RoLL: Robust Low-Rank Learning via Nesterov Momentum

In modern multivariate problems, exploiting the low-rank structure of the coefficient matrix serves as a powerful dimension reduction strategy to enhance statistical efficiency and interpretability, but its effectiveness can be severely compromised by outliers. In this work, we propose a robust low-rank learning (RoLL) framework for a broad range of loss functions, implemented via a fast algorithm that exploits local restricted strong convexity. Theoretically, we establish non-asymptotic error bounds for the obtained fixed-point estimator (not necessarily the global or local optimum), showing that it achieves the minimax optimal rate. Furthermore, we develop a novel information criterion with finite-sample theoretical guarantees for model selection. Extensive experiments demonstrate the effectiveness of our proposed framework in the presence of data anomalies.