From 7fd038c7b0f00f6db411e9c5037133fd09509a8d Mon Sep 17 00:00:00 2001
From: Jean Pouget-Abadie <jean.pougetabadie@gmail.com>
Date: Wed, 29 Apr 2015 20:58:17 -0400
Subject: small reviewer updates

---
 paper/sections/results.tex | 39 +++++++++++++++++++++++----------------
 1 file changed, 23 insertions(+), 16 deletions(-)

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

diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index f3c5366..b156897 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -20,7 +20,12 @@ 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 $s=|S|$. 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 node $i$, also referred to as its 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 node $i$,
+also referred to as its parents. Our main theorem will rely on the now standard
+\emph{restricted eigenvalue condition}.
+{\color{red} Add reference}
 
 \begin{definition}
     Let $\Sigma\in\mathcal{S}_m(\reals)$ be a real symmetric matrix and $S$ be
@@ -29,7 +34,7 @@ write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. Recall, that $\theta_i$ is the
     3\|X_S\|_1\}$. We say that $\Sigma$ satisfies the
     $(S,\gamma)$-\emph{restricted eigenvalue condition} iff:
 \begin{equation}
-    \forall X \in {\cal C(S)}, X^T \Sigma X \geq \gamma \|X\|_2^2 
+    \forall X \in {\cal C(S)}, X^T \Sigma X \geq \gamma \|X\|_2^2
 \tag{RE}
 \label{eq:re}
 \end{equation}
@@ -38,10 +43,12 @@ write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. Recall, that $\theta_i$ is the
 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}. In our
 setting we require that the (RE)-condition holds for the Hessian of the
-log-likelihood function $\mathcal{L}$: it essentially captures the fact that the binary
-vectors of the set of active nodes are not \emph{too} collinear.
+log-likelihood function $\mathcal{L}$: it essentially captures the fact that the
+binary vectors of the set of active nodes are not \emph{too} collinear.
 
-We will also need the following assumption on the inverse link function $f$ of the generalized linear cascade model:
+{\color{red} Rewrite the minimal assumptions necessary}
+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|,
@@ -51,11 +58,11 @@ We will also need the following assumption on the inverse link function $f$ of t
 for some $\alpha\in(0, 1)$ and for all $x\in\reals^m$ such that
 $f(\inprod{\theta^*}{x})\notin\{0,1\}$.
 
-In the voter model, $\frac{f'}{f}(z) = \frac{1}{z}$ and $\frac{f'}{f}(1-z)
-= \frac{1}{1-z}$. Hence (LF) will hold as soon as $\alpha\leq \Theta_{i,j}\leq
-1-\alpha$ for all $(i,j)\in E$ which is always satisfied for some $\alpha$ for non-isolated nodes.
-In the Independent Cascade Model, $\frac{f'}{f}(z)
-= \frac{1}{1-e^{-z}}$ and $\frac{f'}{1-f}(z) = -1$ and (LF) is not restrictive.
+In the voter model, $\frac{f'}{f}(z) = \frac{1}{z}$ and $\frac{f'}{f}(1-z) =
+\frac{1}{1-z}$. Hence (LF) will hold as soon as $\alpha\leq \Theta_{i,j}\leq
+1-\alpha$ for all $(i,j)\in E$ which is always satisfied for some $\alpha$ for
+non-isolated nodes.  In the Independent Cascade Model, $\frac{f'}{f}(z) =
+\frac{1}{1-e^{-z}}$ and $\frac{f'}{1-f}(z) = -1$ and (LF) is not restrictive.
 
 \begin{comment}
 Remember that in this case we have $\Theta_{i,j} = \log(1-p_{i,j})$. These
@@ -189,12 +196,12 @@ solve the Graph Inference problem.
 Under the same assumptions as Theorem~\ref{thm:main}, let $\hat {\cal S}_\eta
 \defeq \{ j \in \{1,\ldots, m\} : \hat{\theta}_j > \eta\}$ for $\eta > 0$. For
 $\epsilon>0$ and $\epsilon < \eta$, let ${\cal S}^*_{\eta + \epsilon} \defeq \{
-i \in \{1,\ldots,m\} :\theta_i^* > \eta +\epsilon \}$ be the set of all true `strong'
-parents. Suppose the number of measurements verifies:
-$ n > \frac{9s\log m}{\alpha\gamma^2\epsilon^2}$.
-Then with probability $1-\frac{1}{m}$, ${\cal S}^*_{\eta + \epsilon} \subseteq
-\hat {\cal S}_\eta \subseteq {\cal S}^*$. In other words we recover all `strong'
-parents and no `false' parents. 
+i \in \{1,\ldots,m\} :\theta_i^* > \eta +\epsilon \}$ be the set of all true
+`strong' parents. Suppose the number of measurements verifies: $ n >
+\frac{9s\log m}{\alpha\gamma^2\epsilon^2}$.  Then with probability
+$1-\frac{1}{m}$, ${\cal S}^*_{\eta + \epsilon} \subseteq \hat {\cal S}_\eta
+\subseteq {\cal S}^*$. In other words we recover all `strong' parents and no
+`false' parents. 
 \end{corollary}
 
 \begin{proof}
-- 
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