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

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