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Deep Neural Networks: A Formulation Via Non-Archimedean Analysis

W. A. Zúñiga-Galindo

TL;DR

This work presents a rigorous framework for hierarchical deep neural networks built over non-Archimedean local fields, realized as finite tree-structured architectures on the ring of integers $\\mathcal{O}_{\\mathbb{K}}$. It develops a discrete DNN formalism using Bruhat-Schwartz test functions, establishes matrix- and backpropagation-based training, and proves robust universal approximation both on $\\mathcal{O}_{\\mathbb{K}}$ and on $[0,1]$ via a $p$-adic embedding and measure-preserving maps. The paper further shows parallelization and extension to open-compact subsets, and connects these networks to Fourier-Walsh analysis through additive characters, enabling harmonic-analytic representations of functions in $L^{\\rho}$ spaces. Collectively, the results fuse non-Archimedean arithmetic, hierarchical function representation, and standard optimization techniques to enable training of tree-like DNNs for hierarchical data and p-adic-inspired computations. The framework broadens the theoretical landscape for neural networks by leveraging $p$-adic structures and offers a pathway for fractal-like or hierarchical data processing in non-Archimedean settings with practical training via backpropagation.

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.

Deep Neural Networks: A Formulation Via Non-Archimedean Analysis

TL;DR

This work presents a rigorous framework for hierarchical deep neural networks built over non-Archimedean local fields, realized as finite tree-structured architectures on the ring of integers . It develops a discrete DNN formalism using Bruhat-Schwartz test functions, establishes matrix- and backpropagation-based training, and proves robust universal approximation both on and on via a -adic embedding and measure-preserving maps. The paper further shows parallelization and extension to open-compact subsets, and connects these networks to Fourier-Walsh analysis through additive characters, enabling harmonic-analytic representations of functions in spaces. Collectively, the results fuse non-Archimedean arithmetic, hierarchical function representation, and standard optimization techniques to enable training of tree-like DNNs for hierarchical data and p-adic-inspired computations. The framework broadens the theoretical landscape for neural networks by leveraging -adic structures and offers a pathway for fractal-like or hierarchical data processing in non-Archimedean settings with practical training via backpropagation.

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.
Paper Structure (19 sections, 12 theorems, 138 equations)

This paper contains 19 sections, 12 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Non-Archimedean local fields
  4. The space of test functions
  5. Non-Archimedean deep neural networks
  6. The spaces $\mathcal{D}^{l}(\mathcal{O}_{\mathbb{K}})$
  7. Positive characteristic versus zero characteristic
  8. Discrete non-Archimedean DNNs
  9. Implementation of discrete non-Archimedean deep neural networks
  10. Matrix form of the DNNs
  11. Stochastic gradient
  12. Back propagation
  13. The discrete non-Archimedean DNNs are robust universal approximators
  14. Parallelization of non-Archimedean DNNs
  15. Approximation of functions on open compact subsets
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\sigma_{M}$ be as before. Assume that $X\in\mathcal{D}^{l-1}\left( \mathcal{O}_{\mathbb{K}}\right)$, $w(\cdot,y)\in \mathcal{D}^{l}\left( \mathcal{O}_{\mathbb{K}}\right)$, $\theta \in\mathcal{D}^{l}\left( \mathcal{O}_{\mathbb{K}}\right)$, for some $l\geq2$. Then, all the continuous non-Archimed

Theorems & Definitions (28)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • ...and 18 more