Final compression

3 files changed, 12 insertions, 6 deletions

diff --git a/paper/sections/intro.tex b/paper/sections/intro.tex
index c89c641..cc29ed7 100644
--- a/paper/sections/intro.tex
+++ b/paper/sections/intro.tex
@@ -73,21 +73,26 @@ However, the best known upper bound to this day is $\O(s^2\log
 m)$~\cite{Netrapalli:2012, Daneshmand:2014}
 
 The contributions of this paper are the following:
+\vspace{-1em}
 \begin{itemize}
     \item we formulate the Graph Inference problem in the context of
         discrete-time influence cascades as a sparse recovery problem for
         a specific type of Generalized Linear Model. This formulation notably
         encompasses the well-studied Independent Cascade Model and Voter Model.
+        \vspace{-0.5em}
     \item we give an algorithm which recovers the graph's edges using $\O(s\log
         m)$ cascades. Furthermore, we show that our algorithm is also able to
         efficiently recover the edge weights (the parameters of the influence
         model) up to an additive error term,
+        \vspace{-0.5em}
     \item we show that our algorithm is robust in cases where the signal to
         recover is approximately $s$-sparse by proving guarantees in the
         \emph{stable recovery} setting.
+        \vspace{-0.5em}
     \item we provide an almost tight lower bound of $\Omega(s\log \frac{m}{s})$
         observations required for sparse recovery.
 \end{itemize}
+\vspace{-0.5em}
 
 The organization of the paper is as follows: we conclude the introduction by a
 survey of the related work. In Section~\ref{sec:model} we present our model of
diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index 5d3a232..ecf5ad6 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -234,6 +234,7 @@ cascades, the graph inference problem becomes a linear inverse problem.
 Bernoulli realization whose parameters are given by applying $f$ to the
 matrix-vector product, where the measurement matrix encodes which nodes are
 ``contagious'' at each time step.}
+\vspace{-1em}
 \end{figure}
 
 Inferring the model parameter $\Theta$ from observed influence cascades is the
diff --git a/paper/sections/results.tex b/paper/sections/results.tex
index e91cad4..6b9fd7a 100644
--- a/paper/sections/results.tex
+++ b/paper/sections/results.tex
@@ -121,10 +121,10 @@ Assume {\bf(LF)} holds for some $\alpha>0$. For any $\delta\in(0,1)$:
 \end{lemma}
 
 The proof of Lemma~\ref{lem:ub} relies crucially on Azuma-Hoeffding's
-inequality, which allows us to handle correlated observations. This departs from
-the usual assumptions made in sparse recovery settings, where the sequence of
-measurements are assumed to be independent from one another. We now show how
-to use Theorem~\ref{thm:main} to recover the support of $\theta^*$, that is, to
+inequality, which allows us to handle correlated observations. This departs
+from the usual assumptions made in sparse recovery settings, that the
+measurements are independent from one another. We now show how to
+use Theorem~\ref{thm:main} to recover the support of $\theta^*$, that is, to
 solve the Network Inference problem.
 
 \begin{corollary}
@@ -225,7 +225,7 @@ Observe that the Hessian of $\mathcal{L}$ can be seen as a re-weighted
     \bigg[x_i^{t+1}\frac{f''f-f'^2}{f^2}(\inprod{\theta^*}{x^t})\\
     -(1-x_i^{t+1})\frac{f''(1-f) + f'^2}{(1-f)^2}(\inprod{\theta^*}{x^t})\bigg]
 \end{multline*}
-If $f$ and $1-f$ are $c$-strictly log-convex~\cite{bagnoli2005log} for $c>0$,
+If $f$ and $(1-f)$ are $c$-strictly log-convex for $c>0$,
 then $ \min\left((\log f)'', (\log (1-f))'' \right) \geq c $. This implies that
 the $(S, \gamma)$-({\bf RE}) condition in Theorem~\ref{thm:main} and
 Theorem~\ref{thm:approx_sparse} reduces to a condition on the \emph{Gram
@@ -267,7 +267,7 @@ 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
+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