Generalized infinite factorization models

Factorization models express a statistical object of interest in terms of a collection of simpler objects. For example, a matrix or tensor can be expressed as a sum of rank-one components. In practice, however, it can be challenging to infer the number of components and the relative impact of the different components. A popular idea is to include infinitely many components whose impact decreases with the component index. This article is motivated by two limitations of such existing methods: (i) lack of careful consideration of the within-component sparsity structure; and (ii) not accommodating grouped variables and other nonexchangeable structures. We propose a general class of infinite factorization models that address these limitations. Theoretical support is provided, practical gains are demonstrated in simulation studies, and an ecology application focusing on modelling bird species occurrence is discussed.