Convergence rate of Bayesian supervised tensor modeling with multiway shrinkage priors

Abstract

This article studies the convergence rate of the posterior for Bayesian low rank supervised tensor modeling with multiway shrinkage priors. Multiway shrinkage priors constitute a new class of shrinkage prior distributions for tensor parameters in Bayesian low rank supervised tensor modeling to regress a scalar response on a tensor predictor with the primary aim to identify cells in the tensor predictor which are predictive of the scalar response. This novel and computationally efficient framework stems from pressing needs in many applications, including functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI). This article shows that the convergence rate is nearly optimal in terms of in-sample predictive accuracy of the Bayesian supervised low rank tensor model with a multiway shrinkage prior distribution when the number of observations grows. The conditions under which this nearly optimal convergence rate is achieved are seen to be very mild. More importantly, the rate is achieved for an easily computable method, even when the true CP/PARAFAC rank of the tensor coefficient corresponding to the tensor predictor is unknown.