Box constrained $\ell_1$ optimization in random linear systems -- asymptotics

TL;DR

This paper analyzes the asymptotic behavior of box constrained $ ext{l}_1$ optimization methods for solving sparse, bounded linear systems, focusing on phase transitions and large deviation principles using probabilistic and geometric approaches.

Contribution

It introduces and characterizes two novel $ ext{l}_1$ adaptation methods for constrained sparse recovery, providing a comprehensive asymptotic analysis with probabilistic and geometric insights.

Findings

Characterization of phase transition phenomena for constrained $ ext{l}_1$ methods

Derivation of large deviation principles for solution probabilities

Comparison of probabilistic and geometric analytical approaches

Abstract

In this paper we consider box constrained adaptations of $\ell_1$ optimization heuristic when applied for solving random linear systems. These are typically employed when on top of being sparse the systems' solutions are also known to be confined in a specific way to an interval on the real axis. Two particular $\ell_1$ adaptations (to which we will refer as the \emph{binary} $\ell_1$ and \emph{box} $\ell_1$) will be discussed in great detail. Many of their properties will be addressed with a special emphasis on the so-called phase transitions (PT) phenomena and the large deviation principles (LDP). We will fully characterize these through two different mathematical approaches, the first one that is purely probabilistic in nature and the second one that connects to high-dimensional geometry. Of particular interest we will find that for many fairly hard mathematical problems a collection of pretty elegant characterizations of their final solutions will turn out to exist.