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

TL;DR

The paper argues that hierarchical data are naturally ultrametric and introduces v-PuNNs, neural networks whose neurons are $p$-adic ball indicators in $\mathbb{Z}_p$, with all weights as $p$-adic numbers under the Transparent Ultrametric Representation Learning (TURL) principle. A Finite Hierarchical Approximation Theorem proves that depth-$K$ v-PuNNs with $N=(p^{K}-1)/(p-1)$ coefficients universally represent any $K$-level tree, enabling exact, interpretable, and efficient modeling of hierarchies. Training is facilitated by Valuation-Adaptive Perturbation Optimization (VAPO), a derivative-free optimization framework that operates directly in the $p$-adic space, and its Adam-based variant, both achieving CPU-based state-of-the-art results on WordNet nouns, Gene Ontology, and NCBI Mammalia taxonomy with zero ultrametric violations. Beyond classification, HiPaQ and Tab-HiPaN demonstrate the framework’s versatility for symbolic invariants in physics/ algebra and controllable generation in tabular data, respectively, highlighting the potential of $p$-adic reasoning as a practical tool for hierarchical reasoning and scientific workflows.

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 $ρ= -0.96$ 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.

Figures (19)

• Figure 1: Force‑directed Euclidean layout of the full WordNet noun hierarchy. Its visual clutter motivates an ultrametric treatment.
• Figure 2: Conceptual landscape of geometric spaces for hierarchical data embedding. v-PuNNs leverage ultrametric spaces for exact tree geometry matching, while Euclidean and hyperbolic approaches provide continuous approximations. Dashed arrows indicate model families associated with each space.
• Figure 3: A depth-3 subtree (shaded) forms a p-adic ball $B\!=B(a,p^{-3})$. The digits $(a_0,a_1,a_2)$ are the successive sibling indices along the root-leaf path and serve as the $p$-adic prefix $\operatorname{pref}_3(a)$gouvea1997p.
• Figure 4: HiPaN architecture. Input $x \in \mathbb{Z}_{p}$ activates characteristic functions of p-adic balls $B_k$, scaled by coefficients $c_{B_k} \in \mathbb{Q}_{p}$. Each coefficient feeds a specialized prediction head for a p-adic digit $d_k$. The digit outputs are combined through p-adic reconstruction: $\hat{x} = \sum_k d_k \cdot p^k$. Below: A depth-1 van der Put basis illustration ($p=5$). Each colored outline is the indicator of a radius-$p^{-1}$ ball; their weighted sum forms a piece-wise constant function.
• Figure 5: Dense, opaque weights (left) versus sparse, structurally grounded $p$-adic atoms (right). HiPaN replaces real weight edges with characteristic functions, learns coefficients in $\mathbb Q_p$, and uses a valuation-aware optimizer, yielding exact subtree attribution.

Theorems & Definitions (37)

• Definition 3.1: Hierarchical data
• Remark
• Remark : Relation to the classical van der Put basis
• Lemma 3.1: Isometry
• proof : Sketch
• Theorem 3.1: Parameter-Subtree Duality
• proof
• Theorem 3.2: van der Put, 1968 (classical form)
• proof
• Definition 4.1: Characteristic-ball neuron