UCSC-SOE-20-09: Bayesian Methods for Tensor Regressions

Rajarshi Guhaniyogi
09/14/2020 05:12 AM
For many applications pertaining to neuroimaging, social science, international relations, chemometrics, genomics and molecular-omics, datasets often involve variables
which are best represented in the form of a multi-dimensional array or tensor, which
extends the familiar two-way data matrix into higher dimensions. Rather than vectorizing tensor-valued variables prior to analysis which results in loss of inference, new
methods have emerged developing regression relationships between variables with either tensor-valued response(s) or predictor(s). Bayesian approaches, in particular, have
shown great promise in applications pertaining to tensor regressions. A remarkable feature of fully Bayesian approaches is that they allow
flexible modeling of tensor-valued
parameters in the regressions involving tensor variables and naturally offer characterization of uncertainty in the parametric and predictive inferences. This paper provides
a review of some relevant Bayesian models on tensor regressions developed in recent
years. We divide methods according to the objective of the analysis. We begin with
tensor regression approaches with a scalar response and a tensor-valued covariate, discuss both parametric and nonparametric modeling options and applications in this
framework. We then address the problem of making inference with a tensor response
and a vector of covariates, with applications including task related brain activation
and connectivity studies. Finally, we offer discussion on Bayesian models involving a
tensor response and a tensor covariate. Discussion of each model is accompanied by
available results on its posterior contraction properties, laying out restrictions on key
model parameters (such as the tensor dimensions) to draw accurate posterior inference.