From 190cd5633e7e3237be118c18d374b5542aa38be4 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Thu, 5 Feb 2015 23:22:36 -0500
Subject: Polishing

---
 paper/sections/discussion.tex | 22 +++++++++++++++++++---
 1 file changed, 19 insertions(+), 3 deletions(-)

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

diff --git a/paper/sections/discussion.tex b/paper/sections/discussion.tex
index 25d9c53..b2947a1 100644
--- a/paper/sections/discussion.tex
+++ b/paper/sections/discussion.tex
@@ -1,7 +1,15 @@
 
 
 \paragraph{Future Work}
-Solving the Graph Inference problem with sparse recovery techniques opens new venues for future work. Firstly, the sparse recovery literature has already studied regularization patterns beyond the $\ell-1$-norm, notably the thresholded and adaptive lasso \cite{vandegeer:2011} \cite{Zou:2006}. Another series of papers that are directly relevant to the Graph Inference setting have shown that confidence intervals can be established for the lasso. Finally, the linear threshold model is a commonly studied diffusion process and can also be cast as a \emph{generalized linear cascade} with inverse link function $z \mapsto \mathbbm{1}_{z > 0}$:
+Solving the Graph Inference problem with sparse recovery techniques opens new
+venues for future work. Firstly, the sparse recovery literature has already
+studied regularization patterns beyond the $\ell_1$-norm, notably the
+thresholded and adaptive lasso \cite{vandegeer:2011, Zou:2006}. Another goal
+would be to obtain confidence intervals for our estimator, similarly to what
+has been obtained for the Lasso in the recent series of papers
+\cite{javanmard2014, zhang2014}.
+
+Finally, the linear threshold model is a commonly studied diffusion process and can also be cast as a \emph{generalized linear cascade} with inverse link function $z \mapsto \mathbbm{1}_{z > 0}$:
 
 \begin{equation}
     \label{eq:lt}
@@ -9,7 +17,15 @@ Solving the Graph Inference problem with sparse recovery techniques opens new ve
     X^{t+1}_j = \text{sign} \left(\inprod{\theta_j}{X^t} - t_j \right)
 \end{equation}
 
-This model therefore falls into the 1-bit compressed sensing model \cite{Boufounos:2008} framework. Several recent papers study the theoretical guarantees obtained for 1-bit compressed sensing with specific measurements \cite{Gupta:2010}, \cite{Plan:2014}. Whilst they obtained bounds of the order ${\cal O}(n \log \frac{n}{s}$), no current theory exists for recovering positive bounded signals from bernoulli hyperplanes. This research direction may provide the first clues to solve the ``active learning'' problem: if we are allowed to adaptively \emph{choose} the source nodes at the beginning of each cascade, can we improve on current results?
+This model therefore falls into the 1-bit compressed sensing model
+\cite{Boufounos:2008} framework. Several recent papers study the theoretical
+guarantees obtained for 1-bit compressed sensing with specific measurements
+\cite{Gupta:2010, Plan:2014}. Whilst they obtained bounds of the order
+${\cal O}(n \log \frac{n}{s}$), no current theory exists for recovering
+positive bounded signals from bernoulli hyperplanes. This research direction
+may provide the first clues to solve the ``adaptive learning'' problem: if we
+are allowed to adaptively \emph{choose} the source nodes at the beginning of
+each cascade, how much can we improve the current results?
 
 \begin{comment}
 The Linear Threshold model can \emph{also} be cast a generalized linear cascade model. However, as we show below, its link function is non-differentiable and necessitates a different analysis. In the Linear Threshold Model, each node $j\in V$ has a threshold $t_j$ from the interval $[0,1]$ and for each node, the sum of incoming weights is less than $1$: $\forall j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$. 
@@ -29,4 +45,4 @@ where we defined again $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$. In o
 \end{equation}
 
 The link function of the linear threshold model is the sign function: $z \mapsto \mathbbm{1}_{z > 0}$. This model therefore falls into the 1-bit compressed sensing model \cite{Boufounos:2008} framework. Several recent papers study the theoretical guarantees obtained for 1-bit compressed sensing with specific measurements \cite{Gupta:2010}, \cite{Plan:2014}. Whilst they obtained bounds of the order ${\cal O}(n \log \frac{n}{s}$), no current theory exists for recovering positive bounded signals from bernoulli hyperplanes. We leave this research direction to future work.
-\end{comment}
\ No newline at end of file
+\end{comment}
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