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

arXiv: https://arxiv.org/abs/2508.01010 · HTML: https://mlsanity.com/paper/2508.01010/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 40 references that are in the map.
- First posted on arXiv 1 Aug 2025.

## Builds on (most cited first, 11 of 11)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Improved Semantic Representations From Tree-Structured Long Short-Term Memory Networks](https://mlsanity.com/paper/1503.00075/) (2015)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Hierarchical Transformers Are More Efficient Language Models](https://mlsanity.com/paper/2110.13711/) (2021)
- [Efficient softmax approximation for GPUs](https://mlsanity.com/paper/1609.04309/) (2016)
- [Graph Transformers: A Survey](https://mlsanity.com/paper/2407.09777/) (2024)
- [View-Consistent Hierarchical 3D Segmentation Using Ultrametric Feature Fields](https://mlsanity.com/paper/2405.19678/) (2024)
- [Hierarchical Neural Networks, p-Adic PDEs, and Applications to Image Processing](https://mlsanity.com/paper/2406.07790/) (2024)

## Built on by (most cited first, 5 of 5)

- [p-adic Character Neural Network](https://mlsanity.com/paper/2603.29905/) (2026)
- [p-adic Manifold Learning and Benchmark Tasks from Impartial Games](https://mlsanity.com/paper/2605.04374/) (2026)
- [p-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences](https://mlsanity.com/paper/2606.06117/) (2026)
- [On Hamming-Lipschitz Type Stability of the Subdominant (Minmax) Ultrametric: Theory and Simple Proofs](https://mlsanity.com/paper/2608.04014/) (2026)
- [Continuous Optimization for p-adic Models](https://mlsanity.com/paper/2609.25501/) (2026)

## Abstract (arXiv)

Conventional deep learning models embed data in Euclidean space ℝd, 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 mathbbZp. 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 sumj=0K-1p,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.