From 6446b4bc986be1e4ff4bd1914efc4a2e6d0b7d12 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Sun, 1 Feb 2015 09:18:22 -0500
Subject: Lower bound: beginning

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+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.
+
+\begin{theorem}
+    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 $\delta<\frac{1}{2}$
+    (over the measurements):
+    \begin{displaymath}
+        \|\hat{\theta}-\theta^*\|_2\leq
+        C\min_{\|\theta\|_0\leq s}\|\theta-\theta^*\|_2
+    \end{displaymath}
+    where $\theta^*$ is the true paramter of the cascade model. Then $n
+    = \Omega(s\log\frac{m}{s}/\log C)$.
+\end{theorem}
+
+This theorem should be contrasted with Corollary XXX: 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.
+
+Formally, let $\mathcal{F}$ be a family such that:
+\begin{itemize}
+    \item
+    \item
+    \item
+\end{itemize}
+and 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}
+    \item Alice chooses $S$ uniformly at random from $\mathcal{F}$ and
+        $t$ uniformly at random from $X$ subject to $\mathrm{supp}(x) = S$
+    \item Let $w\sim\mathcal{N}(0, \alpha \frac{s}{m}I_m)$ and $\theta
+        = t + w$. Since for all $\theta$, $\mathcal{A}$ recovers $\theta$ with
+        probability $1-\delta$ over the measurements, there exists a fixed set
+        of measurements $x_1,\ldots, x_n$ such that with probability
+        $1-\delta$ over $\theta$, $\mathcal{A}$ recovers $\theta$. Alice sends
+        those measurements to Bob.
+    \item Bob uses $\mathcal{A}$ to recover $\hat{\theta}$ from the
+        measurements, then computes $\hat{t}$ the best $\ell_2$ approximation
+        of $\hat{\theta}$ in $X$.
+\end{itemize}
+
+We have the following from XXX and XXX respectively.
+
+\begin{lemma}
+    Let $S'=\mathrm{supp}(\hat{t})$. If $\alpha = \Omega(\frac{1}{C})$, $I(S, S')
+    = \O(n\log C)$.
+\end{lemma}
+
+\begin{lemma}
+    If $\alpha = \Omega(\frac{1}{C})$, $I(S, S') = \Omega(s\log\frac{m}{s})$
+\end{lemma}
+
+Lemmas XXX and XXX together give Theorem XXX.
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