More compression

1 files changed, 13 insertions, 13 deletions

diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index cab27a7..e91cad4 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -242,22 +242,15 @@ this probabilistic model. Several recent papers show that large classes of
 correlated designs obey the restricted eigenvalue property with high
 probability \cite{raskutti:10, rudelson:13}.
 
-The {\bf(RE)}-condition is well-behaved in the following sense: if it holds for
-the expected Hessian matrix $\E[\nabla^2\mathcal{L}(\theta^*)]$, then it holds
-for the finite sample Hessian matrix $\nabla^2\mathcal{L}(\theta^*)$ with high
-probability.
+The {\bf(RE)}-condition has the following concentration property: if it holds
+for the expected Hessian matrix $\E[\nabla^2\mathcal{L}(\theta^*)]$, then it
+holds for the finite sample Hessian matrix $\nabla^2\mathcal{L}(\theta^*)$ with
+high probability.
 
-If $f$ and $1-f$ are strictly log-convex, then the previous observations yields
-the following natural interpretation of the {\bf(RE)}-condition: the expected
-\emph{Gram matrix} $A \equiv \mathbb{E}[X^T X]$ is a matrix whose entry
-$a_{i,j}$ is the average fraction of times node $i$ and node $j$ are infected
-at the same time during several runs of the cascade process. Note that the
-diagonal terms $a_{i,i}$ are simply the average fraction of times the
-corresponding node $i$ is infected.
 
 Therefore, under an assumption which only involves the probabilistic model and
-not the actual observations, Proposition~\ref{prop:fi} allows us to draw the
-same conclusion as in Theorem~\ref{thm:main}.
+not the actual observations, we can obtain the same conclusion as in
+Theorem~\ref{thm:main}:
 
 \begin{proposition}
     \label{prop:fi}
@@ -274,6 +267,13 @@ cascade, which are independent, we can apply Theorem 1.8 from
 \cite{rudelson:13}, lowering the number of required cascades to $s\log m \log^3(
 s\log m)$.
 
+If $f$ and $1-f$ are strictly log-convex, then the previous observations show
+that the quantity $\E[\nabla2\mathcal{L}(\theta^*)]$ in
+Proposition~\ref{prop:fi} can be replaced by the expected \emph{Gram matrix}:
+$A \equiv \mathbb{E}[X^T X]$. This matrix $A$ has a natural interpretation: the
+entry $a_{i,j}$ is the probability that node $i$ and node $j$ are infected at
+the same time during a cascade. In particular, the diagonal term $a_{i,i}$ is
+simply the probability that node $i$ is infected during a cascade.
 %\paragraph{(RE) vs Irrepresentability Condition}
 
 %\cite{Daneshmand:2014} rely on an `incoherence' condition on the hessian of the