  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 ${\mathbf Y}\sim p({\mathbf Y}|{\mathbf
Z})=\prod\limits_{i,j}p(Y_{ij}|Z_{ij})$, 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.
