用多项式回归逼近高维p进函数，精度可控且无需神经网络
$p$-Adic Polynomial Regression as Alternative to Neural Network for Approximating $p$-Adic Functions of Many Variables
- 将高维p进函数分解为一维p进函数的线性叠加
- 任意精度逼近连续p进函数，理论保证可实现
- 适合对数论计算与代数结构感兴趣的研究者
本文提出一种方法，通过连续函数 $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ 的线性叠加来逼近连续函数 $\mathbb{Z}_{p}^{n}\rightarrow\mathbb{Z}_{p}$，并构建了多项式回归模型，可在任意精度下实现此类函数的逼近。给出了该模型的物理意义，并讨论了其训练方法。该模型可作为基于神经网络架构的p进模型的一种简单替代方案。
原文摘要 · Abstract (English)
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.
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