Generalized Bilinear Factorization via Hybrid Vector Message Passing

Generalized bilinear factorization (GBF), in which two matrices are recovered from noisy and typically compressed measurements of their product, arises in various applications such as blind channel-and-signal estimation, image completion, and compressed video foreground and background separation. In this paper, we formulate the GBF problem by unifying several existing bilinear inverse problems, and establish a novel hybrid vector message passing (HVMP) algorithm for GBF. The GBF-HVMP algorithm integrates expectation propagation (EP) and variational message passing (VMP) via variational free energy minimization, and exchanges matrix-variable messages in closed form. GBF-HVMP is advantageous over its counterparts in several aspects. For example, a matrix-variable message can characterize the correlations between the elements of the matrix, which is not possible in scalar-variable message passing; the hybrid of EP and VMP yields closed-form Gaussian messages associated with the bilinear constraints inherent in the GBF problem. We show that damping is unnecessary for GBF-HVMP to ensure convergence. We also show that GBF-HVMP performs close to the replica bound, and significantly outperforms state-of-the-art approaches in terms of both normalized mean squared error (NMSE) performance and computational complexity.