  It is well known that tensor network regression models operate on an
exponentially large feature space, but questions remain as to how effectively
they are able to utilize this space. Using a polynomial featurization, we
propose the interaction decomposition as a tool that can assess the relative
importance of different regressors as a function of their polynomial degree. We
apply this decomposition to tensor ring and tree tensor network models trained
on the MNIST and Fashion MNIST datasets, and find that up to 75% of interaction
degrees are contributing meaningfully to these models. We also introduce a new
type of tensor network model that is explicitly trained on only a small subset
of interaction degrees, and find that these models are able to match or even
outperform the full models using only a fraction of the exponential feature
space. This suggests that standard tensor network models utilize their
polynomial regressors in an inefficient manner, with the lower degree terms
being vastly under-utilized.
