Novel Algorithm for Sparse Solutions to Linear Inverse Problems with Multiple Measurements

TL;DR

This paper introduces a new efficient NESTA-based algorithm for recovering jointly sparse signals from multiple incomplete measurements, improving convergence and detection capabilities over traditional methods.

Contribution

The paper presents novel enhancements to the NESTA-based MMV algorithm, including support knowledge integration, partial support detection with MUSIC, and an iterative hard thresholding approach.

Findings

Achieves recovery of sparse matrices with fewer measurements than traditional methods.

Improves convergence rate through support knowledge and iterative techniques.

Demonstrates effectiveness of the proposed methods under RIP conditions.

Abstract

In this report, a novel efficient algorithm for recovery of jointly sparse signals (sparse matrix) from multiple incomplete measurements has been presented, in particular, the NESTA-based MMV optimization method. In a nutshell, the jointly sparse recovery is obviously superior to applying standard sparse reconstruction methods to each channel individually. Moreover several efforts have been made to improve the NESTA-based MMV algorithm, in particular, (1) the NESTA-based MMV algorithm for partially known support to greatly improve the convergence rate, (2) the detection of partial (or all) locations of unknown jointly sparse signals by using so-called MUSIC algorithm; (3) the iterative NESTA-based algorithm by combing hard thresholding technique to decrease the numbers of measurements. It has been shown that by using proposed approach one can recover the unknown sparse matrix X with () Spark A -sparsity from () Spark A measurements, predicted in Ref. [1], where the measurement matrix denoted by A satisfies the so-called restricted isometry property (RIP). Under a very mild condition on the sparsity of X and characteristics of the A, the iterative hard threshold (IHT)-based MMV method has been shown to be also a very good candidate.