Finding the Homology of Submanifolds with High Confidence from Random Samples

Nonlinear Dimensionality Reduction by Locally Linear Embedding

Roweis, S. T., & Saul, L. K. (2000) · Science · doi:10.1126/science.290.5500.2323 · Cited by 14,965

A Global Geometric Framework for Nonlinear Dimensionality Reduction

Tenenbaum, J. B., Silva, V. D., & Langford, J. C. (2000) · Science · doi:10.1126/science.290.5500.2319 · Cited by 13,585

A Theory of the Learnable

Valiant, L. G. (1984) · ACM eBooks · doi:10.1145/3749740.3749746 · Cited by 4,211

A theory of the learnable

Valiant, L. G. (1984) · Communications of the ACM · doi:10.1145/1968.1972 · Cited by 3,279

Three-dimensional alpha shapes

Edelsbrunner, H., & Mücke, E. P. (1994) · ACM Transactions on Graphics · doi:10.1145/174462.156635 · Cited by 2,488

Elements of Algebraic Topology

Munkres, J. R. (2018) · CRC Press eBooks · doi:10.1201/9780429493911 · Cited by 2,113

Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data

Donoho, D. L., & Grimes, C. (2003) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1031596100 · Cited by 1,704

Computing Persistent Homology

Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591

Handbook of Combinatorics

Graham, R., Grötschel, M., & Lovász, L. (1995) · Medical Entomology and Zoology · Cited by 1,526

ELEMENTS OF ALGEBRAIC TOPOLOGY

Kuratowski, K. (1972) · Elsevier eBooks · doi:10.1016/b978-0-08-016160-0.50028-2 · Cited by 1,290

Expected invariants of simplicial complexes obtained from random point samples

Paik, T., & van Koert, O. (2023) · Archiv der Mathematik · doi:10.1007/s00013-023-01826-5 · Cited by 1

Topological Spaces

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.002 · Cited by 1

Manifold-driven decomposition for adversarial robustness

Zhang, W., Zhang, Y., Hu, X., Yao, Y., Goswami, M., Chen, C., & Metaxas, D. (2024) · Frontiers in Computer Science · doi:10.3389/fcomp.2023.1274695 · Cited by 1

Topological learning in multiclass data sets

Griffin, C. H., Karn, T., & Apple, B. (2024) · Physical review. E · doi:10.1103/physreve.109.024131 · Cited by 2

From samples to persistent stratified homotopy types

Mäder, T., & Waas, L. (2024) · Journal of Applied and Computational Topology · doi:10.1007/s41468-024-00170-z · Cited by 1

Efficient learning with projected histograms

Huang, Z., Kabán, A., & Reeve, H. W. J. (2024) · Data Mining and Knowledge Discovery · doi:10.1007/s10618-024-01063-6 · Cited by 1

Generating Topologically and Geometrically Diverse Manifold Data in Dimensions Four and Below

Hannouch, K. M., & Chalup, S. K. (2024) · International Conference on Digital Image Computing: Techniques and Applications (DICTA) · doi:10.1109/dicta63115.2024.00036 · Cited by 1

Coefficient-to-Basis Network: a fine-tunable operator learning framework for inverse problems with adaptive discretizations and theoretical guarantees

Zhang, Z., Liu, H., Liao, W., & Lin, G. (2025) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2024.0054 · Cited by 1

Geometric Data Manipulation with Clifford Algebras and Möbius Transforms

Guillemard, M., Iske, A., & Zölzer, U. (2015) · Advances in Applied Clifford Algebras · doi:10.1007/s00006-015-0608-z · Cited by 1

Preserving geometric properties in reconstructing regions from internal and nearby points

Davis, E. (2012) · Computational Geometry · doi:10.1016/j.comgeo.2012.01.001 · Cited by 1

Model-based signal recovery: A geometric perspective

Eftekhari, A. (2016) · Digital Collections of Colorado (Colorado State University) · Cited by 1

Approximate Triangulations of Grassmann Manifolds

Knudson, K. P. (2020) · Algorithms · doi:10.3390/a13070172 · Cited by 1

Topological Inference of the Conley Index

Yim, K. M., & Nanda, V. (2023) · Journal of Dynamics and Differential Equations · doi:10.1007/s10884-023-10310-1 · Cited by 1

