Recovery under Side Constraints

TL;DR

This paper explores how structural side constraints affect sparse signal recovery, proposing improved algorithms and measurement designs for various signal types and measurement systems, including nonlinear cases like phase retrieval.

Contribution

It introduces methods to incorporate prior structural information into sparse recovery, enhancing performance and computational efficiency, and extends to nonlinear measurement system design.

Findings

Structural constraints influence recovery guarantees via null space properties.

Algorithms leveraging prior information improve recovery accuracy and reduce complexity.

Design strategies for measurement matrices enhance performance in linear and nonlinear systems.

Abstract

This paper addresses sparse signal reconstruction under various types of structural side constraints with applications in multi-antenna systems. Side constraints may result from prior information on the measurement system and the sparse signal structure. They may involve the structure of the sensing matrix, the structure of the non-zero support values, the temporal structure of the sparse representationvector, and the nonlinear measurement structure. First, we demonstrate how a priori information in form of structural side constraints influence recovery guarantees (null space properties) using L1-minimization. Furthermore, for constant modulus signals, signals with row-, block- and rank-sparsity, as well as non-circular signals, we illustrate how structural prior information can be used to devise efficient algorithms with improved recovery performance and reduced computational complexity. Finally, we address the measurement system design for linear and nonlinear measurements of sparse signals. Moreover, we discuss the linear mixing matrix design based on coherence minimization. Then we extend our focus to nonlinear measurement systems where we design parallel optimization algorithms to efficiently compute stationary points in the sparse phase retrieval problem with and without dictionary learning.