$p$-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences
For researchers in genomic sequence classification, pVR offers a novel alignment-free method that outperforms existing baselines on low-sample datasets, though it underperforms on datasets with point-mutation divergence and saturates in large-sample regimes.
The paper introduces pVR, a topological machine learning framework that combines p-adic numbers with topological data analysis for alignment-free genomic sequence classification. On twelve genomic benchmarks, pVR outperforms four established baselines on three of six low-sample datasets with gains up to 21 percentage points, and outperforms a 500M-parameter transformer model by 6.7 to 11.4 percentage points on three low-sample benchmarks.
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.