Precise Phase Transition of Total Variation Minimization
This solves a long-standing theoretical gap for researchers in compressed sensing, machine learning, and statistics, though it is incremental as it builds on existing conjectures and methods.
The paper fully characterizes the phase transition curve for total variation (TV) minimization in recovering sparse-gradient signals, addressing a previously open problem in compressed sensing and related fields.
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.