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Title: Tensor-on-tensor regression Authors:  Eric Lock - University of Minnesota (United States) [presenting]
Abstract: In neuroimaging analysis and other fields, both predictors and outcomes can take the form of a multi-way array (i.e., a tensor). We propose a framework for the linear prediction of a multi-way array from another multi-way array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar. We describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced CP-rank. We propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge ($L_2$) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. We illustrate the approach with an application to metabolite resonance spectroscopy data.