$p$-Adic Polynomial Regression as Alternative to Neural Network for Approximating $p$-Adic Functions of Many Variables

TL;DR

The paper introduces a $p$-adic polynomial regression framework as a practical alternative to neural networks for approximating $p$-adic-valued functions of many variables. By leveraging Mahler-type representations, it shows that continuous functions on $\mathbb{Z}_p^n$ can be expressed as finite superpositions of univariate basis functions $\omega_k(x)=\binom{x}{k}$ through a $p$-adic embedding $\sum_{i=1}^n p^{i-1}\phi(x_i)$, enabling a multivariate regression form. It then proposes a concrete model $f(\boldsymbol{x},\boldsymbol{w})=\sum_{k=0}^{K} w_k\,\omega_k\left(\sum_{i=1}^{n} p^{i-1}\phi(x_i)\right)$ with a loss $L(\boldsymbol{w})=\dfrac{1}{N}\sum_{a=1}^{N} |l_a(\boldsymbol{w})|_p$ and a training protocol based on train/validation splitting. Owing to the non-Archimedean nature of the loss, the authors advocate a stochastic, non-Markovian optimization scheme with updates $\boldsymbol{w}_{i+1}=\boldsymbol{w}_i+\boldsymbol{\xi}_i(\boldsymbol{w}_i,\beta_i)$ that can guarantee a strict decrease in expectation, offering a principled path to training without standard gradient methods. The work frames a scalable, controllable alternative to neural networks for $p$-adic function approximation with potential relevance to $p$-adic physics and analysis.

Abstract

A method for approximating continuous functions $\mathbb{Z}_{p}^{n}\rightarrow\mathbb{Z}_{p}$ by a linear superposition of continuous functions $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is presented and a polynomial regression model is constructed that allows approximating such functions with any degree of accuracy. A physical interpretation of such a model is given and possible methods for its training are discussed. The proposed model can be considered as a simple alternative to possible $p$-adic models based on neural network architecture.