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diff --git a/paper/sections/assumptions.tex b/paper/sections/assumptions.tex
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--- a/paper/sections/assumptions.tex
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@@ -1,11 +1,16 @@
+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.
+
 \subsection{The Irrepresentability Condition}
 
-\cite{Daneshmand:2014} rely on an `incoherence' condition on the hessian of the likelihood function. It is in fact easy to see that their condition is equivalent to the more commonly called {\it (S, s)-irrepresentability} condition:
+\cite{Daneshmand:2014} rely on an `incoherence' condition on the hessian of the likelihood function. It is in fact easy to see that their condition is equivalent to the more commonly called {\it (S,s)-irrepresentability} condition:
 
 \begin{definition}
-Following similar notation, let $Q^* \nabla^2 f(\theta^*)$. Let $Q_{S^C,S} XXX$, the {\it (S, s)-irrepresentability} condition is defined as:
+Following similar notation, let $Q^* \defeq \nabla^2 f(\theta^*)$. Let $S$ and $S^c$ be the set of indices of all the parents and non-parents respectively and $Q_{S,S}$, $Q_{S^c,S}$, $Q_{S, S^c}$, and $Q_{S^c, S^c}$ the induced sub-matrices.
+
+
+The {\it (S,s)-irrepresentability} condition is defined as:
 \begin{equation}
-blabla
+\nu_{\text{irrepresentable}}(S) \defeq \max_{\tau \in \mathbb{R}^p \ :\ \| \tau \|_{\infty} \leq 1} \|Q_{S^c, S} Q_{S, S}^{-1} \tau\|_{\infty}
 \end{equation}
 \end{definition}