Generalized Bilinear Factorization via Hybrid Vector Message Passing
Hao Jiang, Xiaojun Yuan, Qinghua Guo · IEEE Transactions on Signal Processing · 2024
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.