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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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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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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