v-PuNNs: van der Put Neural Networks for Transparent Ultrametric Representation Learning

TL;DR

v-PuNNs are a novel neural network architecture that models hierarchical data using p-adic numbers, providing exact subtree semantics and state-of-the-art results on benchmarks with ultrametric properties.

Contribution

This paper introduces v-PuNNs, the first neural network architecture based on p-adic numbers for hierarchical data representation, along with a universal approximation theorem and new optimization methods.

Findings

Achieved 99.96% accuracy on WordNet noun hierarchy

Learned perfectly ultrametric metrics with zero triangle violations

Set new state-of-the-art results on multiple hierarchical benchmarks

Abstract

Conventional deep learning models embed data in Euclidean space $\mathbb{R}^d$, a poor fit for strictly hierarchical objects such as taxa, word senses, or file systems. We introduce van der Put Neural Networks (v-PuNNs), the first architecture whose neurons are characteristic functions of p-adic balls in $\mathbb{Z}_p$. Under our Transparent Ultrametric Representation Learning (TURL) principle every weight is itself a p-adic number, giving exact subtree semantics. A new Finite Hierarchical Approximation Theorem shows that a depth-K v-PuNN with $\sum_{j=0}^{K-1}p^{\,j}$ neurons universally represents any K-level tree. Because gradients vanish in this discrete space, we propose Valuation-Adaptive Perturbation Optimization (VAPO), with a fast deterministic variant (HiPaN-DS) and a moment-based one (HiPaN / Adam-VAPO). On three canonical benchmarks our CPU-only implementation sets new state-of-the-art: WordNet nouns (52,427 leaves) 99.96% leaf accuracy in 16 min; GO molecular-function 96.9% leaf / 100% root in 50 s; NCBI Mammalia Spearman $\rho = -0.96$ with true taxonomic distance. The learned metric is perfectly ultrametric (zero triangle violations), and its fractal and information-theoretic properties are analyzed. Beyond classification we derive structural invariants for quantum systems (HiPaQ) and controllable generative codes for tabular data (Tab-HiPaN). v-PuNNs therefore bridge number theory and deep learning, offering exact, interpretable, and efficient models for hierarchical data.