A Local Analysis of Block Coordinate Descent for Gaussian Phase Retrieval
This provides a theoretical guarantee for a nonconvex optimization method in a specific domain, but it is incremental as it builds on existing ADMM frameworks.
The paper tackles the Gaussian phase retrieval problem by analyzing a block coordinate descent algorithm, showing it converges linearly to the global minimizer from a specific initialization point.
While convergence of the Alternating Direction Method of Multipliers (ADMM) on convex problems is well studied, convergence on nonconvex problems is only partially understood. In this paper, we consider the Gaussian phase retrieval problem, formulated as a linear constrained optimization problem with a biconvex objective. The particular structure allows for a novel application of the ADMM. It can be shown that the dual variable is zero at the global minimizer. This motivates the analysis of a block coordinate descent algorithm, which is equivalent to the ADMM with the dual variable fixed to be zero. We show that the block coordinate descent algorithm converges to the global minimizer at a linear rate, when starting from a deterministically achievable initialization point.