Living near the edge: A lower-bound on the phase transition of total variation minimization

TL;DR

This paper derives a tight lower-bound on the measurement threshold for successful total variation minimization in recovering gradient-sparse signals, considering generalized sparsity notions, and confirms its accuracy through simulations.

Contribution

It provides the first tight lower-bound on the TV phase transition threshold that accounts for generalized gradient-sparsity, improving understanding of TV recovery limits.

Findings

The lower-bound closely matches the actual phase transition observed in simulations.

The bound applies to any measurement matrix with a null space uniformly distributed Haar measure.

Generalized sparsity notions significantly influence the phase transition threshold.

Abstract

This work is about the total variation (TV) minimization which is used for recovering gradient-sparse signals from compressed measurements. Recent studies indicate that TV minimization exhibits a phase transition behavior from failure to success as the number of measurements increases. In fact, in large dimensions, TV minimization succeeds in recovering the gradient-sparse signal with high probability when the number of measurements exceeds a certain threshold; otherwise, it fails almost certainly. Obtaining a closed-form expression that approximates this threshold is a major challenge in this field and has not been appropriately addressed yet. In this work, we derive a tight lower-bound on this threshold in case of any random measurement matrix whose null space is distributed uniformly with respect to the Haar measure. In contrast to the conventional TV phase transition results that depend on the simple gradient-sparsity level, our bound is highly affected by generalized notions of gradient-sparsity. Our proposed bound is very close to the true phase transition of TV minimization confirmed by simulation results.