Robust High-Dimensional Low-Rank Matrix Estimation: Optimal Rate and Data-Adaptive Tuning

The matrix lasso, which minimizes a least-squared loss function with the nuclear-norm regularization, offers a generally applicable paradigm for high-dimensional low-rank matrix estimation, but its efficiency is adversely affected by heavy-tailed distributions. This paper introduces a robust procedure by incorporating a Wilcoxon-type rank-based loss function with the nuclear-norm penalty for a unified high-dimensional low-rank matrix estimation framework. It includes matrix regression, multivariate regression and matrix completion as special examples. This procedure enjoys several appealing features. First, it relaxes the distributional …

data efficiency function lasso least loss low matrix norm nuclear paper paradigm rate regularization the matrix type