Random linear systems with sparse solutions -- asymptotics and large deviations

In this paper we revisit random linear under-determined systems with sparse solutions. We consider $\ell_1$ optimization heuristic known to work very well when used to solve these systems. A collection of fundamental results that relate to its performance analysis in a statistical scenario is presented. We start things off by recalling on now classical phase transition (PT) results that we derived in \cite{StojnicCSetam09,StojnicUpper10}. As these represent the so-called breaking point characterizations, we now complement them by analyzing the behavior in a zone around the breaking points in a sense typically used in the study of the large deviation properties (LDP) in the classical probability theory. After providing a conceptual solution to these problems we attack them on a "hardcore" mathematical level attempting/hoping to be able to obtain explicit solutions as elegant as those we obtained in \cite{StojnicCSetam09,StojnicUpper10} (this time around though, the final characterizations were to be expected to be way more involved than in \cite{StojnicCSetam09,StojnicUpper10}, simply, the ultimate goals are set much higher and their achieving would provide a much richer collection of information about the $\ell_1$'s behavior). Perhaps surprisingly, the final LDP $\ell_1$ characterizations that we obtain happen to match the elegance of the corresponding PT ones from \cite{StojnicCSetam09,StojnicUpper10}. Moreover, as we have done in \cite{StojnicEquiv10}, here we also present a corresponding LDP set of results that can be obtained through an alternative high-dimensional geometry approach. Finally, we also prove that the two types of characterizations, obtained through two substantially different mathematical approaches, match as one would hope that they do.