From e794a29f1d777e65f778aaf2ff9c764ce5155f3f Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Tue, 12 May 2015 19:46:27 +0200
Subject: more cosmetic changes

---
 paper/sections/appendix.tex | 25 +++++++++++++++++++++++++
 1 file changed, 25 insertions(+)

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

diff --git a/paper/sections/appendix.tex b/paper/sections/appendix.tex
index 79baf2f..64953a3 100644
--- a/paper/sections/appendix.tex
+++ b/paper/sections/appendix.tex
@@ -74,6 +74,31 @@ the proof.
     Proposition~\ref{prop:fi} follows.
 \end{proof}
 
+\subsection{Lower Bound}
+
+Let us consider an algorithm $\mathcal{A}$ which verifies the recovery
+guarantee of Theorem~\ref{thm:lb}: there exists a probability distribution over
+measurements such that for all vectors $\theta^*$, \eqref{eq:lb} holds w.p.
+$\delta$. This implies by the probabilistic method that for all distribution
+$D$ over vectors $\theta$, there exists an $n\times m$ measurement matrix $X_D$
+with such that \eqref{eq:lb} holds w.p. $\delta$ ($\theta$ is now the random
+variable).
+
+Consider the following distribution $D$: choose $S$ uniformly at random from a
+``well-chosen'' set of $s$-sparse supports $\mathcal{F}$ and $t$ uniformly at
+random from
+$X \defeq\big\{t\in\{-1,0,1\}^m\,|\, \mathrm{supp}(t)\in\mathcal{F}\big\}$.
+Define $\theta = t + w$ where
+$w\sim\mathcal{N}(0, \alpha\frac{s}{m}I_m)$ and $\alpha = \Omega(\frac{1}{C})$.
+
+Consider the following communication game between Alice and Bob: \emph{(1)}
+Alice sends $y\in\reals^m$ drawn from a Bernouilli distribution of parameter
+$f(X_D\theta)$ to Bob. \emph{(2)} Bob uses $\mathcal{A}$ to recover
+$\hat{\theta}$ from $y$.  It can be shown that at the end of the game Bob now
+has a quantity of information $\Omega(s\log \frac{m}{s})$ about $S$. By the
+Shannon-Hartley theorem, this information is also upper-bounded by $\O(n\log
+C)$. These two bounds together imply the theorem.
+
 \subsection{Other continuous time processes binned to ours: proportional
 hazards model}
 
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