Bilinear Adaptive Generalized Vector Approximate Message Passing

Abstract

This paper considers the generalized bilinear recovery problem which aims to jointly recover the vector $\mathbf{b}$ and the matrix $\mathbf{X}$ from componentwise nonlinear measurements $\mathbf{Y}\sim p(\mathbf{Y}|\mathbf{Z})=\prod _{i,j}p(Y_{ij}|Z_{ij})$, where $\mathbf{Z}=\mathbf{A}(\mathbf{b})\mathbf{X}$, $\mathbf{A}(\cdot )$ is a known affine linear function of $\mathbf{b}$, and $p(Y_{ij}|Z_{ij})$ is a scalar conditional distribution which models the general output transform. A wide range of real-world applications, e.g., quantized compressed sensing with matrix uncertainty, blind self-calibration and dictionary learning from nonlinear measurements, one-bit matrix completion, etc., can be cast as the generalized bilinear recovery problem. To address this problem, we propose a novel algorithm called the Bilinear Adaptive Generalized Vector Approximate Message Passing (BAd-GVAMP), which extends the recently proposed Bilinear Adaptive Vector AMP (BAd-VAMP) algorithm to incorporate arbitrary distributions on the output transform. Numerical results on various applications demonstrate the effectiveness of the proposed BAd-GVAMP algorithm.

19 pages, 7 figures