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diff --git a/paper/sections/assumptions.tex b/paper/sections/assumptions.tex index eccd095..d6fed83 100644 --- a/paper/sections/assumptions.tex +++ b/paper/sections/assumptions.tex @@ -4,3 +4,10 @@ Proving the restricted eigenvalue assumption for correlated measurements is non- \subsection{The Irrepresentability Condition} +If our objective is to recover only the support of the graph and to do it exactly , then the irrepresentability condition\footnote{{\it incoherence} in the text} used in \cite{Daneshmand:2014} is believed to be necessary. + +However, assuming we only wish to recover all edges above a certain threshold, bounding the $\ell2$-error allows us to recover the full graph above a certain minimum threshold. Furthermore, it is intuitive that the irrepresentability condition is stronger than our suggested {\bf(RE)} assumption. In fact, by adapting slightly the following result from \cite{vandegeer:2009}, we can show that a `strong' irrepresentability condition directly {\it implies} the {\bf(RE)} condition for $\ell2$-recovery + +\begin{proposition} +If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then the restricted eigenvalue condition holds with constant: $(1 - 3\epsilon)^2 \lambda_{\min}n/s$ +\end{proposition}
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