Alternating linear scheme in a Bayesian framework for low-rank tensor approximation

TL;DR

This paper introduces a Bayesian framework for low-rank tensor approximation that interprets the alternating linear scheme as a probabilistic inference process, enabling noise consideration, prior incorporation, and uncertainty quantification.

Contribution

It develops a novel Bayesian approach to tensor decomposition, providing a probabilistic interpretation and an algorithm using the unscented transform in tensor train format.

Findings

Probabilistic interpretation of ALS algorithm

Incorporation of measurement noise and prior knowledge

Uncertainty quantification of tensor estimates

Abstract

Multiway data often naturally occurs in a tensorial format which can be approximately represented by a low-rank tensor decomposition. This is useful because complexity can be significantly reduced and the treatment of large-scale data sets can be facilitated. In this paper, we find a low-rank representation for a given tensor by solving a Bayesian inference problem. This is achieved by dividing the overall inference problem into sub-problems where we sequentially infer the posterior distribution of one tensor decomposition component at a time. This leads to a probabilistic interpretation of the well-known iterative algorithm alternating linear scheme (ALS). In this way, the consideration of measurement noise is enabled, as well as the incorporation of application-specific prior knowledge and the uncertainty quantification of the low-rank tensor estimate. To compute the low-rank tensor estimate from the posterior distributions of the tensor decomposition components, we present an algorithm that performs the unscented transform in tensor train format.