The Cheeger Constant: from Discrete to ContinuousPreprint

Ery Arias-castro, A., Bruno Pelletier, B., & Pudlo, P. (2010) · arXiv (Cornell University) · Cited by 1

Distributed boundary tracking using alpha and Delaunay-Cech shapesPreprint

Chintakunta, H., & Krim, H. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1302.3982 · Cited by 1

Discretization Effects of Continuous Dictionary in Matching Pursuits: Density, Convergence and Optimization.Preprint

Jacques, L., & Vleeschouwer, C. D. (2007) · Cited by 1

Cyclic network automata and cohomological wavesPreprint

Cai, Y., & Ghrist, R. (2014) · doi:10.5555/2602339.2602363 · Cited by 1

Thresholds for vanishing of `Isolated' faces in random Čech and Vietoris-Rips complexesPreprint

Iyer, S. K., & Yogeshwaran, D. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1802.08224 · Cited by 1

An algebraic geometry perspective on topological data analysisPreprint

Breiding, P. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2001.02098 · Cited by 1

Data analysis from empirical moments and the Christoffel functionPreprint

Pauwels, E., Putinar, M., & Lasserre, J. B. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1810.08480 · Cited by 1

The space of sections of a smooth functionPreprint

Carlsson, G., & Filippenko, B. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2006.12023 · Cited by 5

Surface and length estimation based on Crofton´s formulaPreprint

Aaron, C., Cholaquidis, A., & Fraiman, R. (2020) · HAL · Cited by 1

On Fast Johnson-Lindenstrauss Embeddings of Compact Submanifolds of R N with Boundary.Preprint

Iwen, M., Schmidt, B., & Tavakoli, A. (2021) · arXiv (Cornell University) · Cited by 1

Index

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.014 · Cited by 1

Generative modeling of biological shapes and images using a probabilistic α -shape samplerPreprint

Winn-Nuñez, E. T., Witt, H. A., Bhaskar, D., Huang, R. Y., Reichner, J. S., Wong, I. Y., & Crawford, L. A. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.01.09.574919 · Cited by 1

Functional norms, condition numbers and numerical algorithms in algebraic geometryPreprint

Felipe Cucker, B., Ergür, A. A., & Tonelli-Cueto, J. (2021) · arXiv (Cornell University) · doi:10.1017/fms.2022.89

On the Contractibility of Random Vietoris–Rips Complexes

Müller, T., & Stehlı́k, M. (2022) · Discrete & Computational Geometry · doi:10.1007/s00454-022-00378-9

Topology at a Scale in Metric Spaces

Smale, N. (2012) · Springer eBooks · doi:10.1007/978-3-642-28821-0_16

Cyclic Cellular Automata on Networks and Cohomological Waves

Cai, Y. (2013) · ScholarlyCommons (University of Pennsylvania)

The Reach of Randomly Embedded ManifoldsPreprint

Adler, R. J., Krishnan, S. R., Taylor, J., & Weinberger, S. (2015) · arXiv (Cornell University)

A Bayesian Approach to Manifold Topology Reconstruction

Tevs, A., Wand, M., Ihrke, I., & Seidel, H. (2010) · Max Planck Institute for Plasma Physics

Adaptive Semisupervised InferencePreprint

Azizyan, M., Singh, A., & Wasserman, L. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1111.6410

The Relation between one-dimensional Persistence and Multidimensional Persistence

Luna, G. (2015) · Libraries and Cultural Resources (University of Calgary) · doi:10.11575/prism/28291

A hybrid approach of PTSVM and ELM inspired by samples' geometric distribution structure

Hu, H., Ouyang, P., Pang, L., & Shi, Z. (2014) · IEEE International Conference on Granular Computing (GrC) · doi:10.1109/grc.2014.6982815

Separating features from noise with persistence and statistics

Wang, B. (2010)

Sketching the Support of a Probability Measure

Giesen, J., Laue, S., & Kühne, L. (2014)

Curvature Distortion and Frequency Modulation Maps in Dimensionality Reduction

Guillemard, M., & Iske, A. (2009)

Algorithms for minimally supervised learning

Dasgupta, S. (2013)

Computing Persistent Homology via Discrete Morse Theory

Mller, J. M., & Winter, P. (2013)

Lecture 21 | Mar 18th, 2014

Hu, X. (2014)

