Bilinear Recovery using Adaptive Vector-AMP

Abstract

We consider the problem of jointly recovering the vector $\mathbf{b}$ and the matrix $\mathbf{C}$ from noisy measurements $\mathbf{Y}=\mathbf{A}(\mathbf{b})\mathbf{C}+\mathbf{W}$, where $\mathbf{A}(\cdot )$ is a known affine linear function of $\mathbf{b}$ (i.e., $\mathbf{A}(\mathbf{b})={\mathbf{A}}_0+{\sum }_{i=1}^Q{b}_i{\mathbf{A}}_i$ with known matrices ${\mathbf{A}}_i$). This problem has applications in matrix completion, robust PCA, dictionary learning, self-calibration, blind deconvolution, joint-channel/symbol estimation, compressive sensing with matrix uncertainty, and many other tasks. To solve this bilinear recovery problem, we propose the Bilinear Adaptive Vector Approximate Message Passing (BAd-VAMP) algorithm. We demonstrate numerically that the proposed approach is competitive with other state-of-the-art approaches to bilinear recovery, including lifted VAMP and Bilinear GAMP.