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

arXiv: https://arxiv.org/abs/2012.11228 · HTML: https://mlsanity.com/paper/2012.11228/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Dec 2020.

## Built on by (most cited first, 3 of 3)

- [RecFusion: A Binomial Diffusion Process for 1D Data for Recommendation](https://mlsanity.com/paper/2306.08947/) (2023)
- [Projecting basis functions with tensor networks for Gaussian process regression](https://mlsanity.com/paper/2310.20630/) (2023)
- [Variational Bayesian Inference for Tensor Robust Principal Component Analysis](https://mlsanity.com/paper/2412.18717/) (2024)

## Abstract (arXiv)

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.