From 2ea212ea11dd8cde1c6fc380c1ba136276a22a43 Mon Sep 17 00:00:00 2001 From: jeanpouget-abadie Date: Fri, 30 Jan 2015 10:32:14 -0500 Subject: small changes --- paper/sections/assumptions.tex | 11 ++++++++--- 1 file changed, 8 insertions(+), 3 deletions(-) (limited to 'paper/sections/assumptions.tex') diff --git a/paper/sections/assumptions.tex b/paper/sections/assumptions.tex index a697d38..cff1051 100644 --- a/paper/sections/assumptions.tex +++ b/paper/sections/assumptions.tex @@ -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} -- cgit v1.2.3-70-g09d2