Group Projected Subspace Pursuit for Block Sparse Signal Reconstruction: Convergence Analysis and Applications

TL;DR

This paper analyzes the convergence of the GPSP algorithm for block sparse signal recovery, extending its application, and demonstrating its superior performance in various noisy and noiseless scenarios.

Contribution

It provides a convergence proof for GPSP under BRIP conditions and extends its application to general block sparse recovery tasks.

Findings

GPSP exactly recovers block sparse signals under BRIP with small BRIC.

RMC enhances robustness of GPSP in noisy observations.

GPSP outperforms other algorithms in diverse experimental settings.

Abstract

In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. [HKL+23] (Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), Journal of Computational Physics, 494, 112526) and extend its application to general tasks of block sparse signal recovery. We prove that when the sampling matrix satisfies the Block Restricted Isometry Property (BRIP) with a sufficiently small Block Restricted Isometry Constant (BRIC), GPSP exactly recovers the true block sparse signals. When the observations are noisy, this convergence property of GPSP remains valid if the magnitude of true signal is sufficiently large. GPSP selects the features by subspace projection criterion (SPC) for candidate inclusion and response magnitude criterion (RMC) for candidate exclusion. We compare these criteria with counterparts of other state-of-the-art greedy algorithms. Our theoretical analysis and numerical ablation studies reveal that SPC is critical to the superior performances of GPSP, and that RMC can enhance the robustness of feature identification when observations contain noises. We test and compare GPSP with other methods in diverse settings, including heterogeneous random block matrices, inexact observations, face recognition, and PDE identification. We find that GPSP outperforms the other algorithms in most cases for various levels of block sparsity and block sizes, justifying its effectiveness for general applications.