Compressed Sensing with 1D Total Variation: Breaking Sample Complexity Barriers via Non-Uniform Recovery (iTWIST'20)

TL;DR

This paper demonstrates that non-uniform recovery of gradient-sparse signals via total variation minimization can be achieved with significantly fewer measurements than previously thought, especially when signals have well-separated discontinuities.

Contribution

It provides a rigorous analysis showing that the sample complexity for non-uniform recovery can be reduced to near-linear in the sparsity level for certain signals, breaking the previous square-root barrier.

Findings

Non-uniform recovery succeeds with m ~ s * PolyLog(n) measurements.

Signals with well-separated jumps enable improved recovery guarantees.

The results apply to discretizations of piecewise constant functions.

Abstract

This paper investigates total variation minimization in one spatial dimension for the recovery of gradient-sparse signals from undersampled Gaussian measurements. Recently established bounds for the required sampling rate state that uniform recovery of all $s$-gradient-sparse signals in $\mathbb{R}^n$ is only possible with $m \gtrsim \sqrt{s n} \cdot \text{PolyLog}(n)$ measurements. Such a condition is especially prohibitive for high-dimensional problems, where $s$ is much smaller than $n$. However, previous empirical findings seem to indicate that the latter sampling rate does not reflect the typical behavior of total variation minimization. Indeed, this work provides a rigorous analysis that breaks the $\sqrt{s n}$-bottleneck for a large class of natural signals. The main result shows that non-uniform recovery succeeds with high probability for $m \gtrsim s \cdot \text{PolyLog}(n)$ measurements if the jump discontinuities of the signal vector are sufficiently well separated. In particular, this guarantee allows for signals arising from a discretization of piecewise constant functions defined on an interval. The present paper serves as a short summary of the main results in our recent work [arxiv:2001.09952].