# Learning with the p-adics

arXiv: https://arxiv.org/abs/2512.22692 · HTML: https://mlsanity.com/paper/2512.22692/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 27 references that are in the map.
- First posted on arXiv 27 Dec 2025.

## Builds on (most cited first, 4 of 4)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Ultrametric Fitting by Gradient Descent](https://mlsanity.com/paper/1905.10566/) (2019)
- [On Efficient Low Distortion Ultrametric Embedding](https://mlsanity.com/paper/2008.06700/) (2020)
- [Number Theory Meets Linguistics: Modelling Noun Pluralisation Across 1497 Languages Using 2-adic Metrics](https://mlsanity.com/paper/2211.13124/) (2022)

## Built on by (most cited first, 4 of 4)

- [p-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences](https://mlsanity.com/paper/2606.06117/) (2026)
- [Non-Archimedean Polydisc Spaces and Applications to Optimisation](https://mlsanity.com/paper/2606.07782/) (2026)
- [Signed p-adic Residual Encodings of Finite-Domain All-Different Systems with a Sudoku Case Study](https://mlsanity.com/paper/2609.16063/) (2026)
- [Continuous Optimization for p-adic Models](https://mlsanity.com/paper/2609.25501/) (2026)

## Abstract (arXiv)

Existing machine learning frameworks operate over the field of real numbers (ℝ) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., ℝd). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization. But is this the only possible choice? In this paper, we study the suitability of a radically different field as an alternative to ℝ -- the ultrametric and non-archimedean space of p-adic numbers, mathbbQp. The hierarchical structure of the p-adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the p-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact p-adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.