1 files changed, 13 insertions, 6 deletions
diff --git a/paper/sections/assumptions.tex b/paper/sections/assumptions.tex
index cff1051..58ea977 100644
--- a/paper/sections/assumptions.tex
+++ b/paper/sections/assumptions.tex
@@ -1,4 +1,4 @@
-In this section, we compare the main assumption of \cite{Daneshmand:2014}, commonly known as the {\it irrepresentability condition}, to the restricted eigenvalue condition. We argue that the restricted eigenvalue is weaker and more likely to hold in practical situations.
+In this section, we compare the main assumption of \cite{Daneshmand:2014}, commonly known as the {\it irrepresentability condition}, to the restricted eigenvalue condition. We argue that the restricted eigenvalue is more likely to hold in practical situations.
 
 \subsection{The Irrepresentability Condition}
 
@@ -14,11 +14,16 @@ The {\it (S,s)-irrepresentability} condition is defined as:
 \end{equation}
 \end{definition}
 
-If our objective is to recover the support of the graph exactly, the irrepresentability condition has been shown to be essentially necessary \cite{Zhao:2006}. However, several recent papers \cite{vandegeer:2011}, \cite{Zou:2006}, argue this condition is unrealistic in situations where there is a correlation between variables.
-
-Consider for example the following matrix:
-
-Yet, assuming we only wish to recover all edges above a certain threshold, bounding the $\ell2$-error allows us to recover all edges with weights above a certain minimum threshold under an intuitively weaker {\bf(RE)} condition. In practical scenarios, such as in social networks, where one seeks to recover significant edges, this is a reasonable assumption.
+If our objective is to recover the support of the graph exactly, the irrepresentability condition has been shown to be essentially necessary \cite{Zhao:2006}. However, several recent papers \cite{vandegeer:2011}, \cite{Zou:2006}, argue this condition is unrealistic in situations where there is a correlation between variables. Consider the following simplified example from \cite{vandegeer:2011}:
+\begin{equation}
+\left( 
+\begin{array}{cccc}
+I_s & \rho J \\
+\rho J & I_s \\
+\end{array}
+\right)
+\end{equation}
+where $I_s$ is the $s \times s$ identity matrix, $J$ is the all-ones matrix and $\rho \in \mathbb{R}^+$. It is easy to see that $\nu_{\text{irrepresentable}}(S) = \rho s$ and $\lambda_{\min}(Q) \geq 1 - \rho$, such that for any $\rho > \frac{1}{s}$ and $\rho < 1$, the restricted eigenvalue holds trivially but the (S,s)-irrepresentability does not hold.
 
 As mentioned previously, it is intuitive that the irrepresentability condition is stronger than our suggested {\bf(RE)} assumption. In fact, by adapting slightly a result from \cite{vandegeer:2009}, we can show that a `strong' irrepresentability condition directly {\it implies} the {\bf(RE)} condition for $\ell2$-recovery:
 
@@ -30,6 +35,8 @@ If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then th
 
 \subsection{The Restricted Eigenvalue Condition}
 
+Yet, assuming we only wish to recover all edges above a certain threshold, bounding the $\ell2$-error allows us to recover all edges with weights above a certain minimum threshold under an intuitively weaker {\bf(RE)} condition. In practical scenarios, such as in social networks, where one seeks to recover significant edges, this is a reasonable assumption.
+
 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:
 
 \begin{lemma}