Learned Turbo Message Passing for Affine Rank Minimization and Compressed Robust Principal Component Analysis

This paper is focused on the efficient algorithm design for affine rank minimization (ARM) and compressed robust principal component analysis (CRPCA). Given the proliferation of the literature on the ARM and CRPCA problems, the existing algorithms mostly take a model-based approach in the algorithm design, and so are sensitive to the generation model of the linear measurement operator or to the generation model of the low-rank matrix to be recovered. It is known that, all these algorithms do not work well, e.g., when the low-rank matrix is ill-conditioned. As inspired by the success of learned iterative soft-thresholding (LIST) and learned approximate message passing (LAMP), we develop learning-based message passing algorithms, namely, the learned turbo message passing (LTMP) algorithm for affine rank minimization to cope with the ARM problem and the LTMP algorithm for the CRPCA problem. The LTMP algorithms learn their parameters from data, and hence are robust to the generation models of the linear operator and the low-rank matrix. We derive analytical expressions for the partial derivatives involved in training the LTMP network. Given the large size of the low-rank matrix in a typical ARM/CRPCA problem, these analytical expressions are of essential importance for the development of computationally feasible network training. Numerical results demonstrate that LTMP significantly outperforms the state-of-the-art counterparts for various generation models of the linear operator and the low-rank matrix, especially when the low-rank matrix is ill-conditioned.