appendix.tex')

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{Proposition~\ref{prop:irrepresentability}}
-In the words and notation of Theorem 9.1 in \cite{vandegeer:2009}: 
+\subsection{Proof for different lemmas}
+
+\subsubsection{Bounded gradient}
+\subsubsection{Approximate sparsity proof}
+\subsubsection{RE with high probability}
+
+\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}
-- 
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