Reconstruction of Complex Discrete-Valued Vector via Convex Optimization With Sparse Regularizers

In this paper, we propose a method for the reconstruction of a complex discrete-valued vector from its linear measurements. In particular, we mainly focus on the underdetermined cases, where the number of measurements is less than that of the unknown complex discrete variables, and propose a reconstruction approach of solving an optimization problem called sum of complex sparse regularizers (SCSR) optimization. The sum of sparse regularizers in the objective function can directly utilize the discrete nature of the unknown vector in the complex domain. We also propose an algorithm for the SCSR optimization problem on the basis of alternating direction method of multipliers. For the proposed convex regularizers, we analytically prove that the sequence obtained by the proposed algorithm converges to the optimal solution of the problem. To obtain better reconstruction performance, we further propose an iterative approach named iterative weighted SCSR (IW-SCSR), where we update the parameters in the objective function in each iteration by using the tentative estimate in the previous iteration. Simulation results show that IW-SCSR can reconstruct the complex discrete-valued vector from its underdetermined linear measurements and achieve good performance in the applications of overloaded multiple-input multiple-output signal detection and channel equalization.