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arxiv: 2606.06117 · v1 · pith:PFD4RTPHnew · submitted 2026-06-04 · 🧬 q-bio.QM · cs.LG· math.AT· q-bio.GN

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

Tirtharaj Dash , Gunja Sachdeva This is my paper

Pith reviewed 2026-06-27 22:44 UTC · model grok-4.3

classification 🧬 q-bio.QM cs.LGmath.ATq-bio.GN
keywords p-adic numberstopological data analysisgenomic sequence classificationalignment-free methodsVietoris-Rips complexbi-filtrationmachine learningk-mer analysis
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The pith

pVR encodes genomic sequences via a bi-filtration of p-adic prefix distances and k-mer frequency distances to extract topological features for classification.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents pVR as an alignment-free method that represents each DNA sequence by two distances: a p-adic distance on ordered k-mer prefixes that encodes hierarchical positional information and an L1 distance on k-mer counts that encodes compositional content. These distances together define a bi-filtered Vietoris-Rips complex whose persistent homology summaries become input features for ordinary classifiers. Theoretical results prove stability of the summaries under small metric changes and invariance to the choice of prime, while showing that a single p-adic filtration produces trivial topology. On twelve genomic benchmarks containing 28 to 500 sequences, the resulting features raise accuracy over four standard alignment-free baselines on three of six low-sample tasks and exceed zero-shot embeddings from a 500-million-parameter transformer on three such tasks.

Core claim

pVR jointly parameterizes a bi-filtered Vietoris-Rips complex by a p-adic metric on k-mer prefixes and an L1 metric on k-mer frequencies; the persistent homology of this bi-filtration yields stable, prime-invariant topological summaries that serve as features improving classification accuracy on low-sample genomic datasets relative to alignment-free baselines.

What carries the argument

bi-filtered Vietoris-Rips complex jointly parameterized by p-adic distance on k-mer prefixes and L1 distance on k-mer frequencies

If this is right

  • Outperforms four alignment-free baselines on three of six low-sample datasets by as much as 21 percentage points.
  • 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 topological summaries remain stable under metric perturbations and invariant to the choice of prime p.
  • A single p-adic axis produces uninformative homology while the bi-filtration recovers nontrivial features.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bi-filtration construction could be tested on other ordered sequential data such as protein sequences or parsed text where prefix hierarchies are present.
  • Accuracy may remain limited on problems with high rates of isolated substitutions, suggesting a need for hybrid filtrations that also capture pointwise differences.
  • In regimes where all methods saturate, the topological features might still reduce variance when combined with sequence kernels or learned embeddings.

Load-bearing premise

The p-adic distance on k-mer prefixes captures meaningful hierarchical positional structure in the genomic sequences.

What would settle it

Performance gains disappear or reverse on additional genomic datasets dominated by point-mutation divergence rather than hierarchical prefix structure, such as further SARS-CoV-2 variant collections.

Figures

Figures reproduced from arXiv: 2606.06117 by Gunja Sachdeva, Tirtharaj Dash.

