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$p$-adic Character Neural Network

Tomoki Mihara

Abstract

We propose a new frame work of $p$-adic neural network. Unlike the original $p$-adic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi using a family of characteristic functions indexed by hyperparameters of precision as activation functions, we use a single injective $p$-adic character on the topological Abelian group $\mathbb{Z}_p$ of $p$-adic integers as an activation function. We prove the $p$-adic universal approximation theorem for this formulation of $p$-adic neural network, and reduce it to the feasibility problem of polynomial equations over the finite ring of integers modulo a power of $p$.

$p$-adic Character Neural Network

Abstract

We propose a new frame work of -adic neural network. Unlike the original -adic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi using a family of characteristic functions indexed by hyperparameters of precision as activation functions, we use a single injective -adic character on the topological Abelian group of -adic integers as an activation function. We prove the -adic universal approximation theorem for this formulation of -adic neural network, and reduce it to the feasibility problem of polynomial equations over the finite ring of integers modulo a power of .

Paper Structure

This paper contains 5 sections, 2 theorems, 19 equations, 6 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Convention
  3. $p$-adic Character
  4. Universal Approximation
  5. Formulation

Key Result

Proposition 2.3

Let $\chi$ be a $p$-adic character on $\mathbb{Z}_p$. The the following are equivalent:

Figures (6)

  • Figure : Computation of $p$-adic valuation and $p$-coprime part of $x \in \mathbb{Z}_p$
  • Figure : Preprocessing the computation of the $p$-adic valuations of factorials and $p$-coprime parts of them modulo $p^E$ up to $N!$
  • Figure : Computation of $a^x$ modulo $p^E$ for $a \in (1 + p \mathbb{Z}_p) \cap \mathbb{N}_{< p^E})$ and $x \in \mathbb{N}_{< p^{E-1}}$ using the binomial expansion
  • Figure : Computation of $\exp_p(qx)$ modulo $p^E$ for $a \in (1 + p \mathbb{Z}_p) \cap \mathbb{N}_{< p^E})$ and $x \in \mathbb{N}_{< p^{E-m}}$ using the Taylor expansion
  • Figure : Computation of $a^x$ modulo $p^E$ for $a \in (1 + p \mathbb{Z}_p) \cap \mathbb{N}_{< p^E})$ and $x \in \mathbb{N}_{< p^{E-1}}$ using binary modular method
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1: Universal approximation theorem for a $p$-adic character
  • proof