assumptions section+references

2 files changed, 66 insertions, 17 deletions

diff --git a/paper/sections/assumptions.tex b/paper/sections/assumptions.tex
index 89113b4..7cf9345 100644
--- a/paper/sections/assumptions.tex
+++ b/paper/sections/assumptions.tex
@@ -1,4 +1,24 @@
-In this section, we discuss the main assumption of Theorem~\ref{thm:neghaban} namely the restricted eigenvalue condition. We begin by comparing to the irrepresentability condition considered in \cite{Daneshmand:2014}.
+In this section, we discuss the main assumption of Theorem~\ref{thm:neghaban} namely the restricted eigenvalue condition. We then compare to the irrepresentability condition considered in \cite{Daneshmand:2014}.
+
+\subsection{The Restricted Eigenvalue Condition}
+
+The restricted eigenvalue condition, introduced in \cite{bickel:2009}, is one of the weakest sufficient condition on the design matrix for successful sparse recovery \cite{vandegeer:2009}. Several recent papers show that large classes of correlated designs obey the restricted eigenvalue property with high probability \cite{raskutti:10} \cite{rudelson:13}. Expressing the minimum restricted eigenvalue $\gamma$ as a function of the cascade model parameters is highly non-trivial. However, the restricted eigenvalue property is however well behaved in the following sense:
+
+\begin{lemma}
+\label{lem:expected_hessian}
+Expected hessian analysis!
+\end{lemma}
+
+This result is easily proved by adapting slightly a result from \cite{vandegeer:2009} XXX. Similarly to the analysis conducted in \cite{Daneshmand:2014}, we can show that if the eigenvalue can be showed to hold for the `expected' hessian, it can be showed to hold for the hessian itself. It is easy to see that:
+
+\begin{proposition}
+\label{prop:expected_hessian}
+If result holds for the expected hessian, then it holds for the hessian!
+\end{proposition}
+
+It is most likely possible to remove this extra s factor. See sub-gaussian paper by ... but the calculations are more involved.
+
+
 
 \subsection{The Irrepresentability Condition}
 
@@ -32,20 +52,4 @@ If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then th
 \end{proposition}
 
 
-\subsection{The Restricted Eigenvalue Condition}
-
-In practical scenarios, such as in social networks, recovering only the `significant' edges is a reasonable assumption. This can be done under the less restrictive eigenvalue assumption. Expressing $\gamma$ as a function of the cascade model parameters process is non-trivial. The restricted eigenvalue property is however well behaved in the following sense:
-
-\begin{lemma}
-\label{lem:expected_hessian}
-Expected hessian analysis!
-\end{lemma}
 
-This result is easily proved by adapting slightly a result from \cite{vandegeer:2009} XXX. Similarly to the analysis conducted in \cite{Daneshmand:2014}, we can show that if the eigenvalue can be showed to hold for the `expected' hessian, it can be showed to hold for the hessian itself. It is easy to see that:
-
-\begin{proposition}
-\label{prop:expected_hessian}
-If result holds for the expected hessian, then it holds for the hessian!
-\end{proposition}
-
-It is most likely possible to remove this extra s factor. See sub-gaussian paper by ... but the calculations are more involved.
diff --git a/paper/sparse.bib b/paper/sparse.bib
index 7d88004..d1abd66 100644
--- a/paper/sparse.bib
+++ b/paper/sparse.bib
@@ -225,3 +225,48 @@ URL = {http://dx.doi.org/10.1198/016214506000000735},
   bibsource = {dblp computer science bibliography, http://dblp.org}
 }
 
+@article{bickel:2009,
+author = "Bickel, Peter J. and Ritov, Ya’acov and Tsybakov, Alexandre B.",
+doi = "10.1214/08-AOS620",
+fjournal = "The Annals of Statistics",
+journal = "Ann. Statist.",
+month = "08",
+number = "4",
+pages = "1705--1732",
+publisher = "The Institute of Mathematical Statistics",
+title = "Simultaneous analysis of Lasso and Dantzig selector",
+url = "http://dx.doi.org/10.1214/08-AOS620",
+volume = "37",
+year = "2009"
+}
+
+@article{raskutti:10,
+  author    = {Garvesh Raskutti and
+               Martin J. Wainwright and
+               Bin Yu},
+  title     = {Restricted Eigenvalue Properties for Correlated Gaussian Designs},
+  journal   = {Journal of Machine Learning Research},
+  volume    = {11},
+  pages     = {2241--2259},
+  year      = {2010},
+  url       = {http://portal.acm.org/citation.cfm?id=1859929},
+  timestamp = {Wed, 15 Oct 2014 17:04:32 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/jmlr/RaskuttiWY10},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{rudelson:13,
+  author    = {Mark Rudelson and
+               Shuheng Zhou},
+  title     = {Reconstruction From Anisotropic Random Measurements},
+  journal   = {{IEEE} Transactions on Information Theory},
+  volume    = {59},
+  number    = {6},
+  pages     = {3434--3447},
+  year      = {2013},
+  url       = {http://dx.doi.org/10.1109/TIT.2013.2243201},
+  doi       = {10.1109/TIT.2013.2243201},
+  timestamp = {Tue, 21 May 2013 14:15:50 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/tit/RudelsonZ13},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
\ No newline at end of file