Recovery under side constraints

This chapter 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 representation vector, and the nonlinear measurement structure. First, we demonstrate how a priori information in the form of structural side constraints influence recovery guarantees (null space properties) using $ell$1-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. To this end, we derive a new 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.