Bayesian factorizations of big sparse tensors

TL;DR

This paper introduces a Bayesian tensor factorization method that adapts to sparse, high-dimensional data, improving upon traditional PARAFAC by allowing variable effective ranks across tensor dimensions.

Contribution

It develops a Bayesian approach with priors and an efficient Gibbs sampler for tensor factorization, addressing limitations of existing methods in sparse, large-scale data.

Findings

Posterior concentration rates are established for high-dimensional tensors.

The method outperforms traditional PARAFAC in simulations.

Real data applications demonstrate practical effectiveness.

Abstract

It has become routine to collect data that are structured as multiway arrays (tensors). There is an enormous literature on low rank and sparse matrix factorizations, but limited consideration of extensions to the tensor case in statistics. The most common low rank tensor factorization relies on parallel factor analysis (PARAFAC), which expresses a rank $k$ tensor as a sum of rank one tensors. When observations are only available for a tiny subset of the cells of a big tensor, the low rank assumption is not sufficient and PARAFAC has poor performance. We induce an additional layer of dimension reduction by allowing the effective rank to vary across dimensions of the table. For concreteness, we focus on a contingency table application. Taking a Bayesian approach, we place priors on terms in the factorization and develop an efficient Gibbs sampler for posterior computation. Theory is provided showing posterior concentration rates in high-dimensional settings, and the methods are shown to have excellent performance in simulations and several real data applications.