results section updated

2 files changed, 27 insertions, 1 deletions

diff --git a/paper/paper.tex b/paper/paper.tex
index 86ef9d8..0b54de0 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -46,7 +46,7 @@
 % Therefore, a short form for the running title is supplied here:
 \icmltitlerunning{Cracking Cascades: Sparse Recovery for Graph Inference}
 
-\usepackage{amsmath, amsfonts, amssymb}
+\usepackage{amsmath, amsfonts, amssymb, amsthm}
 \newcommand{\reals}{\mathbb{R}}
 \newcommand{\ints}{\mathbb{N}}
 \DeclareMathOperator{\E}{\mathbb{E}}
@@ -57,6 +57,8 @@
 \newcommand{\inprod}[2]{#1 \cdot #2}
 \newcommand{\defeq}{\equiv}
 
+\newtheorem{theorem}{Theorem}
+
 \begin{document} 
 
 \twocolumn[
diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index e69de29..68c10ff 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -0,0 +1,24 @@
+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}
\ No newline at end of file