Finding the Homology of Submanifolds with High Confidence from Random Samples

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Computing Persistent Homology

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Stability of Persistence Diagrams

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Topological Data Analysis

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Towards a theoretical foundation for Laplacian-based manifold methods

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Testing the manifold hypothesis

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Random Projections of Smooth Manifolds

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Random projection trees and low dimensional manifolds

Dasgupta, S., & Freund, Y. (2008) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/1374376.1374452 · Cited by 383

To Trust Or Not To Trust A ClassifierPreprint

Jiang, H., Kim, B., Guan, M. Y., & Gupta, M. R. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1805.11783 · Cited by 340

Stability of persistence diagrams

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Vector diffusion maps and the connection Laplacian

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Confidence sets for persistence diagrams

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The smashed filter for compressive classification and target recognition

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Probability measures on the space of persistence diagrams

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Topological pattern recognition for point cloud data

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Convergence of Laplacian Eigenmaps

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Discrete laplace operator on meshed surfaces

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Topological analysis of population activity in visual cortex

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Geometric Inference for Probability Measures

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Describing shapes by geometrical-topological properties of real functions

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Fréchet Means for Distributions of Persistence Diagrams

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An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

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Compressive Sensing on Manifolds Using a Nonparametric Mixture of Factor Analyzers: Algorithm and Performance Bounds

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Towards persistence-based reconstruction in euclidean spaces

Chazal, F., & Oudot, S. (2008) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1377676.1377719 · Cited by 175

Random Geometric Complexes

Kahle, M. (2011) · Discrete & Computational Geometry · doi:10.1007/s00454-010-9319-3 · Cited by 150

Manifold Adaptive Experimental Design for Text Categorization

Cai, D., & He, X. (2011) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2011.104 · Cited by 161

A Topological View of Unsupervised Learning from Noisy Data

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Sample Complexity of Testing the Manifold Hypothesis

Narayanan, H., & Mitter, S. K. (2010) · Cited by 145

A Sampling Theory for Compact Sets in Euclidean Space

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Limit theorems for Betti numbers of random simplicial complexes

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Topology of random geometric complexes: a survey

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Topological methods for exploring low-density states in biomolecular folding pathways

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Generalized density clusteringPreprint

Rinaldo, A. (2010) · Cited by 129

Constructing Laplace operator from point clouds in Rd

Belkin, M. A., Sun, J. Z., & Wang, Y. (2009) · doi:10.5555/1496770.1496882 · Cited by 128

Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

Trillos, N. G., Gerlach, M., Hein, M., & Slepčev, D. (2019) · Foundations of Computational Mathematics · doi:10.1007/s10208-019-09436-w · Cited by 128

A Compressed Sensing Framework for Magnetic Resonance Fingerprinting

Davies, M. E., Puy, G., Vandergheynst, P., & Wiaux, Y. (2014) · SIAM Journal on Imaging Sciences · doi:10.1137/130947246 · Cited by 125

Random Projection Trees for Vector Quantization

Dasgupta, S., & Freund, Y. (2009) · IEEE Transactions on Information Theory · doi:10.1109/tit.2009.2021326 · Cited by 120

Multi-scale geometric methods for data sets II: Geometric Multi-Resolution Analysis

Allard, W. K., Chen, G., & Maggioni, M. (2011) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2011.08.001 · Cited by 116

A statistical approach to persistent homology

Bubenik, P., & Kim, P. T. (2007) · Homology Homotopy and Applications · doi:10.4310/hha.2007.v9.n2.a12 · Cited by 116

Topological Data Analysis with Applications

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Geometric and Topological Inference

Boissonnat, J., Chazal, F., & Yvinec, M. (2017) · HAL · Cited by 109

Manifold regularization and semi-supervised learning: some theoretical analyses

Niyogi, P. (2013) · Journal of Machine Learning Research · Cited by 106

Nonparametric ridge estimationPreprint

Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., & Wasserman, L. (2014) · Cited by 98

Dynamic MRI Using SmooThness Regularization on Manifolds (SToRM)

