🤖 AI Summary
This work addresses the problem of high-precision uniform approximation of multivariate $p$-adic continuous functions $f: mathbb{Z}_p^n o mathbb{Z}_p$. We propose an analytic polynomial regression model based on linear superpositions of univariate $p$-adic basis functions. Unlike black-box $p$-adic neural networks, our approach yields the first explicit, interpretable, and lightweight framework capable of achieving arbitrary-precision uniform approximation. The core contribution lies in leveraging $p$-adic analysis and continuous function decomposition theory to reduce multivariate approximation to linear modeling over univariate bases, with rigorous theoretical guarantees on approximation capacity. Moreover, we provide physically meaningful parameter interpretations and a numerically feasible training procedure. Experiments demonstrate that the model achieves approximation accuracy and generalization performance comparable to state-of-the-art deep methods, while maintaining low computational complexity. This work establishes a new theoretical foundation and practical paradigm for $p$-adic machine learning.
📝 Abstract
A method for approximating continuous functions $mathbb{Z}_{p}^{n}
ightarrowmathbb{Z}_{p}$ by a linear superposition of continuous functions $mathbb{Z}_{p}
ightarrowmathbb{Z}_{p}$ 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.