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arXiv · 2602.02573

Product Interaction: An Algebraic Formalism for Deep Learning Architectures

Haonan Dong·Chun-Wun Cheng·Angelica I. Aviles-Rivero
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Abstract

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.

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Haonan Dong, Chun-Wun Cheng, Angelica I. Aviles-Rivero. 2026-01-31. Product Interaction: An Algebraic Formalism for Deep Learning Architectures. https://arxiv.org/abs/2602.02573

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