Low-rank regularization in two-sided matrix regression

The two-sided matrix regression model Y = A^∗ XB^∗ + E aims at predicting Y by taking into account both linear links between column features of X, via the unknown matrix B^∗, and also among the row features of X, via the matrix A^∗. We propose low-rank predictors in this high-dimensional matrix regression model via rank-penalized and nuclear norm-penalized least squares. Both criteria are non jointly convex; however, we propose explicit predictors based on SVD and show optimal prediction bounds. We give sufficient conditions for consistent rank selector. We also propose a fully data-driven rank-adaptive procedure. Simulation results confirm the good prediction and the rank-consistency results under data-driven explicit choices of the tuning parameters and the scaling parameter of the noise.