{"ID":2888097,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.01010","arxiv_id":"2508.01010","title":"v-PuNNs: van der Put Neural Networks for Transparent Ultrametric Representation Learning","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.","short_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 $\\mathbb{Z}_p$...","url_abs":"https://arxiv.org/abs/2508.01010","url_pdf":"https://arxiv.org/pdf/2508.01010v2","authors":"[\"Gnankan Landry Regis N'guessan\"]","published":"2025-08-01T18:23:38Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.AI\"]","methods":"[]","has_code":false}