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The unit-modulus least squares (UMLS) problem has a wide spectrum ofapplications in signal processing, e.g., phase-only beamforming, phaseretrieval, radar code design, and sensor network localization. Scalablefirst-order methods such as projected gradient descent (PGD) have recently beenstudied as a simple yet efficient approach to solving the UMLS problem.Existing results on the convergence of PGD for UMLS often focus on globalconvergence to stationary points. As a non-convex problem, only a sublinearconvergence rate has been established. However, these results do not explainthe fast convergence of PGD frequently observed in practice. This manuscriptpresents a novel analysis of convergence of PGD for UMLS, justifying the linearconvergence behavior of the algorithm near the solution. By exploiting thelocal structure of the objective function and the constraint set, we establishan exact expression for the convergence rate and characterize the conditionsfor linear convergence. Simulations show that our theoretical analysiscorroborates numerical examples. Furthermore, variants of PGD with adaptivestep sizes are proposed based on the new insight revealed in our convergenceanalysis. The variants show substantial acceleration in practice.

On Local Linear Convergence of Projected Gradient Descent for Unit-Modulus Least Squares