  We theoretically and experimentally investigate tensor-based regression and
classification. Our focus is regularization with various tensor norms,
including the overlapped trace norm, the latent trace norm, and the scaled
latent trace norm. We first give dual optimization methods using the
alternating direction method of multipliers, which is computationally efficient
when the number of training samples is moderate. We then theoretically derive
an excess risk bound for each tensor norm and clarify their behavior. Finally,
we perform extensive experiments using simulated and real data and demonstrate
the superiority of tensor-based learning methods over vector- and matrix-based
learning methods.
