Phase Transitions in Recovery of Structured Signals from Corrupted Measurements

TL;DR

This paper provides a theoretical framework explaining the sharp phase transitions observed in recovering structured signals from corrupted measurements, using Gaussian process tools to identify precise transition points.

Contribution

It introduces a rigorous analysis of phase transitions in signal recovery, linking geometric measures to the success of convex programming methods, and compares constrained and penalized procedures.

Findings

Identifies exact phase transition locations using Gaussian widths and distances.

Shows the relationship between constrained and penalized recovery procedures.

Proposes an optimal parameter selection strategy for penalized recovery.

Abstract

This paper is concerned with the problem of recovering a structured signal from a relatively small number of corrupted random measurements. Sharp phase transitions have been numerically observed in practice when different convex programming procedures are used to solve this problem. This paper is devoted to presenting theoretical explanations for these phenomenons by employing some basic tools from Gaussian process theory. Specifically, we identify the precise locations of the phase transitions for both constrained and penalized recovery procedures. Our theoretical results show that these phase transitions are determined by some geometric measures of structure, e.g., the spherical Gaussian width of a tangent cone and the Gaussian (squared) distance to a scaled subdifferential. By utilizing the established phase transition theory, we further investigate the relationship between these two kinds of recovery procedures, which also reveals an optimal strategy (in the sense of Lagrange theory) for choosing the tradeoff parameter in the penalized recovery procedure. Numerical experiments are provided to verify our theoretical results.