$p$-adic Character Neural Network

TL;DR

This paper introduces a novel $p$-adic neural network framework using a single $p$-adic character as an activation function, proving a universal approximation theorem and linking it to polynomial equations over finite rings.

Contribution

It presents a new $p$-adic neural network model with a single activation function and establishes its universal approximation capabilities.

Findings

Proves the $p$-adic universal approximation theorem.

Reduces the approximation problem to polynomial equations over finite rings.

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 $p$.