Product Interaction: An Algebraic Formalism for Deep Learning Architectures

In this paper, we introduce product interactions, an algebraic formalism in which neural network layers are constructed from compositions of a multiplication operator defined over suitable algebras. Product interactions provide a principled way to generate and organize algebraic expressions by increasing interaction order. Our central observation is that algebraic expressions in modern neural networks admit a unified construction in terms of linear, quadratic, and higher-order product interactions. Convolutional and equivariant networks arise as symmetry-constrained linear product interactions, while attention and Mamba correspond to higher-order product interactions.