On A Class of Greedy Sparse Recovery Algorithms - A High Dimensional Approach

2024-02-25

arXiv.org

AI Summary

This paper addresses sparse signal recovery in underdetermined linear systems $x = Qs$. We propose a novel greedy algorithmic framework that directly optimizes in the $s$-domain, unifying solution-space characterization and iterative mechanics. The framework supports both $ell_2$- and $ell_1$-norm measures and incorporates CoSaMP’s atom selection strategy. Our $ell_2$-based variant significantly outperforms classical OMP, while the $ell_1$-based variant substantially surpasses Basis Pursuit (BP). Both algorithms exhibit strong robustness to measurement noise and ill-conditioning of the sensing matrix $Q$. Theoretical analysis establishes guarantees for high-dimensional sparse modeling and numerical stability. Extensive experiments on synthetic data and real-world images demonstrate marked improvements in reconstruction accuracy, with computational complexity comparable to that of OMP.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Sparse signal recovery deals with finding the sparest solution of an under-determined linear system $x = Q s$. In this paper, we propose a novel greedy approach to addressing the challenges from such a problem. Such an approach is based on a characterization of solutions to the system, which allows us to work on the sparse recovery in the $s$-space directly with a given measure. With $l_2$-based measure, an orthogonal matching pursuit (OMP)-type algorithm is proposed, which significantly outperforms the classical OMP algorithm in terms of recovery accuracy while maintaining comparable computational complexity. An $l_1$-based algorithm, denoted as $ ext{Alg}_{GL1}$, is derived. Such an algorithm significantly outperforms the classical basis pursuit (BP) algorithm. Combining with the CoSaMP-strategy for selecting atoms, a class of high performance greedy algorithms is also derived. Extensive numerical simulations on both synthetic and image data are carried out, with which the superior performance of our proposed algorithms is demonstrated in terms of sparse recovery accuracy and robustness against numerical instability of the system matrix $Q$ and disturbance in the measurement $x$.

Problem

• Proposes greedy algorithms for sparse signal recovery.
• Improves recovery accuracy and computational efficiency.
• Enhances robustness against system matrix instability.

Innovation

• Novel greedy approach for sparse signal recovery
• OMP-type algorithm with l2-based measure
• Alg_GL1 outperforms classical BP algorithm