Topological Data Analysis

Also known as TDA

Topological Data Analysis studies features such as connected components, loops, and higher-dimensional holes, and how they persist as data are viewed across scales. Persistent homology, Mapper and Reeb graphs, Euler characteristic transforms, and higher-order Laplacians turn these structures into summaries for clustering, classification, time-series analysis, and scientific discovery. Current work improves filtrations, statistical comparisons, and faster computation, while integrating topological features with neural networks and other machine-learning models. A central challenge is ensuring that measured structure reflects meaningful topology rather than noise, sampling density, or representation choices, while preserving stability and scaling to large datasets.

Latest papers

67 papers6 papers in the last four weeks, up 20% on the four weeks before. 0.0% of all new papers.

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.
Persistent Entropy as a Detector of Phase Transitions
Feb 8, 2026Marcos Gutierrez-del-Pozo, Eduardo Paluzo-Hidalgo, Matteo RuccoPhase Transitions Persistent Homology
Exact and Approximate Range Queries in Ball Mapper
Jan 4, 2026Jay-Anne Bulauan, John Rick ManzanaresApproximation Algorithms Topological Data Analysis
Independent Researcher · Dioscuri Centre in Topological Data Analysis, Institute of Mathematics of the Polish Academy of Sciences International Environmental Doctoral School of the University of Silesia in Katowice
Finite Topological Space Filtrations: A Topological Framework for Data Analysis
Dec 29, 2025Selçuk KayacanClustering Persistent Homology
Bahçeşehir University, Faculty of Engineering and Natural Sciences, Istanbul, Turkey
From Topology to Retrieval: Decoding Embedding Spaces with Unified Signatures
Nov 27, 2025Florian Rottach, William Rudman, Bastian Rieck +2Representation Geometry in Language Models Text Embedding Evaluation
University of Tübingen · Boehringer Ingelheim GmbH, Biberach (Riss), Germany · The University of Texas at Austin +1
Topological data analysis using persistent discrete homology
Jun 17, 2025Chris Kapulkin, Nathan KershawNoise Robustness Persistent Homology
Topology-enhanced machine learning for speech signal processing
Nov 26, 2023Pingyao Feng, Qingrui Qu, Haiyu Zhang +4Interpretable ML Speech Processing
Department of Mathematics, Southern University of Science and Technology, Shenzhen, China. · Department of Mathematics, North Carolina State University, Raleigh, NC, USA. · School of Intelligence Science and Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen, China. +1

Feb 16, 2026J/K move · Enter open · S save

Multi-dimensional Persistent Sheaf Laplacians for Image Analysis

Xiang Xiang Wang, Guo-Wei Wei

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.

Topological Representation Learning Image Classification Topological Data Analysis Dimensionality Reduction

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We propose a multi-dimensional persistent sheaf Laplacian (MPSL) framework on simplicial complexes for image analysis. The proposed method is motivated by the strong sensitivity of commonly used dimensionality reduction techniques, such as principal component analysis (PCA), to the choice of reduced dimension. Rather than selecting a single reduced dimension or averaging results across dimensions, we exploit complementary advantages of multiple reduced dimensions. At a given dimension, image samples are regarded as simplicial complexes, and persistent sheaf Laplacians are utilized to extract a multiscale localized topological spectral representation for individual image samples. Statistical summaries of the resulting spectra are then aggregated across scales and dimensions to form multiscale multi-dimensional image representations. We evaluate the proposed framework on the COIL20 and ETH80 image datasets using standard classification protocols. Experimental results show that the proposed method provides more stable performance across a wide range of reduced dimensions and achieves consistent improvements to PCA-based baselines in moderate dimensional regimes.