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A Rice method proof of the Null-Space Property over the Grassmannian
The Null-Space Property (NSP) is a necessary and sufficient condition for the
recovery of the largest coefficients of solutions to an under-determined system
of linear equations. Interestingly, this property governs also the success and
the failure of recent developments in high-dimensional statistics, signal
processing, error-correcting codes and the theory of polytopes. Although this
property is the keystone of -minimization techniques, it is an open
problem to derive a closed form for the phase transition on NSP. In this
article, we provide the first proof of NSP using random processes theory and
the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds
for NSP with respect to unitarily invariant distributions. Furthermore, we
derive a simple sufficient condition for NSP.Comment: 18 Pages, some Figure