Smoothing and Decomposition for Analysis Sparse Recovery

TL;DR

This paper introduces smoothing and decomposition techniques combined with MFISTA for analysis sparse recovery, providing convergence guarantees and demonstrating faster MRI image reconstruction.

Contribution

It proposes two novel methods, smoothing-based and decomposition-based MFISTA, for analysis sparse recovery with theoretical guarantees and practical efficiency.

Findings

Smoothing-based MFISTA converges faster than decomposition-based in practice.

The methods can recover signals sparse in redundant tight frames.

Performance bounds are established under a modified RIP condition.

Abstract

We consider algorithms and recovery guarantees for the analysis sparse model in which the signal is sparse with respect to a highly coherent frame. We consider the use of a monotone version of the fast iterative shrinkage- thresholding algorithm (MFISTA) to solve the analysis sparse recovery problem. Since the proximal operator in MFISTA does not have a closed-form solution for the analysis model, it cannot be applied directly. Instead, we examine two alternatives based on smoothing and decomposition transformations that relax the original sparse recovery problem, and then implement MFISTA on the relaxed formulation. We refer to these two methods as smoothing-based and decomposition-based MFISTA. We analyze the convergence of both algorithms, and establish that smoothing- based MFISTA converges more rapidly when applied to general nonsmooth optimization problems. We then derive a performance bound on the reconstruction error using these techniques. The bound proves that our methods can recover a signal sparse in a redundant tight frame when the measurement matrix satisfies a properly adapted restricted isometry property. Numerical examples demonstrate the performance of our methods and show that smoothing-based MFISTA converges faster than the decomposition-based alternative in real applications, such as MRI image reconstruction.