$p$-adic Character Neural Network
This work addresses a theoretical problem in mathematical machine learning for researchers interested in p-adic methods, but it appears incremental as it modifies an existing p-adic neural network framework.
The authors tackled the problem of constructing a p-adic neural network by proposing a new framework that uses a single injective p-adic character as the activation function, proving a p-adic universal approximation theorem and reducing it to a feasibility problem over finite rings.
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 $p$.