From 692c93dcd1be86bae445dfbbbcf2ddb263653c52 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Wed, 4 Feb 2015 18:02:17 -0500
Subject: More rewriting of 3.1, move assumptions back in section 3

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 paper/sections/assumptions.tex | 46 ------------------------------------------
 1 file changed, 46 deletions(-)

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

diff --git a/paper/sections/assumptions.tex b/paper/sections/assumptions.tex
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-In this section, we discuss the main assumption of Theorem~\ref{thm:neghaban} namely the restricted eigenvalue condition. We then compare to the irrepresentability condition considered in \cite{Daneshmand:2014}.
-
-\subsection{The Restricted Eigenvalue Condition}
-
-The restricted eigenvalue condition, introduced in \cite{bickel:2009}, is one of the weakest sufficient condition on the design matrix for successful sparse recovery \cite{vandegeer:2009}. Several recent papers show that large classes of correlated designs obey the restricted eigenvalue property with high probability \cite{raskutti:10} \cite{rudelson:13}. 
-
-Expressing the minimum restricted eigenvalue $\gamma$ as a function of the cascade model parameters is highly non-trivial. Yet, the restricted eigenvalue property is however well behaved in the following sense: under reasonable assumptions, if the population matrix of the hessian $\mathbb{E} \left[\nabla^2 {\cal L}(\theta) \right]$, corresponding to the \emph{Fisher Information Matrix} of the Cascade Model as a function of $\Theta$, verifies the restricted eigenvalue property, then the finite sample hessian also verifies the restricted eigenvalue property with overwhelming probability. It is straightforward to show this holds when $n \geq C s^2 \log m$ \cite{vandegeer:2009}, where $C$ is an absolute constant. By adapting Theorem 8 \cite{rudelson:13}\footnote{this result still needs to be confirmed!}, this can be reduced to:
-
-$$n \geq C s \log m \log^3 \left( \frac{s \log m}{C'} \right)$$
-
-where $C, C'$ are constants not depending on $(s, m, n)$.
-
-
-\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:
-
-\begin{definition}
-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. Consider the following constant:
-\begin{equation}
-\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}
-The (S,s)-irrepresentability holds if $\nu_{\text{irrepresentable}}(S) < 1 - \epsilon$ for $\epsilon > 0$
-\end{definition}
-
-This condition has been shown to be essentially necessary for variable selection \cite{Zhao:2006}. 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}
-\nonumber
-\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.
-
-It is intuitive that the irrepresentability condition is stronger than the {\bf(RE)} assumption. In fact, a slightly modified result from \cite{vandegeer:2009} shows that a `strong' irrepresentability condition directly {\it implies} the {\bf(RE)} condition for $\ell2$-recovery:
-
-\begin{proposition}
-\label{prop:irrepresentability}
-If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then the restricted eigenvalue condition holds with constant  $\gamma_n \geq \frac{ (1 - 3(1 -\epsilon))^2 \lambda_{\min}^2}{4s}n$, where $\lambda_{\min} > 0$ is the smallest eigenvalue of $Q^*_{S,S}$, on which the results of \cite{Daneshmand:2014} also depend.
-\end{proposition}
-
-
-
-- 
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