Compressed Sensing with Upscaled Vector Approximate Message Passing

TL;DR

This paper enhances the VAMP algorithm for compressed sensing by integrating conjugate gradient methods, enabling scalable, stable, and efficient large-scale image reconstruction with fewer iterations.

Contribution

It introduces a novel warm-starting scheme and theoretical models for CG-VAMP, significantly improving scalability and performance in large-scale problems.

Findings

WS-CG-VAMP requires fewer CG iterations for similar or better reconstruction quality.

The proposed methods enable stable and efficient large-scale image reconstruction.

Theoretical models accurately predict the behavior of the warm-started algorithm.

Abstract

The Recently proposed Vector Approximate Message Passing (VAMP) algorithm demonstrates a great reconstruction potential at solving compressed sensing related linear inverse problems. VAMP provides high per-iteration improvement, can utilize powerful denoisers like BM3D, has rigorously defined dynamics and is able to recover signals measured by highly undersampled and ill-conditioned linear operators. Yet, its applicability is limited to relatively small problem sizes due to the necessity to compute the expensive LMMSE estimator at each iteration. In this work we consider the problem of upscaling VAMP by utilizing Conjugate Gradient (CG) to approximate the intractable LMMSE estimator. We propose a rigorous method for correcting and tuning CG withing CG-VAMP to achieve a stable and efficient reconstruction. To further improve the performance of CG-VAMP, we design a warm-starting scheme for CG and develop theoretical models for the Onsager correction and the State Evolution of Warm-Started CG-VAMP (WS-CG-VAMP). Additionally, we develop robust and accurate methods for implementing the WS-CG-VAMP algorithm. The numerical experiments on large-scale image reconstruction problems demonstrate that WS-CG-VAMP requires much fewer CG iterations compared to CG-VAMP to achieve the same or superior level of reconstruction.