Bilinear Recovery using Adaptive Vector-AMP

TL;DR

This paper introduces BAd-VAMP, an algorithm for jointly recovering vectors and matrices from noisy measurements in bilinear models, applicable to various inverse problems, and demonstrates its competitive performance.

Contribution

The paper proposes the BAd-VAMP algorithm for bilinear recovery, advancing state-of-the-art methods in this domain.

Findings

BAd-VAMP performs competitively with existing bilinear recovery algorithms.

The method is applicable to diverse tasks like matrix completion and blind deconvolution.

Numerical results validate the effectiveness of the proposed approach.

Abstract

We consider the problem of jointly recovering the vector $\boldsymbol{b}$ and the matrix $\boldsymbol{C}$ from noisy measurements $\boldsymbol{Y} = \boldsymbol{A}(\boldsymbol{b})\boldsymbol{C} + \boldsymbol{W}$, where $\boldsymbol{A}(\cdot)$ is a known affine linear function of $\boldsymbol{b}$ (i.e., $\boldsymbol{A}(\boldsymbol{b})=\boldsymbol{A}_0+\sum_{i=1}^Q b_i \boldsymbol{A}_i$ with known matrices $\boldsymbol{A}_i$). This problem has applications in matrix completion, robust PCA, dictionary learning, self-calibration, blind deconvolution, joint-channel/symbol estimation, compressive sensing with matrix uncertainty, and many other tasks. To solve this bilinear recovery problem, we propose the Bilinear Adaptive Vector Approximate Message Passing (BAd-VAMP) algorithm. We demonstrate numerically that the proposed approach is competitive with other state-of-the-art approaches to bilinear recovery, including lifted VAMP and Bilinear GAMP.