Alternating linear scheme in a Bayesian framework for low-rank tensor approximation
This work provides a probabilistic interpretation of the alternating linear scheme (ALS) for low-rank tensor approximation, which is beneficial for researchers and practitioners working with multiway data by enabling uncertainty quantification and noise consideration.
The paper tackles the problem of finding a low-rank representation for a given tensor by solving a Bayesian inference problem. This approach allows for the consideration of measurement noise, incorporation of prior knowledge, and uncertainty quantification of the low-rank tensor estimate.
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.