Precise Phase Transition of Total Variation Minimization

TL;DR

This paper rigorously characterizes the phase transition curve of total variation minimization in recovering sparse-gradient signals, advancing understanding in compressed sensing and convex optimization.

Contribution

It provides the first complete characterization of the phase transition for TV minimization, linking it to AMP algorithm conjectures and high-dimensional convex geometry.

Findings

Established the phase transition curve for TV minimization

Linked TV minimization phase transition to AMP algorithm conjecture

Connected denoising MSE to high-dimensional convex geometry

Abstract

Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery methods such as $\ell_1$ minimization and nuclear norm minimization are well understood through recent years' research. However, rigorously characterizing the phase transition of total variation (TV) minimization in recovering sparse-gradient signal is still open. In this paper, we fully characterize the phase transition curve of the TV minimization. Our proof builds on Donoho, Johnstone and Montanari's conjectured phase transition curve for the TV approximate message passing algorithm (AMP), together with the linkage between the minmax Mean Square Error of a denoising problem and the high-dimensional convex geometry for TV minimization.