Approximate Message Passing With Unitary Transformation for Robust Bilinear Recovery

• Published in: IEEE transactions on signal processing Vol. 69; pp. 617 - 630
• Main Authors: Yuan, Zhengdao, Guo, Qinghua, Luo, Man
• Format: Journal Article
• Language: English
• Published: New York IEEE 2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximate message passing; Approximation algorithms; Bayesian analysis; bilinear recovery; compressive sensing; Covariance matrices; dictionary learning; Gaussian noise; Inference algorithms; Matrix decomposition; Message passing; Propagation; Recovery; Robustness; Self calibration; Signal processing algorithms; Statistical inference; unitary transformation
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2020.3044847

Recently, several promising approximate message passing (AMP) based algorithms have been developed for bilinear recovery with model <inline-formula><tex-math notation="LaTeX">\boldsymbol{Y}=\sum _{\boldsymbol{k}=1}^{\boldsymbol{K}} \boldsymbol{b}_{\boldsymbol{k}} \boldsymbol{A}_{\boldsymbol{k}} \boldsymbol{C}+\boldsymbol{W}</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">\lbrace \boldsymbol{b}_{\boldsymbol{k}}\rbrace</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">\boldsymbol{C}</tex-math></inline-formula> are jointly recovered with known <inline-formula><tex-math notation="LaTeX">\boldsymbol{A}_k</tex-math></inline-formula> from the noisy measurements <inline-formula><tex-math notation="LaTeX">\boldsymbol{Y}</tex-math></inline-formula>. The bilinear recovery problem has many applications such as dictionary learning, self-calibration, compressive sensing with matrix uncertainty, etc. In this work, we propose a new approximate Bayesian inference algorithm for bilinear recovery, where AMP with unitary transformation (UTAMP) is integrated with belief propagation (BP), variational inference (VI) and expectation propagation (EP) to achieve efficient approximate inference. It is shown that, compared to state-of-the-art bilinear recovery algorithms, the proposed algorithm is much more robust and faster, leading to remarkably better performance.