Bayes method for low rank tensor estimation

Abstract

We investigate the statistical convergence rate of a Bayesian low-rank tensor estimator, and construct a Bayesian nonlinear tensor estimator. The problem setting is the regression problem where the regression coefficient forms a tensor structure. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate of the Bayes tensor estimator is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a fast learning rate is achieved without any strong convexity of the observation. Moreover, we extend the tensor estimator to a nonlinear function estimator so that we estimate a function that is a tensor product of several functions.