On the Error in Phase Transition Computations for Compressed Sensing

TL;DR

This paper investigates the inaccuracies in phase transition computations for compressed sensing, revealing limitations of existing bounds and proposing a new, more accurate error bound applicable to overcomplete dictionaries.

Contribution

It identifies shortcomings of previous error bounds in low-dimensional models and introduces a novel bound that improves accuracy, especially for overcomplete dictionaries.

Findings

Existing bounds become large in low-dimensional models.

The new bound significantly reduces estimation gap.

The new bound applies to overcomplete dictionary settings.

Abstract

Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has become standard to replace it with an upper-bound. To ensure that this technique is suitable, [1] has introduced an upper-bound on the gap between the statistical dimension and its approximation. In this work, we first show that the error bound in [1] in some low-dimensional models such as total variation and $\ell_1$ analysis minimization becomes poorly large. Next, we develop a new error bound which significantly improves the estimation gap compared to [1]. In particular, unlike the bound in [1] that is not applicable to settings with overcomplete dictionaries, our bound exhibits a decaying behavior in such cases.