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Testing the manifold hypothesis
Open accessAbstract
The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for testing the existence of a manifold that fits a probability distribution supported in a separable Hilbert space, only using i.i.d. samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution P supported on the unit ball of a separable Hilbert space H . Let G ( d , V , τ ) be the set of submanifolds of the unit ball of H whose volume is at most V and reach (which is the supremum of all r such that any point at a distance less than r has a unique nearest point on the manifold) is at least τ . Let L ( M , P ) denote the mean-squared distance of a random point from the probability distribution P to M . We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d. random samples from P as input and determines which of the following two is true (at least one must be): There exists M ∈ G ( d , C V , τ C ) such that L ( M , P ) ≤ C ϵ . There exists no M ∈ G ( d , V / C , C τ ) such that L ( M , P ) ≤ ϵ C . The answer is correct with probability at least 1 − δ .
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“According to the manifold hypothesis [66], most natural data lies on manifolds with significantly reduced intrinsic dimensionality.”
From Diffusion Models: A Comprehensive Survey of Methods and Applications (2024) · cites this paper as [66]
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“Data with manifold structures are ubiquitous in machine learning.As the manifold hypothesis [66] posits, natural data often reside on manifolds with lower intrinsic dimensionality.”
From Diffusion Models: A Comprehensive Survey of Methods and Applications (2024) · cites this paper as [66]
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Cites
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
Carlsson, G. (2009) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-09-01249-x · Cited by 2,302
Estimation of Dependences Based on Empirical Data
Vapnik, V. N. (2006) · Information science and statistics · doi:10.1007/0-387-34239-7 · Cited by 2,252
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
Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment
Zhang, Z., & Zha, H. (2004) · SIAM Journal on Scientific Computing · doi:10.1137/s1064827502419154 · Cited by 1,504
Uniform Central Limit Theorems
Dudley, R. M. (1999) · Cambridge University Press eBooks · doi:10.1017/cbo9780511665622 · Cited by 808
Introduction to Statistical Learning Theory
Bousquet, O., Boucheron, S., & Lugosi, G. (2004) · Advanced Lectures on Machine Learning · doi:10.1007/978-3-540-28650-9_8 · Cited by 687
Finding the homology of submanifolds with high confidence from random samples
Niyogi, P., Smale, S., & Weinberger, S. (2008) · Discrete & Computational Geometry · doi:10.1007/s00454-008-9053-2 · Cited by 523
Bradley, P., & Mangasarian, O. L. (2000) · Journal of Global Optimization · doi:10.1023/a:1008324625522 · Cited by 406
Pattern Analysis and Machine Intelligence
Yang, J., Zheng, Y., & Gong, C. (2025) · Communications in computer and information science · doi:10.1007/978-981-96-3349-4 · Cited by 390
Research works printed but not linked: 28
- Amari, Shun-ichi, 2000, Methods of information geometry, 191, 10.1090/mmono/191 · DOI
- Baraniuk, Richard G., 2009, Random projections of smooth manifolds, Found. Comput. Math., 9, 51, 10.1007/s10208-007-9011-z · DOI
- Belkin, Mikhail, 2003, Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput., 15, 1373, 10.1162/089976603321780317 · DOI
- Brand, M., 2002, Charting a manifold, NIPS, 15, 985
- Genovese, Christopher R., 2012, Minimax manifold estimation, J. Mach. Learn. Res., 13, 1263
- Coifman, Ronald R., 2005, Geometric diffusions as a tool for harmonic analysis and structure definition of data. Part II: Multiscale methods, Proc. Natl. Acad. Sci. USA, 102, 7432, 10.1073/pnas.0500896102 · DOI
- Cox, Trevor F., 1994, Multidimensional scaling, 59
- David, Guy, 1993, Analysis of and on uniformly rectifiable sets, 38, 10.1090/surv/038 · DOI
- Devroye, Luc, 1996, A probabilistic theory of pattern recognition, 10.1007/978-1-4612-0711-5 · DOI · Paper page
- Federer, Herbert, 1959, Curvature measures, Trans. Amer. Math. Soc., 93, 418, 10.2307/1993504 · DOI