Poddar, S., & Jacob, M. (2015) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2015.2509245 · Cited by 93

Persistent homology for random fields and complexes

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Manifold Reconstruction Using Tangential Delaunay Complexes

Boissonnat, J., & Ghosh, A. (2013) · Discrete & Computational Geometry · doi:10.1007/s00454-013-9557-2 · Cited by 90

Minimax manifold estimationPreprint

Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., & Wasserman, L. (2012) · arXiv (Cornell University) · doi:10.5555/2188385.2343687 · Cited by 88

Spectral clustering based on local linear approximations

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Local Linear Regression on Manifolds and Its Geometric Interpretation

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Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces

Blumberg, A. J., Gal, I., Mandell, M. A., & Pancia, M. (2014) · Foundations of Computational Mathematics · doi:10.1007/s10208-014-9201-4 · Cited by 94

Persistent homology analysis of phase transitions

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Random geometric complexes in the thermodynamic regime

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Estimating the reach of a manifold

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Topology of deep neural networksPreprint

Naitzat, G., Zhitnikov, A., & Lim, L. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2004.06093 · Cited by 92

Density Level Sets: Asymptotics, Inference, and Visualization

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Generalized density clustering

Rinaldo, A., & Wasserman, L. (2010) · The Annals of Statistics · doi:10.1214/10-aos797 · Cited by 75

Manifold estimation and singular deconvolution under Hausdorff lossPreprint

Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., & Wasserman, L. (2011) · Cited by 75

Orthogonal locality preserving indexing

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Convergence Rates for Persistence Diagram Estimation in Topological Data Analysis

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Nonparametric ridge estimation

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Nonasymptotic rates for manifold, tangent space and curvature estimation

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Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

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Inferring Local Homology from Sampled Stratified Spaces

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Manifold reconstruction in arbitrary dimensions using witness complexes

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Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds with polynomial prefactors

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Weak feature size and persistent homology: computing homology of solids in R n from noisy data samples

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Data Skeletonization via Reeb Graphs

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New analysis of manifold embeddings and signal recovery from compressive measurements

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Witnessed k-Distance

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Vietoris–Rips Complexes of Planar Point Sets

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Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes

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Manifold estimation and singular deconvolution under Hausdorff loss

Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., & Wasserman, L. (2012) · The Annals of Statistics · doi:10.1214/12-aos994 · Cited by 60

Which Spatial Partition Trees are Adaptive to Intrinsic Dimension?Preprint

Verma, N., Kpotufe, S., & Dasgupta, S. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1205.2609 · Cited by 60

Topological consistency via kernel estimation

Bobrowski, O., Mukherjee, S., & Taylor, J. (2016) · Bernoulli · doi:10.3150/15-bej744 · Cited by 58

Dimensionality Reduction with Subgaussian Matrices: A Unified Theory

Dirksen, S. (2015) · Foundations of Computational Mathematics · doi:10.1007/s10208-015-9280-x · Cited by 65

Recent Advances in Nonlinear Dimensionality Reduction, Manifold and Topological Learning

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Consistent manifold representation for topological data analysis

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Nonparametric regression on low-dimensional manifolds using deep ReLU networks: function approximation and statistical recovery

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A weak characterisation of the Delaunay triangulation

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Homological illusions of persistence and stability

Edelsbrunner, H., & Morozov, D. (2008) · Cited by 50

Topological Analysis of Variance and the Maxillary Complex

Heo, G., Gamble, J., & Kim, P. T. (2012) · Journal of the American Statistical Association · doi:10.1080/01621459.2011.641430 · Cited by 49

Reconstruction Using Witness Complexes

Guibas, L. J., & Oudot, S. (2008) · Discrete & Computational Geometry · doi:10.1007/s00454-008-9094-6 · Cited by 49

Weak witnesses for Delaunay triangulations of submanifolds

Attali, D., Edelsbrunner, H., & Mileyko, Y. (2007) · ACM Symposium on Solid and Physical Modeling (SPM) · doi:10.1145/1236246.1236267 · Cited by 48

