The article introduces a new non‑archimedean metric, the cophenetic metric, defined on persistent homology classes of all degrees. It demonstrates that zeroth persistent homology combined with this metric and various hierarchical clustering algorithms yields statistically verifiable, commensurate topological information on multiple datasets. The resulting clusters, evaluated by silhouette score and Rand index, perform well, and the metric enables visualization of inter‑relations among persistent homology classes across all degrees via rooted trees.
arXiv:2506. 15020v2 Announce Type: replace-cross Abstract: We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods.
arXiv:2512.23348v3 Announce Type: replace-cross Abstract: We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construc...
arXiv:2505. 04346v2 Announce Type: replace Abstract: Clustering aims at partitioning data points into groups of similar objects without knowing about the class labels.
arXiv:2503. 03156v4 Announce Type: replace-cross Abstract: We propose DiRe, a force-directed dimensionality reduction framework designed to preserve global structure and homological features while remaining practical on modern hardware.
The paper introduces a unified pipeline that classifies univariate time series by first converting them into graphs using one of five constructions from three families (visibility, transition, proximity). The resulting graph is turned into a dissimilarity matrix, from which a Vietoris–Rips filtration produces persistence diagrams that are vectorized via persistence landscapes and topological summary statistics. Experiments on twelve UCR benchmarks reveal that no single graph construction dominates, diffusion distance consistently outperforms shortest-path metrics, and persistence-based features remain robust to noise.
arXiv:2512. 16558v3 Announce Type: replace Abstract: Clustering is a cornerstone of modern data analysis.
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.
arXiv:2606. 06117v1 Announce Type: cross Abstract: We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines $p$-adic numbers with topological data analysis.
arXiv:2609. 31469v1 Announce Type: cross Abstract: Shapley values, a solution concept from cooperative game theory, have recently become a standard tool for feature credit allocation in machine learning.
The paper demonstrates that persistent homology, a tool from topological data analysis, can objectively and interpretable distinguish artistic styles. It shows that PH can differentiate between artists from different currents, between artists within the same current, and can separate an artist’s real works from AI‑generated images in that artist’s style.
arXiv:2606. 11911v1 Announce Type: cross Abstract: Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction.
arXiv:2609.37177v1 Announce Type: cross Abstract: Persistent homology (PH) is a frequently used tool for extracting and preserving topological information from image data, particularly in image segme...