From 6491c0bccc42818cd9d71c1c6fd4bd9af953db2e Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Fri, 30 Jan 2015 00:00:15 -0500
Subject: reworded beginning of intro

---
 paper/sections/assumptions.tex | 11 +++++++++--
 1 file changed, 9 insertions(+), 2 deletions(-)

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

diff --git a/paper/sections/assumptions.tex b/paper/sections/assumptions.tex
index 8b7e3ee..a697d38 100644
--- a/paper/sections/assumptions.tex
+++ b/paper/sections/assumptions.tex
@@ -25,11 +25,18 @@ If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then th
 
 \subsection{The Restricted Eigenvalue Condition}
 
-Expressing the restricted eigenvalue assumption for correlated measurements as parameters of the graph and the cascade diffusion process is non-trivial. Under reasonable assumptions on the graph parameters, we can show a very crude ${\cal O}(N)$-lower bound for $\gamma_n$ by exploiting only the first set of measurements, where only the source nodes are active. Note that even though we waste a lot of information, we obtain similar asymptotic behavior than previous work.
+Expressing the restricted eigenvalue assumption for correlated measurements as parameters of the graph and the cascade diffusion process is non-trivial. The restricted eigenvalue property is however well behaved in the following sense:
 
-Similarly to the analysis conducted in \cite{Daneshmand:2014}, we can show that if the eigenvalue can be showed to hold for the `expected' hessian, it can be showed to hold for the hessian itself.
+\begin{lemma}
+\label{lem:expected_hessian}
+Expected hessian analysis!
+\end{lemma}
+
+This result is easily proved by adapting slightly a result from \cite{vandegeer:2009} XXX. Similarly to the analysis conducted in \cite{Daneshmand:2014}, we can show that if the eigenvalue can be showed to hold for the `expected' hessian, it can be showed to hold for the hessian itself. It is easy to see that:
 
 \begin{proposition}
 \label{prop:expected_hessian}
 If result holds for the expected hessian, then it holds for the hessian!
 \end{proposition}
+
+It is most likely possible to remove this extra s factor. See sub-gaussian paper by ... but the calculations are more involved.
-- 
cgit v1.2.3-70-g09d2

