  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. However, in practice, it can be challenging to
infer the relative impact of the different components as well as the number of
components. A popular idea is to include infinitely many components having
impact decreasing with the component index. This article is motivated by two
limitations of existing methods: (1) lack of careful consideration of the
within component sparsity structure; and (2) no accommodation for grouped
variables and other non-exchangeable structures. We propose a general class of
infinite factorization models that address these limitations. Theoretical
support is provided, practical gains are shown in simulation studies, and an
ecology application focusing on modelling bird species occurrence is discussed.

Methodology