Robust Multi-Task Regression with Shifting Low-Rank Patterns

We consider the problem of multi-task regression with time-varying low-rank patterns, where the collected data may be contaminated by heavy-tailed distributions and/or outliers. Our approach is based on a piecewise robust multi-task learning formulation, in which a robust loss function—not necessarily to be convex, but with a bounded derivative—is used, and each piecewise low-rank pattern is induced by a nuclear norm regularization term. We propose using the composite gradient descent algorithm to obtain stationary points within a data segment and employing the dynamic programming algorithm to determine the optimal segmentation. The theoretical properties of the detected number and time points of pattern shifts are studied under mild conditions. Numerical results confirm the effectiveness of our method.