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This paper proposes an algebraic framework that treats multiplication as a core architectural operation, aiming to unify convolutional, attention, state-space, and equivariant networks under one design principle. Its most concrete claims are that Mamba corresponds to a higher self-interaction order and that a simplified-algebra layer beats Mamba on CIFAR-10. Readers should weigh the unifying framing as a research hypothesis still needing broader benchmarks, though it may offer a useful lens for architecture design.
Product Interaction: An Algebraic Formalism for Deep Learning Architectures
arXiv:2602.02573v2 Announce Type: replace-cross Abstract: Modern neural architectures are largely developed as distinct empirical constructions, despite recurring principles across convolutional, attention, state-space, and equivariant models. We introduce Product Interactions, an algebraic formalism that unifies these architectures under a common construction principle and turns their design into a systematic procedure. The core shift is representing data as algebraic elements and treating multiplication, rather than linear transformation alone, as a primitive architectural operation. Architectures are generated by composing multiplication operators, with input dependence organized by self-interaction order. Within this formalism, we derive convolutional and equivariant networks as symmetry-constrained first-order product interactions, recover Mamba from state-space models as an increase in self-interaction order from one to three, and establish attention as a third-order interaction from composing the same operator. Equivariance is incorporated by constraining the algebraic product via symmetry: translation equivariance yields convolution, while SO(3)-compatibility produces the Clebsch--Gordan product and tensor-field interactions. This perspective recasts architecture design as an algebraic construction problem: the choice of algebra defines admissible interactions, symmetry constrains their structure, and composition generates higher-order architectures. The framework is thus generative. Following these principles, we construct a simplified-algebra dynamical layer that improves both accuracy and throughput over Mamba on CIFAR-10. Experiments show that interaction order affects accuracy non-monotonically and symmetry constraints boost learnability. Product Interactions provides a mathematical methodology for analyzing existing architectures and systematically designing new ones.
Abstract:Modern neural architectures are largely developed as distinct empirical constructions, despite recurring principles across convolutional, attention, state-space, and equivariant models. We introduce Product Interactions, an algebraic formalism that unifies these architectures under a common construction principle and turns their design into a systematic procedure. The core shift is representing data as algebraic elements and treating multiplication, rather than linear transformation alone, as a primitive architectural operation. Architectures are generated by composing multiplication operators, with input dependence organized by self-interaction order. Within this formalism, we derive convolutional and equivariant networks as symmetry-constrained first-order product interactions, recover Mamba from state-space models as an increase in self-interaction order from one to three, and establish attention as a third-order interaction from composing the same operator. Equivariance is incorporated by constraining the algebraic product via symmetry: translation equivariance yields convolution, while SO(3)-compatibility produces the Clebsch--Gordan product and tensor-field interactions. This perspective recasts architecture design as an algebraic construction problem: the choice of algebra defines admissible interactions, symmetry constrains their structure, and composition generates higher-order architectures. The framework is thus generative. Following these principles, we construct a simplified-algebra dynamical layer that improves both accuracy and throughput over Mamba on CIFAR-10. Experiments show that interaction order affects accuracy non-monotonically and symmetry constraints boost learnability. Product Interactions provides a mathematical methodology for analyzing existing architectures and systematically designing new ones.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2602.02573 [cs.LG]
(or arXiv:2602.02573v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2602.02573 Focus to learn more arXiv-issued DOI via DataCite
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