- Hastie, Trevor, 1989, Principal curves, J. Amer. Statist. Assoc., 84, 502, 10.1080/01621459.1989.10478797 · DOI
- Hotelling, H., 1933, Analysis of a complex of statistical variables into principal components, J. Educational Psychology, 24, 417, 10.1037/h0071325 · DOI
- Johnson, William B., 1984, Extensions of Lipschitz mappings into a Hilbert space, 189, 10.1090/conm/026/737400 · DOI
- Ledoux, Michel, 2001, The concentration of measure phenomenon, 89, 10.1090/surv/089 · DOI
- 2012, Manifold learning theory and applications
- Martinetz, Thomas, 1994, Topology representing networks, Neural Netw., 7, 507, 10.1016/0893-6080(94)90109-0 · DOI
- Massart, Pascal, 2000, Some applications of concentration inequalities to statistics, Ann. Fac. Sci. Toulouse Math. (6), 9, 245, 10.5802/afst.961 · DOI
- Narasimhan, Raghavan, 1965, Lectures on topics in analysis
- Narayanan, Hariharan, 2010, On the sample complexity of testing the manifold hypothesis, NIPS, 1786
- Newton, Nigel J., 2013, Infinite-dimensional manifolds of finite-entropy probability measures, 713, 10.1007/978-3-642-40020-9_79 · DOI
- Ozertem, Umut, 2011, Locally defined principal curves and surfaces, J. Mach. Learn. Res., 12, 1249
- Pearson, K., 1901, On lines and planes of closest fit to systems of points in space, Philos. Mag., 2, 559, 10.1080/14786440109462720 · DOI
- Roweis, S. T., 2002, Global coordination of local linear models, Adv. Neural Inf. Process. Syst., 14, 889
- Shawe-Taylor, J., 2004, Kernel methods for pattern analysis, 10.1017/CBO9780511809682 · DOI
- Smola, Alex J., 1999, Regularized principal manifolds, 214, 10.1007/3-540-49097-3_17 · DOI
- Tenenbaum, J. B., 2000, A Global Geometric Framework for Nonlinear Dimensionality Reduction, Science, 290, 2319, 10.1126/science.290.5500.2319 · DOI · Paper page
- Vaidya, Pravin M., 1996, A new algorithm for minimizing convex functions over convex sets, Math. Programming, 73, 291, 10.1016/0025-5610(92)00021-S · DOI
- Weinberger, Kilian Q., 2006, Unsupervised learning of image manifolds by semidefinite programming, Int. J. Comput. Vision, 70, 77, 10.1007/s11263-005-4939-z · DOI
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Diffusion Models: A Comprehensive Survey of Methods and Applications
Yang, L., Zhang, Z., Song, Y., Hong, S., Xu, R., Zhao, Y., Zhang, W., Cui, B., & Yang, M.-H. (2024) · ACM Computing Surveys · doi:10.1145/3626235 · Cited by 1,704
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Li, J., Du, X., & Martins, J. R. R. A. (2022) · Progress in Aerospace Sciences · doi:10.1016/j.paerosci.2022.100849 · Cited by 398
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Kumar, T., Brennan, R., Mileo, A., & Bendechache, M. (2024) · IEEE Access · doi:10.1109/access.2024.3470122 · Cited by 399
A Survey of Topological Machine Learning Methods
Hensel, F., Moor, M., & Rieck, B. (2021) · Frontiers in Artificial Intelligence · doi:10.3389/frai.2021.681108 · Cited by 209
Brown, C. F., Kazmierski, M. R., Pasquarella, V. J., Rucklidge, W. J., Samsikova, M., Zhang, C., Shelhamer, E., Lahera, E., Wiles, O., Ilyushchenko, S., et al. (2025) · arXiv · doi:10.48550/arxiv.2507.22291 · Cited by 172
Metadata
- Authors
- Fefferman, C., Mitter, S., & Narayanan, H.
- Affiliations
- Princeton University (US)
- Massachusetts Institute of Technology (US)
- University of Washington (US)
- Venue
- Journal of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2016-02-09
- Volume
- 29
- Issue
- 4
- Pages
- 983-1049
- ISSN
- 0894-0347 (ISSN-L)
- 1088-6834
- OpenAlex
- W1593775104
- Cited by
- 430
- References
- 11
- Citations by year
- 2026: 50 · 2025: 72 · 2024: 57 · 2023: 63 · 2022: 49 · 2021: 62 · 2020: 34 · 2019: 29 · 2018: 8 · 2017: 3
- FWCI
- 4.20
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of the American Mathematical Society (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (EECS -1135843 " Smart Power Systems of the Future : Foundations for Understanding Volatility and Improving Operational Reliabili)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Machine Learning and Algorithms
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- manifold hypothesis, manifold learning, separable Hilbert space, i.i.d. samples, high-dimensional data
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