Figure 1
Figure 1. Figure 1: Three snapshots of a Vietoris–Rips filtration on four points [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Proposition 8 on the five-point configuration {a, b, c, d, e} at (ϵp, ϵc) = (1, 0.4), where e is compositionally close to every point but p-adically far. (a) dp-only: tetrahedron on {a, b, c, d} with e isolated (β1 = 0). (b) dc-only: the 4-cycle a–c–b–d–a is filled through e (β1 = 0). (c) Bi-filtration: e’s edges are excluded, leaving the 4-cycle unfilled (β1 = 1). The cycle is invisible to… view at source ↗
Figure 3
Figure 3. Figure 3: The pVR pipeline. Each sequence passes through two branches: a p-adic (hierarchical) branch gives the distance matrix Dp, and a compositional (L1) branch gives Dc. The two matrices parameterise a bi-filtered Vietoris–Rips complex, from which per-sequence degree profiles are extracted; these, with the p-adic histograms and k-mer frequencies, form the feature vector for a standard classifier. The ∩ symbol de… view at source ↗
Figure 4
Figure 4. Figure 4: Cosine-UMAP projections of pVR features (left) and Nucleotide Transformer v2 zero-shot embeddings (right) on Influenza HA-small (N = 59, four subtypes). pVR produces visibly subtype-separated clus￾ters; NT v2 embeddings show only weak separation. Equivalent figures for the other two benchmarks are included within our code repository. Ablation. We classify with one feature group at a time, using XGBoost thr… view at source ↗
Figure 5
Figure 5. Figure 5: Distance matrices (left) and Betti heatmaps (right) for two low-sample datasets. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces pVR, a topological machine learning framework for alignment-free genomic sequence classification. It encodes DNA sequences using a p-adic distance on k-mer prefixes (capturing hierarchical positional structure) and an L1 distance on k-mer frequencies (capturing compositional content), jointly parameterizing a bi-filtered Vietoris-Rips complex whose topological summaries serve as features for standard classifiers. Theoretical guarantees are claimed for stability under metric perturbations, invariance to prime choice, and the necessity of the bi-filtration (single p-adic axis being topologically trivial). On twelve genomic benchmarks (28-500 sequences, 3-7 classes), pVR outperforms four alignment-free baselines on three of six low-sample datasets (gains up to 21 pp), underperforms on a SARS-CoV-2 variant set violating the hierarchical assumption, saturates with larger samples, and beats zero-shot 500M-parameter Nucleotide Transformer embeddings by 6.7-11.4 pp on three low-sample tasks. Code is publicly available.

Significance. If the results hold, the work offers a novel integration of p-adic valuations into TDA for genomics that exploits hierarchical structure in sequences, with demonstrated utility in low-sample regimes where it surpasses both classical alignment-free methods and large frozen embeddings. The public codebase is a clear strength, enabling direct verification of the bi-filtration construction and empirical claims.

minor comments (3)
  1. The abstract and results section should explicitly reference the specific theorems or propositions establishing stability, prime invariance, and single-axis triviality (currently only asserted).
  2. Table or figure reporting the per-benchmark accuracies should include standard deviations or statistical significance tests to support the 'up to 21 percentage points' gains.
  3. The k-mer length k and filtration parameters are listed as free parameters; a brief sensitivity analysis or default selection protocol would strengthen the reproducibility claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed summary of our work and the positive evaluation of its significance, particularly regarding the integration of p-adic valuations with TDA for low-sample genomic classification and the public codebase. The recommendation for minor revision is noted. No major comments were provided in the report, so we have no points requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a new p-adic distance on k-mer prefixes and an L1 distance on frequencies, jointly parameterizing a bi-filtered Vietoris-Rips complex whose topological summaries are used as features for off-the-shelf classifiers. Theoretical claims (stability, prime invariance, single-axis triviality) are stated as results of the construction itself rather than fitted or renamed inputs. No equations reduce a claimed prediction to a parameter fit by construction, and no load-bearing premise rests on self-citation chains. Public code supplies an independent verification path. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The approach builds on established mathematical structures from number theory and topology, with the main addition being the bi-filtration parameterized by the two distances.

free parameters (2)
  • k-mer length k
    Parameter for encoding sequences into k-mers, likely selected based on data.
  • filtration parameters
    Parameters for the bi-filtration construction.
axioms (2)
  • standard math p-adic numbers form a metric space suitable for hierarchical structures
    Invoked for the p-adic distance on k-mer prefixes.
  • standard math Vietoris-Rips complexes are stable under small perturbations of the metric
    Basis for the stability guarantee.
invented entities (1)
  • p-adic bi-filtration no independent evidence
    purpose: To jointly capture hierarchical positional and compositional information in sequences via two distances
    Newly introduced construction in the paper to address limitations of single-axis filtrations.

pith-pipeline@v0.9.1-grok · 5825 in / 1470 out tokens · 32183 ms · 2026-06-27T22:44:36.903938+00:00 · methodology

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