From 403268e4d19d7f9c87479f6fac37dff0175c0d60 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Sun, 25 Jan 2015 13:34:20 -0500
Subject: results updated

---
 paper/sections/results.tex | 17 ++++++++---------
 1 file changed, 8 insertions(+), 9 deletions(-)

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

diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index 68c10ff..6871fbe 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -1,20 +1,19 @@
 There have been a series of papers arguing that the Lasso is an inappropriate variable selection method (see H.Zou and T.Hastie, Sarah van de Geer ...). In fact, the irrepresentability condition, though essentially necessary for variable selection, rarely holds in practical situations where correlation between variable occurs. We defer an extended analysis of this situation to section ...
 
-Our approach is different. Rather than trying to perform variable selection by finding $\{j: \theta_j \neq 0\}$, we seek to obtain oracle inequalities by upper-bounding $\|\hat \theta - \theta^* \|_2$. It is easy to see that by thresholding $\hat \theta$, one recovers all `strong' parents with no false positives, as shown in Theorem~\ref{...}
-
-We cite the following Theorem from \cite{Negahban:2009}:
-
-\begin{theorem}
-Suppose that the true vector $\theta^*$ is exactly s-sparse with support S and that the following restricted eigenvalue {\bf(RE) } on the Hessian holds:
+Our approach is different. Rather than trying to perform variable selection by finding $\{j: \theta_j \neq 0\}$, we seek to obtain oracle inequalities by upper-bounding $\|\hat \theta - \theta^* \|_2$. It is easy to see that by thresholding $\hat \theta$, one recovers all `strong' parents with no false positives, as shown in Theorem~\ref{...}. Interestingly, obtaining oracle inequalities depends on a the following restricted eigenvalue condition\footnote{which is less restrictive? Irrepresentability implies compatibility? But do we have comptability and do they have irrepresentability?} for a symetric matrix $\Sigma$ and set ${\cal C} := \{X \in \mathbb{R}^p : \|X_{S^c}\|_1 \leq 3 \|X_S\|_1 \}$
 
 \begin{equation}
-\forall \Delta \ s.t. \ \|\Delta_{S^c}\|_1 \leq 3 \|\Delta_S\|_1  \|\nabla^2 f(\theta^*) \Delta \|^2 \geq \gamma_n \|Delta\|_2^2 \quad \text{\bf RE}
+\nonumber
+\forall X \in {\cal C}, \| \Sigma X \|^2 \geq \gamma_n \|X\|_2^2 \qquad \ \quad \text{\bf (RE)}
 \end{equation}
 
-Then, by solving Eq.~\ref{...} for $\lambda_n \geq 2 \|\nabla f(\theta^*)\|_{\infty}$ we have:
+We cite the following Theorem from \cite{Negahban:2009}:
+
+\begin{theorem}
+Suppose that the true vector $\theta^*$ has support S and that the {\bf(RE)} assumption holds for the hessian $\nabla^2 f(\theta^*)$, then by solving Eq.~\ref{...} for $\lambda_n \geq 2 \|\nabla f(\theta^*)\|_{\infty}$ we have:
 
 \begin{equation}
-\|\hat \theta - \theta^* \|_2 \leq \frac{\sqrt{s}\lambda_n}{\gamma_n}
+\|\hat \theta - \theta^* \|_2 \leq \frac{\sqrt{|S|}\lambda_n}{\gamma_n}
 \end{equation}
 \end{theorem}
 
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