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.
Categories: cs.IT math.IT