Bilinear Adaptive Generalized Vector Approximate Message Passing

TL;DR

This paper introduces BAd-GVAMP, an algorithm for generalized bilinear recovery problems involving nonlinear measurements, extending previous methods to handle arbitrary output distributions and demonstrating effectiveness across multiple applications.

Contribution

The paper proposes BAd-GVAMP, a novel algorithm that extends BAd-VAMP to accommodate arbitrary output distributions in generalized bilinear recovery tasks.

Findings

BAd-GVAMP effectively handles nonlinear measurements in various applications.

Numerical results show superior performance of BAd-GVAMP over existing methods.

The algorithm is versatile for applications like quantized compressed sensing and matrix completion.

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