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

• URL: http://arxiv.org/abs/2606.06117v1
• Date: Thu, 04 Jun 2026 13:05:36 GMT
• Title: $p$-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences
• Abstract summary: pVR is a machine learning framework for alignment-free genomic sequence classification.<n>It combines $p$-adic numbers with topological data analysis.
• Score: 1.0165640083594571
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $L_1$ 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.

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