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.
By \.Ismail G\"uzel, Atabey Kaygun
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...
By Sel\c{c}uk Kayacan
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.
By Alexander Kolpakov, Igor Rivin
arXiv:2505. 04346v2 Announce Type: replace Abstract: Clustering aims at partitioning data points into groups of similar objects without knowing about the class labels.
By Arghya Pratihar, Kushal Bose, Swagatam Das
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.
By \.Ismail G\"uzel
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.
By Reetikaa Reddy Munnangi, Barbara Giunti
arXiv:2609.39242v1 Announce Type: new
Abstract: Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is...
By Enrico Maria Ferrari, Clemens Bannwart, Matteo Biagetti
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...
By Alexander H. Berger, Marco Fontana, Daniel Rueckert, Johannes C. Paetzold, Laurin Lux, Ulrich Bauer
The paper introduces a topological method for detecting Hopf bifurcations directly from scalar time series. It combines delay-coordinate reconstruction with persistent homology, using the maximum persistence of one‑dimensional homology classes as a scalar descriptor of cyclic structure. Finite‑resolution persistence bounds are derived for the supercritical Hopf case, and a derivative‑based estimator localizes the critical parameter; the method is tested on the Hopf normal form, Lorenz system, and a reduced Belousov–Zhabotinsky model.
By Jhonathan Barrios, Y\'asser Ech\'avez, Carlos F. \'Alvarez
arXiv:2606. 17531v1 Announce Type: new Abstract: We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions.
By Matias de Jong van Lier, Shizuo Kaji, Keunsu Kim
arXiv:2608. 18570v1 Announce Type: cross Abstract: Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology.
By Brandon Robinson, Shimal Harichurn, Fabian Ruehle, Sergei Gukov, Rak-Kyeong Seong, Miranda C. N. Cheng
arXiv:2606. 16990v1 Announce Type: new Abstract: While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales.
By Jernej Grlj, Aaron D. Lauda