Tensor on tensor regression with tensor normal errors and tensor network states on the regression parameter

With the growing interest in tensor regression models and decompositions, the tensor normal distribution offers a flexible and intuitive way to model multi-way data and error dependence. In this paper we formulate two regression models where the responses and covariates are both tensors of any number of dimensions and the errors follow a tensor normal distribution. The first model uses a CANDECOMP/PARAFAC (CP) structure and the second model uses a Tensor Chain (TC) structure, and in both cases we derive Maximum Likelihood Estimators (MLEs) and their asymptotic distributions. Furthermore we formulate a tensor on tensor regression model with a Tucker structure on the regression parameter and estimate the parameters using least squares. Aditionally, we find the fisher information matrix of the covariance parameters in an independent sample of tensor normally distributed variables with mean 0, and show that this fisher information also applies to the covariances in the multilinear tensor regression model \cite{Hoff2014} and tensor on tensor models with tensor normal errors regardless of the structure on the regression parameter