Sample Complexity of Total Variation Minimization

TL;DR

This paper develops a new theoretical upper-bound for Total Variation minimization that closely approximates the empirical phase transition curve, improving understanding of its success and failure conditions in signal recovery.

Contribution

It introduces a novel asymptotically sharp upper-bound for TV minimization's phase transition, enhancing theoretical insights into its recovery performance.

Findings

New upper-bound closely matches empirical phase transition curve

Improves upon previous bounds by Kabanava

Numerical results validate the bound's accuracy

Abstract

This work considers the use of Total variation (TV) minimization in the recovery of a given gradient sparse vector from Gaussian linear measurements. It has been shown in recent studies that there exist a sharp phase transition behavior in TV minimization in asymptotic regimes. The phase transition curve specifies the boundary of success and failure of TV minimization for large number of measurements. It is a challenging task to obtain a theoretical bound that reflects this curve. In this work, we present a novel upper-bound that suitably approximates this curve and is asymptotically sharp. Numerical results show that our bound is closer to the empirical TV phase transition curve than the previously known bound obtained by Kabanava.