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.
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: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:2507.21135v2 Announce Type: replace Abstract: We demonstrate how Quantum Cognition Machine Learning (QCML) encodes data as quantum geometry. In QCML, features of the data are represented by lea...
arXiv:2608. 11321v1 Announce Type: cross Abstract: We study spectral clustering in the presence of a confounding latent geometry.
arXiv:2607. 03692v1 Announce Type: new Abstract: Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling.
The survey reviews frequency‑domain techniques for time‑series analysis, covering classical Fourier methods to modern neural operators. It identifies three main research challenges: preserving causal structure during spectral transformations, quantifying uncertainty in learned frequency representations, and performing topology‑aware analysis for non‑Euclidean data. By reviewing over 100 studies, the authors propose a unified taxonomy, establish standardized benchmarks, and highlight gaps in geometric deep learning and quantum‑enhanced spectral analysis.
arXiv:2607. 08987v1 Announce Type: new Abstract: Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures.
The paper introduces DOODL, a framework that learns a dictionary of spectral dynamics to represent related dynamical systems as points on a low‑dimensional manifold in operator space. By constraining operator estimation to this learned manifold, DOODL provides compact, interpretable embeddings and enables fast, accurate operator estimation from short, partially observed trajectories. Experiments on metastable Langevin dynamics and turbulent plasma simulations show that DOODL achieves one to two orders of magnitude lower errors than independent estimation methods, especially in low‑data regimes.
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:2505. 04346v2 Announce Type: replace Abstract: Clustering aims at partitioning data points into groups of similar objects without knowing about the class labels.
arXiv:2605. 13834v2 Announce Type: replace-cross Abstract: In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective.
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.