An Arbitrary Mode-3 Dimensional Tensor-Tensor Product for Tensor Train Decomposition From Interaction Perspective

Recently, the classical tensor-tensor product (T-product) has attracted considerable attention for capturing the interactions between tensor factors. However, the mode-3 consistency in the T-product restricts its flexibility and expressive ability. To break the restriction, we suggest a tensor-tensor product for arbitrary mode-3 dimension (termed as Art-product) which enables us to flexibly and expressively capture the interactions between tensor factors. Concretely, by leveraging the exclusive hierarchical nonlinear transforms along the third mode, two tensor factors with inconsistency dimensions are first transformed into the corresponding latent factors with consistency dimensions. The face-wise product is then performed between these latent factors with consistency dimensions. Empowered with this Art-product, we can readily deconstruct and reconstruct new tensor network decomposition from an interaction perspective. As a representative example, we redesign the tensor train decomposition which can benefit from the advantage of the Art-product. Extensive experiments on multi-spectral images, color videos, and light field data sustain the superiority of tensor train decomposition equipped with Art-product over classic tensor decomposition.