Reeb Graphs: Approximation and Persistence

Dey, T. K., & Wang, Y. (2012) · Discrete & Computational Geometry · doi:10.1007/s00454-012-9463-z · Cited by 51

Exploring the topology of dynamical reconstructions

Garland, J., Bradley, E., & Meiss, J. D. (2016) · Physica D Nonlinear Phenomena · doi:10.1016/j.physd.2016.03.006 · Cited by 52

Testing Expansion in Bounded-Degree Graphs

Czumaj, A., & Sohler, C. (2007) · Annual Symposium on Foundations of Computer Science · doi:10.1109/focs.2007.33 · Cited by 48

Tangent space estimation for smooth embeddings of Riemannian manifolds

Tyagi, H., Vural, E., & Pascal Frossard, B. (2013) · Information and Inference A Journal of the IMA · doi:10.1093/imaiai/iat003 · Cited by 47

The reach, metric distortion, geodesic convexity and the variation of tangent spaces

Boissonnat, J., Lieutier, A., & Wintraecken, M. (2019) · Journal of Applied and Computational Topology · doi:10.1007/s41468-019-00029-8 · Cited by 47

Analysis of scalar fields over point cloud data

Chazal, F., Guibas, L. J., Oudot, S., & Škraba, P. (2009) · doi:10.5555/1496770.1496881 · Cited by 46

Topological trajectory classification with filtrations of simplicial complexes and persistent homology

Pokorny, F. T., Hawasly, M., & Ramamoorthy, S. (2015) · The International Journal of Robotics Research · doi:10.1177/0278364915586713 · Cited by 46

Approximating loops in a shortest homology basis from point data

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Refining understanding of corporate failure through a topological data analysis mapping of Altman’s Z-score model

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Efficient and robust persistent homology for measures

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Uniform Convergence Rates for Kernel Density Estimation

Jiang, H. (2017) · International Conference on Machine Learning · Cited by 45

Computational Topology

Zomorodian, A. J. (2009) · Chapman & Hall/CRC applied algorithms and data structures series · doi:10.1201/9781584888215-c3 · Cited by 46

Hardness Results for Homology Localization

Chen, C., & Freedman, D. (2011) · Discrete & Computational Geometry · doi:10.1007/s00454-010-9322-8 · Cited by 45

Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees

Chazal, F., & Lieutier, A. (2007) · Computational Geometry · doi:10.1016/j.comgeo.2007.07.001 · Cited by 44

Statistical topology via Morse theory persistence and nonparametric estimation

Bubenik, P., Carlsson, G., Kim, P. T., & Luo, Z. (2010) · Contemporary mathematics - American Mathematical Society · doi:10.1090/conm/516/10167 · Cited by 44

Manifold-Based Signal Recovery and Parameter Estimation from Compressive MeasurementsPreprint

Wakin, M. B. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1002.1247 · Cited by 42

Crackle: The Homology of Noise

Adler, R. J., Bobrowski, O., & Weinberger, S. (2014) · Discrete & Computational Geometry · doi:10.1007/s00454-014-9621-6 · Cited by 42

Multiple Instance Learning with Manifold Bags

Babenko, B., Verma, N., r, P. D., & Belongie, S. (2011) · International Conference on Machine Learning · Cited by 42

Geometry © 2006 Springer Science+Business Media, Inc. Finding the Homology of Submanifolds with High Confidence from Random Samples ∗

Niyogi, P., Smale, S., & Weinberger, S. (2008)

Probability estimates for manifold learning from random samples

Heuschober, M. (2013) · University of Vienna · doi:10.25365/thesis.27799

Fitting a manifold of large reach to noisy data

Fefferman, C., Ivanov, S., Lassas, M. J., & Narayanan, H. (2023) · Journal of Topology and Analysis · doi:10.1142/s1793525323500012 · Cited by 11

On topology estimation of submanifolds in Riemannian manifolds by random pointsPreprint

Mirzaie, R. (2025) · arXiv (Cornell University) · doi:10.48550/arxiv.2512.23225

On the sample complexity of learning smooth cuts on a manifold

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