Stable shape comparison by persistent homology

Ferri, M., Frosini, P., & Landi, C. (2011) · CINECA IRIS Institutial research information system (University of Pisa)

Convergence of Finite Integral Method for Poisson Equation and Corresponding EigenproblemsPreprint

Shi, Z., Sun, J. Z., & Li, Z. (2013) · arXiv (Cornell University)

Algorithms and hardness results in computational homology

Freedman, D. Z., & Chen, C. (2009)

Noisy data clusters are hollowPreprint

Léonard, F. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1506.03318

Tangent space variation on submanifolds.Preprint

Boissonnat, J., Dyer, R., & Ghosh, A. (2015) · arXiv (Cornell University)

Some Topics in Computational Topology

Wang, Y. (2014)

Adaptive Metrics for Adaptive Samples

Cavanna, N. J., & Sheehy, D. R. (2020) · Algorithms · doi:10.3390/a13080200

Reconstruction d'ensembles compacts 3D

Cazals, F., & Cohen‐Steiner, D. (2009) · HAL

Epilogue II.C. Background Independence and Problem of Time at Deeper Levels of Structure

Anderson, E. J. (2017) · Fundamental theories of physics · doi:10.1007/978-3-319-58848-3_38

Reconstruction of compacta by finite approximations and inverse persistence

Ruiz, D. M., & Morón, M. A. (2020) · Revista Matemática Complutense · doi:10.1007/s13163-020-00356-w · Cited by 3

Grid Methods in Computational Real Algebraic (and Semialgebraic) Geometry

Felipe Cucker, B. (2018) · Chinese Annals of Mathematics Series B · doi:10.1007/s11401-018-1070-8

On some global topological aspects of manifold learning

Manton, J. H., & Bihan, N. L. (2017) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2017.8296276

polyTop: Software for Computing Topology of Smooth Real Surfaces

Brake, D. A., Hauenstein, J. D., & Regan, M. H. (2018) · Lecture notes in computer science · doi:10.1007/978-3-319-96418-8_47

Sparse Filtered NervesPreprint

Blaser, N., & Brun, M. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1810.02149

An Adaptive Grid Algorithm for Computing the Homology Group of Semialgebraic SetPreprint

Jiadong, H. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1903.02388

What Happens to a Manifold Under a Bi-Lipschitz Map?Preprint

Eftekhari, A., & Wakin, M. B. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1512.06906

Cyclic network automata and cohomological waves

Cai, Y., & Ghrist, R. (2014) · ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN) · doi:10.1109/ipsn.2014.6846754

Witnessed k-Distance

Guibas, L. J., Mérigot, Q., & Morozov, D. (2011) · eScholarship (California Digital Library) · doi:10.48550/arxiv.1102.4972

Stable Manifold Embeddings with Structured Random MatricesPreprint

Yap, H. L., Wakin, M. B., & Rozell, C. J. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1209.3312

Topological and Geometric Reconstruction of Metric Graphs in R n .Preprint

Fasy, B. T., Komendarczyk, R., Majhi, S., & Wenk, C. (2019) · arXiv (Cornell University)

Stratifications on the Ran Space

Lazovskis, J. (2021) · Order · doi:10.1007/s11083-021-09568-1

Complex Links and Algebraic MultiplicitiesPreprint

Helmer, M., & Nanda, V. (2020) · arXiv (Cornell University)

Computational Analysis of Deformable Manifolds: from Geometric Modelling to Deep LearningPreprint

Schonsheck, S. C. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2009.01786

Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural NetworksPreprint

Chen, M., Liu, H., Liao, W., & Zhao, T. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2011.01797

Approximating the derivative of manifold-valued functions

Hielscher, R., & Lippert, L. (2022) · Journal of Approximation Theory · doi:10.1016/j.jat.2022.105832

Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood GraphsPreprint

Green, A., Balakrishnan, S., & Tibshirani, R. J. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2111.07394

Determining homology of an unknown space from a sample

Brun, M., Pascual, B. G., & Salbu, L. M. (2023) · European Journal of Mathematics · doi:10.1007/s40879-023-00683-4

Persistent Homology

Patrangenaru, V., & Ellingson, L. (2015) · CRC Press eBooks · doi:10.1201/b18969-33

Manifold Reconstruction

Cheng, S.-W. (2014) · Encyclopedia of Algorithms · doi:10.1007/978-3-642-27848-8_714-1

