Bilinear Adaptive Generalized Vector Approximate Message Passing

Abstract

This paper considers the generalized bilinear recovery problem, which aims to jointly recover the vector b and the matrix X from componentwise nonlinear measurements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\text {Y}}\sim p({\text {Y}}|{\text {Z}})=\prod \limits _{i,j}p(Y_{ij}|Z_{ij})$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\text {Z}}={\text {A}}({\text {b}}){\text {X}}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\text {A}}(\cdot)$ </tex-math></inline-formula> is a known affine linear function of b, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p(Y_{ij}|Z_{ij})$ </tex-math></inline-formula> is a scalar conditional distribution that 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, and joint channel and data decoding, 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 algorithm to incorporate arbitrary distributions on the output transform. The numerical results on various applications demonstrate the effectiveness of the proposed BAd-GVAMP algorithm.