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In \cite{Netrapalli:2012}, the authors explicitate a lower bound of
$\Omega(s\log\frac{n}{s})$ on the number of cascades necessary to achieve good
support recovery with constant probability under a \emph{correlation decay}
assumption. In this section, we will consider the stable sparse recovery
setting of Section~\ref{sec:relaxing_sparsity}. 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 \emph{linear} inverse
problems in \cite{pw11, pw12, bipw11}.
\begin{theorem}
\label{thm:lb}
Let us consider a cascade model of the form \eqref{eq:glm} 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{equation}
\label{eq:lb}
\|\hat{\theta}-\theta^*\|_2\leq
C\min_{\|\theta\|_0\leq s}\|\theta-\theta^*\|_2
\end{equation}
where $\theta^*$ is the true parameter of the cascade model. Then $n
= \Omega(s\log\frac{m}{s}/\log C)$.
\end{theorem}
This theorem should be contrasted with Theorem~\ref{thm:approx_sparse}: up to an additive
$s\log s$ factor, the number of measurements required by our algorithm is
tight. The proof of Theorem~\ref{thm:lb} follows an approach similar to \cite{pw12}.
We only present a sketch of the proof here and refer the reader to their paper
for more details.
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 $y$.
\end{itemize}
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.
\begin{comment}
\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}
\label{lemma:lowerinf}
If $\alpha = \Omega(\frac{1}{C})$, $I(S, S') = \Omega(s\log\frac{m}{s})$.
\end{lemma}
Lemmas \ref{lemma:lowerinf} and \ref{lemma:upperinf} together give Theorem
\ref{thm:lb}.
Formally, let $\mathcal{F}\subset \{S\subset [1..m]\,|\, |S|=s\}$ be a family
of $s$-sparse supports such that:
\begin{itemize}
\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}
such a family can be obtained by considering a linear code (see details in
\cite{pw11}. Let .
\end{comment}
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