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.
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.
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.
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:2608. 06276v1 Announce Type: cross Abstract: Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure.
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...
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.
The paper proposes using concepts from low‑dimensional topology—specifically Morse theory and cobordism—to enhance graph diffusion models, introducing the MG‑Diff pipeline. It provides theoretical guarantees that the Morse‑theoretic guidance remains stable under small perturbations when a positive decision‑gap exists. The authors demonstrate the approach on spatio‑temporal graph forecasting and graph regeneration, suggesting broader potential for topology in machine learning.
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.
arXiv:2606. 09806v1 Announce Type: cross Abstract: We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains.
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.
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.