  We describe a basic correspondence between linear algebraic structures within
vector embeddings in artificial neural networks and conditional independence
constraints on the probability distributions modeled by these networks. Our
framework aims to shed light on the emergence of structural patterns in data
representations, a phenomenon widely acknowledged but arguably still lacking a
solid formal grounding. Specifically, we introduce a characterization of
compositional structures in terms of "interaction decompositions," and we
establish necessary and sufficient conditions for the presence of such
structures within the representations of a model.
