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This research proposes a novel mathematical framework for deep neural networks using non-Archimedean analysis and tree-like architectures derived from local fields. The study demonstrates that these theoretically constructed models serve as robust universal approximators, expanding the theoretical boundaries of how DNN structures can be formulated beyond standard Euclidean spaces. While highly abstract, this work contributes directly to Machine Learning theory by exploring alternative algebraic foundations for neural network design.
Deep Neural Networks: A Formulation Via Non-Archimedean Analysis
arXiv:2402.00094v3 Announce Type: replace-cross Abstract: We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.
Abstract:We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.
Neural and Evolutionary Computing (cs.NE); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)
arXiv:2402.00094 [cs.NE]
(or arXiv:2402.00094v3 [cs.NE] for this version)
From: W. A. Zuniga-Galindo [view email] [v1] Wed, 31 Jan 2024 14:49:44 UTC (25 KB) [v2] Mon, 30 Mar 2026 13:43:44 UTC (26 KB) [v3] Mon, 15 Jun 2026 15:15:21 UTC (26 KB)
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