Convergence rate of Bayesian tensor estimator: Optimal rate without restricted strong convexity

TL;DR

This paper establishes the near optimal statistical convergence rate of a Bayesian low-rank tensor estimator in regression tasks, without requiring strong convexity, and demonstrates adaptivity to unknown tensor rank.

Contribution

It provides the first analysis showing near optimal convergence rates for Bayesian tensor estimation without strong convexity assumptions, with adaptivity to unknown rank.

Findings

Achieves near optimal convergence rate without strong convexity

Demonstrates adaptivity to unknown tensor rank

Applicable to practical problems like collaborative filtering and spatio-temporal data analysis

Abstract

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.