Partial $\ell_1$ optimization in random linear systems -- finite dimensions

In this paper we provide a complementary set of results to those we present in our companion work \cite{Stojnicl1HidParasymldp} regarding the behavior of the so-called partial $\ell_1$ (a variant of the standard $\ell_1$ heuristic often employed for solving under-determined systems of linear equations). As is well known through our earlier works \cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13}, the partial $\ell_1$ also exhibits the phase-transition (PT) phenomenon, discovered and well understood in the context of the standard $\ell_1$ through Donoho's and our own works \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}. \cite{Stojnicl1HidParasymldp} goes much further though and, in addition to the determination of the partial $\ell_1$'s phase-transition curves (PT curves) (which had already been done in \cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13}), provides a substantially deeper understanding of the PT phenomena through a study of the underlying large deviations principles (LDPs). As the PT and LDP phenomena are by their definitions related to large dimensional settings, both sets of our works, \cite{StojnicICASSP10knownsupp,StojnicTowBettCompSens13} and \cite{Stojnicl1HidParasymldp}, consider what is typically called the asymptotic regime. In this paper we move things in a different direction and consider finite dimensional scenarios. Basically, we provide explicit performance characterizations for any given collection of systems/parameters dimensions. We do so for two different variants of the partial $\ell_1$, one that we call exactly the partial $\ell_1$ and another one, possibly a bit more practical, that we call the hidden partial $\ell_1$.