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

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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 subproblems 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.

Original languageEnglish
Pages (from-to)A1116-A1144
JournalSIAM Journal on Scientific Computing
Issue number3
Publication statusPublished - 2022


  • alternating linear scheme
  • Bayesian inference
  • low-rank approximation
  • tensor decomposition
  • tensor train

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