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Random linear systems with sparse solutions -- finite dimensions

In our companion work \cite{Stojnicl1RegPosasymldp} we revisited random under-determined linear systems with sparse solutions. The main emphasis was on the performance analysis of the $\ell_1$ heuristic in the so-called asymptotic regime, i.e. in the regime where the systems' dimensions are large. Through an earlier sequence of work \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}, it is now well known that in such a regime the $\ell_1$ exhibits the so-called \emph{phase transition} (PT) phenomenon. \cite{Stojnicl1RegPosasymldp} then went much further and established the so-called \emph{large deviations principle} (LDP) type of behavior that characterizes not only the breaking points of the $\ell_1$'s success but also the behavior in the entire so-called \emph{transition zone} around these points. Both of these concepts, the PTs and the LDPs, are in fact defined so that one can use them to characterize the asymptotic behavior. In this paper we complement the results of \cite{Stojnicl1RegPosasymldp} by providing an exact detailed analysis in the non-asymptotic regime. Of course, not only are the non-asymptotic results complementing those from \cite{Stojnicl1RegPosasymldp}, they actually are the ones that ultimately fully characterize the $\ell_1$'s behavior in the most general sense. We introduce several novel high-dimensional geometry type of strategies that enable us to eventually determine the $\ell_1$'s behavior.