From 3fd4b1e192143f6ff71104b4a553c2da3ab28bcc Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Fri, 6 Feb 2015 12:44:05 -0500
Subject: Section 3

---
 paper/sections/results.tex | 29 +++++++++++++++++++++--------
 1 file changed, 21 insertions(+), 8 deletions(-)

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

diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index 0c1cc3b..6c8a35a 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -310,14 +310,27 @@ case to the assumption made in the Lasso analysis of \cite{bickel:2009}.
 \paragraph{(RE) with high probability}
 
 The Generalized Linear Cascade model yields a probability distribution over the
-set observed nodes $x^t$. It is then natural to ask whether the restricted
-eigenvalue condition is likely to occur under this probabilistic model. Several
-recent papers show that large classes of correlated designs obey the restricted
-eigenvalue property with high probability \cite{raskutti:10}
-\cite{rudelson:13}. 
-
-Expressing the minimum restricted eigenvalue $\gamma$ as a function of the
-cascade model parameters is highly non-trivial. Yet, the restricted eigenvalue
+observed sets of infeceted nodes $(x^t)_{t\in\mathcal{T}}$. It is then natural
+to ask whether the restricted eigenvalue condition is likely to occur under
+this probabilistic model. Several recent papers show that large classes of
+correlated designs obey the restricted eigenvalue property with high
+probability \cite{raskutti:10, rudelson:13}. 
+
+In our case, we can show that if (RE)-condition holds for the expected Hessian
+matrix $\E[\nabla^2\mathcal{L}(\theta^*)]$, then it holds for the finite sample
+Hessian matrix $\nabla^2\mathcal{L}(\theta)$ with high probability. Note that
+the expected Hessian matrix is exactly the Fisher Information matrix of the
+generalized linear cascade model which captures the amount of information about
+$\theta$ conveyed by the random observations. Therefore, under an assumption
+which only involves the probabilistic model and not the actual observations, we
+can reformulate Theorem~\ref{thm:main}.
+
+We will need the following additional assumptions on the inverse link
+function $f$:
+
+
+
+Yet, the restricted eigenvalue
 property is however well behaved in the following sense: under reasonable
 assumptions, if the population matrix of the hessian $\mathbb{E} \left[\nabla^2
 {\cal L}(\theta) \right]$, corresponding to the \emph{Fisher Information
-- 
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