Topological Representation Learning

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The field studies how to encode and learn data structure that ordinary distances or pairwise graphs can miss, such as loops, connected components, and relations among groups of points. Common approaches use persistent homology and persistence diagrams, spectral features from graph or sheaf Laplacians, and neural networks on simplicial or cellular complexes; these can serve as input features, training objectives, or architectural building blocks. The goal is to make representations more faithful to underlying shape and connectivity, supporting tasks from graph and point-cloud learning to medical imaging and multimodal models. Results are promising across varied benchmarks, but computational cost and whether models learn genuine topology rather than dataset-specific combinatorial patterns remain open challenges.

Latest papers

67 papers11 papers in the last four weeks, up 22% on the four weeks before. 0.1% of all new papers.

Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport
Mar 16, 2026Matteo PegoraroTopological Representation Learning Persistent Homology
Institute of Computing, Faculty of Informatics, Università della Svizzera Italiana
TopoCL: Topological Contrastive Learning for Medical Imaging
Mar 15, 2026Guangyu Meng, Pengfei Gu, Peixian Liang +3Medical Image Representation Learning Contrastive Learning
Dept. of Computer Science and Engineering, University of Notre Dame · The University of Texas Rio Grande Valley · Dept. of IT, Analytics, and Operations, University of Notre Dame
TopoOR: A Unified Topological Scene Representation for the Operating Room
Mar 10, 2026Tony Danjun Wang, Ka Young Kim, Tolga Birdal +2Higher-Order GNNs Topological Representation Learning
Technical University of Munich, Munich, Germany · Munich Center for Machine Learning, Germany · Kyung Hee University, South Korea +1
Torus embeddings
Mar 3, 2026Dan StowellTopological Representation Learning
Tilburg University, Tilburg, The Netherlands · Naturalis Biodiversity Centre, Leiden, The Netherlands
What Topological and Geometric Structure Do Biological Foundation Models Learn? Evidence from 141 Hypotheses
Feb 25, 2026Ihor KendiukhovNeural Representation Geometry Genomic Foundation Models
Department of Computer Science, University of Tübingen, Tübingen, Germany
SOM-VQ: Topology-Aware Tokenization for Interactive Generative Models
Feb 24, 2026Alessandro Londei, Denise Lanzieri, Matteo BenatiVector-Quantized Variational Autoencoder Topological Representation Learning
Sony Computer Science Laboratories - Rome. Joint Initiative CREF-SONY, Centro Ricerche Enrico Fermi. Via Panisperna 89/A, 00184, Rome, Italy · Department of Computer, Automatic and Management Engineering. Sapienza University, Via Ariosto 25, Rome, Italy
Multi-dimensional Persistent Sheaf Laplacians for Image Analysis
Feb 16, 2026Xiang Xiang Wang, Guo-Wei WeiTopological Representation Learning Image Classification
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. · Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824, USA. · Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA.

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Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport

Matteo Pegoraro

Institute of Computing, Faculty of Informatics, Università della Svizzera Italiana

Topological Representation Learning Persistent Homology Optimal Transport

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We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure

μ

on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function

S(μ)\in C(\mathbb{S}^2)

, and the map is stable with respect to 1-Wasserstein partial transport distance

\mathrm{POT}_1

. Moreover, to the best of our knowledge, persistence spheres are the first explicit representation used in topological machine learning for which continuity of the inverse on the image is established at every compactly supported target. Recent bounded-cardinality bi-Lipschitz embedding results in partial transport spaces, despite being powerful, are not given by the kind of explicit summary map considered here. Our construction is rooted in convex geometry: for positive measures, the defining ReLU integral is the support function of the lift zonoid. Building on~\cite{pegoraro2025persistence}, we refine the definition to better match the

\mathrm{POT}_1

deletion mechanism, encoding partial transport via a signed diagonal augmentation. In particular, for integrable

μ

, the uniform norm between

S(0)

and

S(μ)

depends only on the persistence of

μ

, without any need of ad-hoc re-weightings, reflecting optimal transport to the diagonal at persistence cost. This yields a parameter-free representation at the level of measures (up to numerical discretization), while accommodating future extensions where

μ

is a smoothed measure derived from PDs (e.g., persistence intensity functions~\citep{wu2024estimation}). Across clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds, the updated persistence spheres are competitive and often improve upon persistence images, persistence landscapes, persistence splines, and sliced Wasserstein kernel baselines.