# p-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences

arXiv: https://arxiv.org/abs/2606.06117 · HTML: https://mlsanity.com/paper/2606.06117/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 37 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Incorporating Domain Knowledge into Deep Neural Networks](https://mlsanity.com/paper/2103.00180/) (2021)
- [v-PuNNs: van der Put Neural Networks for Transparent Ultrametric Representation Learning](https://mlsanity.com/paper/2508.01010/) (2025)
- [Learning with the p-adics](https://mlsanity.com/paper/2512.22692/) (2025)
- [Biology-informed neural networks learn nonlinear representations from omics data to improve genomic prediction and interpretability](https://mlsanity.com/paper/2510.14970/) (2025)
- [BIRDNet: Mining and Encoding Boolean Implication Knowledge Graphs as Interpretable Deep Neural Networks](https://mlsanity.com/paper/2605.28739/) (2026)

## Abstract (arXiv)

We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines p-adic numbers with topological data analysis. Each DNA sequence is encoded along two complementary axes: a p-adic distance on k-mer prefixes, which captures hierarchical positional structure, and a compositional L1 distance on k-mer frequencies, which captures local sequence content. The two distances jointly parameterise a bi-filtered Vietoris--Rips complex, and per-sequence topological summaries from this bi-filtration serve as features for standard machine learning classifiers. We establish theoretical guarantees for the construction: stability under metric perturbations and invariance to the choice of prime, alongside a result that explains why a single p-adic axis is topologically uninformative and why the bi-filtration recovers nontrivial homology. On twelve genomic benchmarks (28 to 500 sequences, 3 to 7 classes), pVR outperforms four established alignment-free baselines on three of six low-sample datasets, with gains of up to 21 percentage points; it underperforms only on a SARS-CoV-2 variant benchmark whose point-mutation divergence violates the hierarchical assumption, and all methods saturate in the large-sample regime. pVR also outperforms zero-shot frozen embeddings from the 500M-parameter Nucleotide Transformer v2 by 6.7 to 11.4 percentage points on three low-sample benchmarks. The pVR codebase is publicly available at https://github.com/MAHI-Group/pVR.