1 files changed, 38 insertions, 3 deletions
diff --git a/notes/presentation/beamer_2.tex b/notes/presentation/beamer_2.tex
index 40dc769..2a42a3b 100644
--- a/notes/presentation/beamer_2.tex
+++ b/notes/presentation/beamer_2.tex
@@ -292,7 +292,7 @@ By thresholding $\hat \theta_i$, if $n > C' s \log m$, we recover the support of
 
 \begin{block}{Correlation}
 \begin{itemize}
-\item Measurements are correlated, which is unusual and good for us.
+\item Positive result despite correlated measurements \smiley
 \item Independent measurements $\implies$ taking one measurement per cascade.
 \end{itemize}
 \end{block}
@@ -309,10 +309,45 @@ By thresholding $\hat \theta_i$, if $n > C' s \log m$, we recover the support of
 %%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{frame}
-Slides about expected hessian
-TODO: slide about matrice de gram!
+\frametitle{Restricted Eigenvalue Condition}
+
+\begin{block}{From Hessian to Expected Hessian}
+\begin{itemize}
+\item If $n > C' s^2 \log m$ and $(S, \gamma)$-RE holds for $\mathbb{E}({\cal H})$, then $(S, \gamma/2)$-RE holds for ${\cal H}$
+\item $\mathbb{E}({\cal H})$ only depends on $p_{\text{init}}$ and $(\theta_j)_j$
+\end{itemize}
+\end{block}
+
+\begin{block}{From Hessian to Gram Matrix}
+\begin{itemize}
+\item If $\log f$ and $\log 1 -f$ are strictly log-concave with constant $c$, then if $(S, \gamma)$-RE holds for $\mathbb{E}({\cal H})$, then $(S, c \gamma)$-RE holds for the gram matrix $X X^T$
+\item Gram Matrix has natural interpretation:
+\begin{itemize}
+\item Diagonal : average number of times node is infected
+\item Outer-diagonal : average number of times pair of nodes is infected {\emph together}
+\end{itemize}
+\end{itemize}
+\end{block}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Future Work}
+\begin{itemize}
+\item Better lower bounds
+\item Active Learning
+\item Lower bound restricted eigenvalues of expected gram matrix
+\item Confidence Intervals
+\item Show that $n > C' s \log m$ measurements are necessary w.r.t. expected hessian.
+\item Linear Threshold model $\rightarrow$ 1-bit compressed sensing formulation
+\end{itemize}
 \end{frame}
 
+
+%%%%%%%%%%%%%%%%%
+
 \bibliography{../../paper/sparse}
 \bibliographystyle{apalike}