Analysis of Approximate Message Passing with a Class of Non-Separable Denoisers

TL;DR

This paper extends the analysis of approximate message passing algorithms to include non-separable denoisers, specifically sliding-window types, demonstrating that their performance can still be accurately predicted by a new form of state evolution.

Contribution

It introduces a novel state evolution framework for AMP with non-separable denoisers, advancing understanding of AMP in dependent signal scenarios.

Findings

State evolution accurately predicts AMP with non-separable denoisers.

Non-separable denoisers improve performance in dependent signal recovery.

Framework applicable to compressive image reconstruction.

Abstract

Approximate message passing (AMP) is a class of efficient algorithms for solving high-dimensional linear regression tasks where one wishes to recover an unknown signal \beta_0 from noisy, linear measurements y = A \beta_0 + w. When applying a separable denoiser at each iteration, the performance of AMP (for example, the mean squared error of its estimates) can be accurately tracked by a simple, scalar iteration referred to as state evolution. Although separable denoisers are sufficient if the unknown signal has independent and identically distributed entries, in many real-world applications, like image or audio signal reconstruction, the unknown signal contains dependencies between entries. In these cases, a coordinate-wise independence structure is not a good approximation to the true prior of the unknown signal. In this paper we assume the unknown signal has dependent entries, and using a class of non-separable sliding-window denoisers, we prove that a new form of state evolution still accurately predicts AMP performance. This is an early step in understanding the role of non-separable denoisers within AMP, and will lead to a characterization of more general denoisers in problems including compressive image reconstruction.