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.
