Theoretical Linear Convergence of Deep Unfolding Network for Block-Sparse Signal Recovery

TL;DR

This paper introduces Ada-BlockLISTA, a deep unfolding network for block-sparse signal recovery, with proven linear convergence and improved performance over existing methods by leveraging block structure.

Contribution

It presents a novel deep unfolding network that exploits block-sparse structure and provides theoretical guarantees of linear convergence and exact recovery.

Findings

Proves linear convergence rate of Ada-BlockLISTA.

Demonstrates better recovery performance than existing algorithms.

Guarantees exact recovery at higher sparsity levels.

Abstract

In this paper, we consider the recovery of the high-dimensional block-sparse signal from a compressed set of measurements, where the non-zero coefficients of the recovered signal occur in a small number of blocks. Adopting the idea of deep unfolding, we explore the block-sparse structure and put forward a block-sparse reconstruction network named Ada-BlockLISTA, which performs gradient descent on every single block followed by a block-wise shrinkage. Furthermore, we prove the linear convergence rate of our proposed network, which also theoretically guarantees exact recovery for a potentially higher sparsity level based on underlyingblock structure. Numerical results indicate that Ada-BlockLISTA yields better signal recovery performance compared with existing algorithms, which ignore the additional block structure in the signal model.