Phase Transition of Convex Programs for Linear Inverse Problems with Multiple Prior Constraints

Abstract: A sharp phase transition emerges in convex programs when solving the linear inverse problem, which aims to recover a structured signal from its linear measurements. This paper studies this phenomenon in theory under Gaussian random measurements. Different from previous studies, in this paper, we consider convex programs with multiple prior constraints. These programs are encountered in many cases, for example, when the signal is sparse and its $\ell_2$ norm is known beforehand, or when the signal is sparse and non-negative simultaneously. Given such a convex program, to analyze its phase transition, we introduce a new set and a new cone, called the prior restricted set and prior restricted cone, respectively. Our results reveal that the phase transition of a convex problem occurs at the statistical dimension of its prior restricted cone. Moreover, to apply our theoretical results in practice, we present two recipes to accurately estimate the statistical dimension of the prior restricted cone. These two recipes work under different conditions, and we give a detailed analysis for them. To further illustrate our results, we apply our theoretical results and the estimation recipes to study the phase transition of two specific problems, and obtain computable formulas for the statistical dimension and related error bounds. Simulations are provided to demonstrate our results.