From c37c4ea9e94862902d294d74e48b8ff0072d1ca6 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Sun, 1 Feb 2015 21:11:14 -0500
Subject: Completing Lower Bound section

---
 paper/sections/lowerbound.tex | 47 ++++++++++++++++++++++++++-----------------
 1 file changed, 28 insertions(+), 19 deletions(-)

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

diff --git a/paper/sections/lowerbound.tex b/paper/sections/lowerbound.tex
index bbd8566..fdee069 100644
--- a/paper/sections/lowerbound.tex
+++ b/paper/sections/lowerbound.tex
@@ -1,10 +1,12 @@
-In this section, we will consider the stable sparse recovery of section XXX.
-Our goal is to obtain an information-theoretic lower bound on the number of
-measurements necessary to approximately recover the parameter $\theta$ of
-a cascade model from observed cascades. Similar lower bounds were obtained for
-sparse linear inverse problems in XXXs.
+In this section, we will consider the stable sparse recovery setting of
+Section~\ref{sec:relaxing}.  Our goal is to obtain an information-theoretic
+lower bound on the number of measurements necessary to approximately recover
+the parameter $\theta$ of a cascade model from observed cascades. Similar lower
+bounds were obtained for sparse linear inverse problems in \cite{pw11, pw12,
+bipw11}
 
 \begin{theorem}
+    \label{thm:lb}
     Let us consider a cascade model of the form XXX and a recovery algorithm
     $\mathcal{A}$ which takes as input $n$ random cascade measurements and
     outputs $\hat{\theta}$ such that with probability $\delta<\frac{1}{2}$
@@ -17,23 +19,27 @@ sparse linear inverse problems in XXXs.
     = \Omega(s\log\frac{m}{s}/\log C)$.
 \end{theorem}
 
-This theorem should be contrasted with Corollary XXX: up to an additive
+This theorem should be contrasted with Corollary~\ref{cor:relaxing}: up to an additive
 $s\log s$ factor, the number of measurements required by our algorithm is
 tight.
 
-The proof of Theorem XXX follows an approach similar to XXX. Let us consider an
-algorithm $\mathcal{A}$ as in the theorem. Intuitively, $\mathcal{A}$ allows
-Alice and Bob to send $\Omega(s\log\frac{m}{s})$ quantity of information over
-a Gaussian channel. By the Shannon-Hartley theorem, this quantity of
-information is $O(n\log C)$.  These two bounds together give the theorem.
+The proof of Theorem~\ref{thm:lb} follows an approach similar to \cite{pw12}.
+Let us consider an algorithm $\mathcal{A}$ as in the theorem. Intuitively,
+$\mathcal{A}$ allows Alice and Bob to send $\Omega(s\log\frac{m}{s})$ quantity
+of information over a Gaussian channel. By the Shannon-Hartley theorem, this
+quantity of information is $O(n\log C)$.  These two bounds together give the
+theorem.
 
-Formally, let $\mathcal{F}$ be a family such that:
+Formally, let $\mathcal{F}\subset \{S\subset [1..m]\,|\, |S|=s\}$ be a family
+of $s$-sparse supports such that:
 \begin{itemize}
-    \item
-    \item
-    \item
+    \item $|S\Delta S'|\geq s$ for $S\neq S'\in\mathcal{F}$ .
+    \item $\P_{S\in\mathcal{F}}[i\in S]=\frac{s}{m}$ for all $i\in [1..m]$ and
+        when $S$ is chosen uniformly at random in $\mathcal{F}$
+    \item $\log|\mathcal{F}| = \Omega(s\log\frac{m}{s})$
 \end{itemize}
-and let $X = \big\{t\in\{-1,0,1\}^m\,|\, \mathrm{supp}(t)\in\mathcal{F}\big\}$. 
+such a family can be obtained by considering a linear code (see details in
+\cite{pw11}. Let $X = \big\{t\in\{-1,0,1\}^m\,|\, \mathrm{supp}(t)\in\mathcal{F}\big\}$. 
 
 Consider the following communication game between Alice and Bob.
 \begin{itemize}
@@ -50,15 +56,18 @@ Consider the following communication game between Alice and Bob.
         of $\hat{\theta}$ in $X$.
 \end{itemize}
 
-We have the following from XXX and XXX respectively.
+We have the following from \cite{pw11} and \cite{pw12} respectively.
 
 \begin{lemma}
+    \label{lemma:upperinf}
     Let $S'=\mathrm{supp}(\hat{t})$. If $\alpha = \Omega(\frac{1}{C})$, $I(S, S')
     = \O(n\log C)$.
 \end{lemma}
 
 \begin{lemma}
-    If $\alpha = \Omega(\frac{1}{C})$, $I(S, S') = \Omega(s\log\frac{m}{s})$
+    \label{lemma:lowerinf}
+    If $\alpha = \Omega(\frac{1}{C})$, $I(S, S') = \Omega(s\log\frac{m}{s})$.
 \end{lemma}
 
-Lemmas XXX and XXX together give Theorem XXX.
+Lemmas \ref{lemma:lowerinf} and \ref{lemma:upperinf} together give Theorem
+\ref{thm:lb}.
-- 
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