Convergence guarantee for the EPILOG ADMM algorithm

Develop a convergence guarantee for the ADMM-based proximal splitting algorithm constructed for optimization with the EPILOG regularizer.

Background

The paper introduces EPILOG, a regularizer that transfers structural priors from an auxiliary variable to the log-magnitude of a complex-valued signal, and derives variable-wise proximity operators for its associated bivariate function. These operators are used to construct an ADMM-based algorithm for audio signal recovery under a Parseval-tight discrete Gabor transform assumption.

Because the EPILOG function is not jointly convex in the complex signal variable and the auxiliary log-magnitude variable, the paper does not provide a convergence analysis for the resulting alternating-minimization and dual-update scheme. Establishing a convergence guarantee would formalize the conditions under which the proposed algorithm converges.