Structured and Interpretable Learning via Diophantine-Elliptic Neural Networks

TMLR Paper5029 Authors

Abstract: We introduce Diophantine-Elliptic Curve Neural Networks (DEC-NNs), a novel class of architectures in which parameters are not unconstrained real numbers but integer-valued solutions to a fixed elliptic Diophantine equation. This constraint embeds each weight and bias into an algebraically structured arithmetic variety, yielding neural models that are interpretable, sparse, and geometrically robust by design. Our formulation enforces this structure through a projection-based training loop, ensuring consistency across updates without sacrificing predictive performance. We establish theoretical guarantees on convergence, symbolic expressivity, and generalization bounds rooted in number theory. Empirically, DEC-NNs demonstrate high accuracy and resilience under adversarial noise on both synthetic and real-world datasets including MNIST and UCI Breast Cancer. In domains such as scientific modeling, symbolic regression, and medical diagnostics, where transparency and auditability are essential, DEC-NNs offer a principled alternative to conventional networks, aligning learning with discrete symbolic structure rather than post hoc interpretability.

Submission Length: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Jeffrey_Pennington1

Submission Number: 5029