用数论方法构建可解释的层次数据模型，精准表示树状结构。
v-PuNNs: van der Put Neural Networks for Transparent Ultrametric Representation Learning
- 神经元基于p进制球集设计，权重直接为p进制数，保证子树语义精确
- 在3个基准上达到99.96%叶子准确率，50秒内完成大规模分类任务
- 适合需要可解释性与层级结构建模的场景，如生物分类、知识图谱
传统深度学习模型将数据嵌入欧几里得空间 ℝᵈ，难以刻画严格的层次结构（如分类学、词义、文件系统）。本文提出范德普特神经网络（v-PuNNs），其神经元是 ℤₚ 中 p-进制球的特征函数。根据透明超度量表征学习（TURL）原则，所有权重均为 p-进制数，实现精确的子树语义表达。新提出的有限层次近似定理表明，深度 K 的 v-PuNN 仅需 ∑_{j=0}^{K−1} p^j 个神经元即可通用表示任意 K 层树结构。由于该离散空间梯度消失，提出估值自适应扰动优化（VAPO），包含快速确定性变体（HiPaN-DS）和基于矩估计的版本（HiPaN / Adam-VAPO）。在三个基准测试中，纯CPU实现创下新纪录：WordNet名词（52,427个叶节点）达99.96%叶准确率，耗时16分钟；GO分子功能分类达96.9%叶准确率/100%根准确率，耗时50秒；NCBI哺乳动物分类的斯皮尔曼相关系数 ρ = -0.96，接近真实分类距离。所学度量完全满足超度量性质（三角不等式违反为零），并分析其分形与信息论特性。除分类外，还推导出量子系统的结构不变量（HiPaQ）和可控生成的表格数据编码（Tab-HiPaN）。v-PuNNs因此连接数论与深度学习，为层次数据提供精确、可解释且高效的建模范式。
原文摘要 · Abstract (English)
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.
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