  In this paper, we investigate the statistical convergence rate of a Bayesian
low-rank tensor estimator. Our problem setting is the regression problem where
a tensor structure underlying the data is estimated. This problem setting
occurs in many practical applications, such as collaborative filtering,
multi-task learning, and spatio-temporal data analysis. The convergence rate is
analyzed in terms of both in-sample and out-of-sample predictive accuracies. It
is shown that a near optimal rate is achieved without any strong convexity of
the observation. Moreover, we show that the method has adaptivity to the
unknown rank of the true tensor, that is, the near optimal rate depending on
the true rank is achieved even if it is not known a priori.
