  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.
