Sparse Vector and Low Rank Recovery Phase Transitions: Uncovering the Explicit Relations

Abstract

We investigate the two primary categories of structured recovery problems, namely Compressed Sensing (CS) and Low Rank Recovery (LRR). Our focus is on the performance analysis of their two tightest convex relaxation based heuristics, the so-called $\ell _{1}$ and the nuclear norm ( $\ell _{1}^{*}$ ) minimizations. We examine two standard types of phase transitions (PTs): 1) general PT, obtained by enforcing sparsity as a fundamental form of structuring, and 2) nonnegative PT, achieved by imposing nonnegativity as an additional form of structuring alongside sparsity. We establish explicit relations between the CS and LRR PTs. Our analysis reveals that the nonnegative PT essentially interpolates between the general and the binary CS PT, in a manner that can be explicitly characterized. Quite surprisingly, although the phase transitions themselves admit fairly complicated mathematical formulations, their relations can be expressed in a very neat and elegant way. This ultimately allows to quickly assess and compare the effects additional presence/absence of the nonnegativity has on $\ell _{1}$ and $\ell _{1}^{*}$ .