  The two-sided matrix regression model $Y = A^*X B^* +E$ 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.
