From 30606c850bc585848aa7cb809590a4e920822197 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Sun, 1 Feb 2015 20:53:25 -0500
Subject: linear threshold model

---
 paper/sections/linear_threshold.tex | 16 +++++++---------
 1 file changed, 7 insertions(+), 9 deletions(-)

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

diff --git a/paper/sections/linear_threshold.tex b/paper/sections/linear_threshold.tex
index 5eb7cf9..56febe1 100644
--- a/paper/sections/linear_threshold.tex
+++ b/paper/sections/linear_threshold.tex
@@ -1,21 +1,19 @@
-In the Linear Threshold Model, each node $j\in V$ has a threshold $t_j$ from the interval $[0,1]$. Furthermore, there is a weight $\Theta_{i,j}\in[0,1]$ for each edge $(i,j)$, such that the sum of incoming weights is less than $1$: $\forall j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$. 
+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 $1$: $\forall j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$. 
 
-Nodes can be either susceptible or active. At each time step, each susceptible
+Nodes have two states: susceptible or active. At each time step, each susceptible
 node $j$ becomes active if the sum of the incoming weights from active parents is greater than $j$'s threshold $t_j$. Nodes remain active until the end of the cascade, which is reached when no new susceptible nodes become active.
 
 As such, the source nodes are chosen, the process is entirely deterministic. Let $X^t$ be the indicator variable of the set of active nodes at time step $t-1$, then:
 \begin{equation}
-    X^{t+1}_j = \left[\sum_{i=1}^m \Theta_{i,j}X^t_i > t_j\right]
-    = \left[\inprod{\theta_j}{X^t} > t_j \right]
+    X^{t+1}_j = \mathbbm{1}_{\sum_{i=1}^m \Theta_{i,j}X^t_i > t_j}
+    = \mathbbm{1}_{\inprod{\theta_j}{X^t} > t_j}
 \end{equation}
-where we defined again $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$ and $[\cdot]$ is the boolean evaluator. In other words, for every step in the linear threshold model, we observe the following signal:
+where we defined again $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$. In other words, for every step in the linear threshold model, we observe the following signal:
 
 \begin{equation}
     \label{eq:lt}
     \tag{LT}
-    y^t_j = \text{sign} \left(\inprod{\theta_j}{X^t} - t_j \right)
+    X^{t+1}_j = \text{sign} \left(\inprod{\theta_j}{X^t} - t_j \right)
 \end{equation}
 
-The linear threshold model can therefore be cast as a variant of the recently introduced 1-bit compressed sensing model \cite{Boufounos:2008}. Several recent papers study the theoretical guarantees obtained for 1-bit compressed sensing with specific measurements \cite{Gupta:2010}, \cite{Plan:2014}. Others study adaptive mechanisms \cite{Jacques:2013}.
-
-Support recovery results in the 1-bit compressed sensing model rely on the fact that well-chosen random hyperplanes segment a compact set into small cells. Knowing which side of the hyperplanes you are on situates you in one of these cells of bounded size. Whilst they obtained bounds are also 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.
\ No newline at end of file
+The linear threshold model can therefore be cast as a variant of the recently introduced 1-bit compressed sensing model \cite{Boufounos:2008}. 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.
\ No newline at end of file
-- 
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