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

TL;DR

This paper provides a finite-dimensional analysis of box-constrained $\, ext{l}_1$ optimization in random linear systems, complementing previous asymptotic results and validating findings through numerical simulations.

Contribution

It offers the first finite-dimensional performance characterizations for binary and box $\, ext{l}_1$ heuristics, bridging the gap between asymptotic theory and practical finite systems.

Findings

Finite-dimensional performance characterizations match asymptotic predictions.

Numerical simulations confirm theoretical accuracy.

Results enhance understanding of $\, ext{l}_1$ methods in practical settings.

Abstract

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.