small changes

1 files changed, 3 insertions, 5 deletions

diff --git a/paper/sections/linear_threshold.tex b/paper/sections/linear_threshold.tex
index 56febe1..ff33c07 100644
--- a/paper/sections/linear_threshold.tex
+++ b/paper/sections/linear_threshold.tex
@@ -1,9 +1,7 @@
-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 $1$: $\forall j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$. 
+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 $1$: $\forall j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$. 
 
 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:
+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 = \mathbbm{1}_{\sum_{i=1}^m \Theta_{i,j}X^t_i > t_j}
     = \mathbbm{1}_{\inprod{\theta_j}{X^t} > t_j}
@@ -16,4 +14,4 @@ where we defined again $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$. In o
     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}. 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
+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.
\ No newline at end of file