From 04801a715ac1f72c0fe21dea475674783c92a906 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Thu, 5 Feb 2015 10:52:27 -0500
Subject: small changes

---
 paper/sections/results.tex | 9 +++------
 1 file changed, 3 insertions(+), 6 deletions(-)

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

diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index 9c356c2..805f6b1 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -20,7 +20,7 @@ a set $\mathcal{T}$ of observations. We will write $n\defeq |\mathcal{T}|$.
 \label{sec:main_theorem}
 
 In this section, we analyze the case where $\theta^*$ is exactly sparse. We
-write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. In our context, $S$ is the set of all nodes susceptible to influence our node $i$. In other words, if $\theta^*$ is exactly $s$-sparse, then node $i$ has at most $s$ parents. Our main theorem will rely on the now standard \emph{restricted eigenvalue condition}.
+write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. Recall, that $\theta_i$ is the vector of weights for all edges \emph{directed at} the node we are solving for. In other words, $S$ is the set of all nodes susceptible to influence our node $i$, also referred to as its parents. Our main theorem will rely on the now standard \emph{restricted eigenvalue condition}.
 
 \begin{definition}
     Let $\Sigma\in\mathcal{S}_m(\reals)$ be a real symmetric matrix and $S$ be
@@ -35,11 +35,8 @@ write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. In our context, $S$ is the set
 \end{equation}
 \end{definition}
 
-A discussion of this assumption in the context of generalized linear cascade
-models can be found in Section~\ref{sec:re}.
-
-We will also need the following assumption on the inverse link function $f$ of
-the generalized linear cascade model:
+This condition, well-known in the sparse recovery literature, captures the idea that the binary vectors of active nodes are not colinear with each other, an essentially necessary condition for recovering $\theta$. In fact, the $(S,\gamma)$-{\bf(RE)} assumption is closely linked to the \emph{Fisher Information Matrix}, which captures the amount of information carried by an observable random variable (here the set of active nodes) about an unknown parameter $\theta$ on which the random variable depends.
+A discussion of the $(S,\gamma)$-{\bf(RE)} assumption in the context of generalized linear cascade models can be found in Section~\ref{sec:re}. We will also need the following assumption on the inverse link function $f$ of the generalized linear cascade model:
 \begin{equation}
     \tag{LF}
     \max\left(\left|\frac{f'}{f}(\inprod{\theta^*}{x})\right|,
-- 
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