Box constrained $\ell_1$ optimization in random linear systems -- finite dimensions

Our companion work \cite{Stojnicl1BnBxasymldp} considers random under-determined linear systems with box-constrained sparse solutions and provides an asymptotic analysis of a couple of modified $\ell_1$ heuristics adjusted to handle such systems (we refer to these modifications of the standard $\ell_1$ as binary and box $\ell_1$). Our earlier work \cite{StojnicISIT2010binary} established that the binary $\ell_1$ does exhibit the so-called phase-transition phenomenon (basically the same phenomenon well-known through earlier considerations to be a key feature of the standard $\ell_1$, see, e.g. \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}). Moreover, in \cite{StojnicISIT2010binary}, we determined the precise location of the co-called phase-transition (PT) curve. On the other hand, in \cite{Stojnicl1BnBxasymldp} we provide a much deeper understanding of the PTs and do so through a large deviations principles (LDP) type of analysis. In this paper we complement the results of \cite{Stojnicl1BnBxasymldp} by leaving the asymptotic regime naturally assumed in the PT and LDP considerations aside and instead working in a finite dimensional setting. Along the same lines, we provide for both, the binary and the box $\ell_1$, precise finite dimensional analyses and essentially determine their ultimate statistical performance characterizations. On top of that, we explain how the results created here can be utilized in the asymptotic setting, considered in \cite{Stojnicl1BnBxasymldp}, as well. Finally, for the completeness, we also present a collection of results obtained through numerical simulations and observe that they are in a massive agreement with our theoretical calculations.