From cec29dcbf5a01dade73d8eb8af7a4d106123bd1c Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Sun, 25 Jan 2015 13:03:18 -0500
Subject: results section updated

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 paper/sections/results.tex | 24 ++++++++++++++++++++++++
 1 file changed, 24 insertions(+)

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diff --git a/paper/sections/results.tex b/paper/sections/results.tex
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+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:
+
+\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}
+\end{equation}
+
+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}
+\end{equation}
+\end{theorem}
+
+
+\subsection{Virtues of Oracle Inequalities}
+
+\subsection{The Irrepresentability Condition}
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