Manifold Reconstruction

Cheng, S.-W. (2016) · Encyclopedia of Algorithms · doi:10.1007/978-1-4939-2864-4_714

Minimax optimal regression over Sobolev spaces via Laplacian Eigenmaps on neighbourhood graphs

Green, A., Balakrishnan, S., & Tibshirani, R. J. (2023) · Information and Inference A Journal of the IMA · doi:10.1093/imaiai/iaad034

Triangulation of Submanifolds

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.008

Good Triangulations

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.006

Delaunay Complexes

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.005

Stability of Distance Functions

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.010

Delaunay Filtrations

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.007

Simplicial Complexes

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.003

Homology Inference

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.012

Bibliography

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.013

Reconstruction of Submanifolds

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.009

Introduction

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.001

Convex Polytopes

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.004

Distance to Probability Measures

Boissonnat, J., Chazal, F., & Yvinec, M. (2018) · Cambridge University Press eBooks · doi:10.1017/9781108297806.011

Convergence of Laplacian Eigenmaps and Its Rate for Submanifolds with Singularities

Aino, M. (2024) · Discrete & Computational Geometry · doi:10.1007/s00454-024-00667-5

Some Lower Bounds on the Reach of an Algebraic Variety

Valle, C. L., & Tonelli-Cueto, J. (2024) · International Symposium on Symbolic and Algebraic Computation · doi:10.1145/3666000.3669693

High-Dimensional Bayesian Optimization via Random Projection of Manifold Subspaces

Nguyen, Q.-A. H., & Tran, T. H. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-70371-3_17

Machine learning from a mathematical perspective

Rajendra, P., Ravi, P. V. N. H., & Meenakshi, K. (2024) · AIP conference proceedings · doi:10.1063/5.0225046 · Cited by 1

Neural Topic Model with Distance Awareness

Chen, S., Zhao, H., Huynh, V. H., Phung, D., & Cai, J. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-78189-6_22

Fitting an immersed submanifold to data via Sussmann’s orbit theorem

Hanson, J., & Raginsky, M. (2025) · Mathematics of Control Signals and Systems · doi:10.1007/s00498-025-00410-2

Optimal Approximation of Smooth Functions on Riemannian Manifolds Using Deep Relu Neural NetworksPreprint

Demetrio Labate, A., & Shi, J. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5117796

Local Topological Structures and Global Connectivity Patterns in a Trachelospermum jasminoides : a Pilot StudyPreprint

Tozzi, A. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.02.04.636494

LieDetect: Detection of Representation Orbits of Compact Lie Groups From Point Clouds

Ennes, H., & Tinarrage, R. (2025) · Foundations of Computational Mathematics · doi:10.1007/s10208-025-09728-4

Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications

Zhang, Y., & Kileel, J. (2025) · Foundations of Computational Mathematics · doi:10.1007/s10208-025-09735-5

Strange Random Topology of the Circle

Lim, U. (2025) · Discrete & Computational Geometry · doi:10.1007/s00454-025-00794-7

Training dynamics of GANs through the lens of persistent homology

Corradini, B. T., Cullen, B. N., Gallegati, C., Marziali, S., D’Inverno, G. A., Bianchini, M., & Scarselli, F. (2025) · Neurocomputing · doi:10.1016/j.neucom.2025.131976

Confidence regions for filamentary structures

Qiao, W. (2025) · Information and Inference A Journal of the IMA · doi:10.1093/imaiai/iaaf030

A geometric framework for multivariate jump locations estimation

Henneuse, H. (2025) · Brazilian Journal of Probability and Statistics · doi:10.1214/25-bjps637

Inferring a Cell Structure on the Space of Cyclooctane Conformations

Bauer, U., & Lenzen, F. (2026) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-032-01279-1_14

Delaunay-Like Triangulation of Smooth Orientable Submanifolds by $$\ell _1$$-Norm Minimization

Attali, D., & Lieutier, A. (2026) · Algorithmica · doi:10.1007/s00453-025-01367-z

Authors’ reply to discussion of ‘Statistical exploration of the Manifold Hypothesis’ by Whiteley et al.

Whiteley, N., Gray, A. E., & Rubin‐Delanchy, P. (2026) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1093/jrsssb/qkag056

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

Narayanan, H., & Niyogi, P. (2009) · Conference on Learning Theory · Cited by 15