We introduce Bayesian multi-tensor factorization, a model that is the first Bayesian formulation for joint factorization of multiple matrices and tensors. The research problem generalizes the joint matrix---tensor factorization problem to arbitrary sets of tensors of any depth, including matrices, can be interpreted as unsupervised multi-view learning from multiple data tensors, and can be generalized to relax the usual trilinear tensor factorization assumptions. The result is a factorization of the set of tensors into factors shared by any subsets of the tensors, and factors private to individual tensors. We demonstrate the performance against existing baselines in multiple tensor factorization tasks in structural toxicogenomics and functional neuroimaging.
When all the tensors have @math and are paired in the first mode, our framework reduces to the group factor analysis (GFA) problem presented by @cite_14 . GFA has been generalized to allow pairings between arbitrary data modes under the name collective matrix factorization (CMF) , which the formulation in generalizes to tensors.
@cite_14
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We introduce a factor analysis model that summarizes the dependencies between observed variable groups, instead of dependencies between individual variables as standard factor analysis does. A group may correspond to one view of the same set of objects, one of many data sets tied by co-occurrence, or a set of alternative variables collected from statistics tables to measure one property of interest. We show that by assuming group-wise sparse factors, active in a subset of the sets, the variation can be decomposed into factors explaining relationships between the sets and factors explaining away set-specific variation. We formulate the assumptions in a Bayesian model which provides the factors, and apply the model to two data analysis tasks, in neuroimaging and chemical systems biology.