From bca54732ef95d8a71437d76039b7b99b3e72ba29 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Sun, 7 Dec 2014 15:25:25 -0500
Subject: RIP constant section

---
 notes/reportYaron.tex | 28 ++++++++++++++++++++++++++++
 1 file changed, 28 insertions(+)

(limited to 'notes/reportYaron.tex')

diff --git a/notes/reportYaron.tex b/notes/reportYaron.tex
index d39d060..6341f98 100644
--- a/notes/reportYaron.tex
+++ b/notes/reportYaron.tex
@@ -173,6 +173,34 @@ In general, the smaller $\delta_k$ is, the better we can recover $k$-sparse vect
 
 \subsection{Verifying the RIP constants}
 
+Though it is not yet clear which normalization to choose for the columns of matrix $M$, nor whether we should include only the first step of every cascade or the cascade in its entirety, we ran simple experiments to show that the measurement matrix $M$ has `well-behaved' RIP constants.
+
+\begin{table}
+\centering
+\begin{tabular}{c | c | c | c }
+c = 100 & c = 1000 & c = 100 & c = 1000 \\
+$p_\text{init}$ = .1 & $p_\text{init}$ = .1 & $p_\text{init}$ = .05 & $p_\text{init}$ = .05 \\
+\hline
+$\delta_4 = .54$ & $\delta_4 = .37$ & $\delta_4 = .43$ & $\delta_4 = .23$ \\
+\end{tabular}
+\caption{Results on dataset of 22 nodes extracted from the Karate club network. $p_\text{init}$ is the probability that a node is part of the seed set. $c$ is the number of cascades. $\delta_4$ is the RIP constant for $4$-sparse vectors. Cascades are generated using the independent cascade model.}
+\end{table}
+
+Finding a non-trivial RIP constant is strongly NP-hard, and as such we were unable to test on networks with more than 25 nodes. To compute the $k-th$ RIP constant ($\delta_k$ with our notation), we must extract $k$ columns of $M$, and compute the following programs on the resulting matrices $(M_k)$ (there are $\binom{n}{k}$ of them!):
+\begin{align}
+\label{eq:lower_bound}
+\min \qquad & \| M_k \vec x \|_2 \\
+\| \vec x\|_2 = 1 & \nonumber
+\end{align}
+\begin{align}
+\label{eq:upper_bound}
+\max \qquad & \| M_k \vec x \|_2 \\
+\| \vec x\|_2 = 1 & \nonumber
+\end{align}
+
+Note how choosing the minimum of Eq~\ref{eq:lower_bound} over all extracted matrices returns the lowest possible $1 - \delta^-_k$ and choosing the maximum of Eq~\ref{eq:upper_bound} over all extracted matrices returns the highest possible $1 + \delta^+_k$. The smallest admissible $k-$RIP constant is therefore: $\max(\delta^+_k, \delta^-_k)$. Equations \ref{eq:lower_bound} and \ref{eq:upper_bound} can be solved efficiently since they correspond to the smallest and greatest eigenvalues of $M^TM$ respectively.
+
+In \cite{candes}, 
 
 \subsection{Testing our algorithm}
 
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