Finding the Homology of Submanifolds with High Confidence from Random Samples

Niyogi, Partha; Smale, Stephen T.; Weinberger, Shmuel

doi:10.1007/s00454-008-9053-2

“…From [27], we have Lemma 26 (Normal variation) Let p, q be two points in M with p − q ≤ α lfs(p), α < 1 2 , and θ = ∠N p N q . We then have 2 sin θ 2 ≤ 1 − √ 1 − 4α, and the weaker bound sin θ ≤ 4α.…”

Section: Hausdorff Distancementioning

confidence: 99%

“…The output of those methods is a triangulated surface that approximates M. This triangulated surface is usually extracted from a 3-dimensional of the ambient space (typically a grid or a triangulation). Although rather inoffensive in 3-dimensional space, such data structures depend exponentially on the dimension of the ambient space, and all attempts to extend those geometric approaches to more general manifolds has led to algorithms whose complexities depend exponentially on d [27,11,14]. The problem in higher dimensions is also of great practical interest in data analysis and machine learning.…”

Section: Introductionmentioning

confidence: 99%

Manifold Reconstruction Using Tangential Delaunay Complexes

Boissonnat

¹

,

Ghosh

²

2013

Discrete Comput Geom

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We give a provably correct algorithm to reconstruct a k-dimensional manifold embedded in d-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [6,19,20], and the technique of sliver removal by weighting the sample points [13]. Differently from previous methods, we do not construct any subdivision of the embedding d-dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension d while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension k. To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold.

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“…The output of those methods is a triangulated surface that approximates M. This triangulated surface is usually extracted from a 3-dimensional of the ambient space (typically a grid or a triangulation). Although rather inoffensive in 3-dimensional space, such data structures depend exponentially on the dimension of the ambient space, and all attempts to extend those geometric approaches to more general manifolds has led to algorithms whose complexities depend exponentially on d [27,11,14].…”

Section: Introductionmentioning

confidence: 99%

Manifold reconstruction using tangential Delaunay complexes

Boissonnat

¹

,

Ghosh

²

2010

Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry

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We give a provably correct algorithm to reconstruct a k-dimensional manifold embedded in d-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [6,19,20], and the technique of sliver removal by weighting the sample points [13]. Differently from previous methods, we do not construct any subdivision of the embedding d-dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension d while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension k. To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold.

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“…Numerical constants will be denoted by C and their value may change at every instance. [87]). A result showing that the removal of a small number of outliers as described here has a negligible effect on the covariance matrices we considered may be found in Appendix E of [32].…”

Section: Some Preliminary Definitionmentioning

confidence: 99%

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

Little

¹

,

Maggioni

²

,

Rosasco

³

2017

Applied and Computational Harmonic Analysis

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Large data sets are often modeled as being noisy samples from probability distributions µ in R D , with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider sets that are locally well approximated by k-dimensional planes, with k ≪ D, with k-dimensional manifolds isometrically embedded in R D being a special case. Samples from µ are furthermore corrupted by D-dimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem well-suited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r, with a fixed center and varying r, and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M. As an application, we present an algorithm for estimating the intrinsic dimension of M.

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Finding the Homology of Submanifolds with High Confidence from Random Samples

Cited by 342 publications

References 18 publications

Manifold Reconstruction Using Tangential Delaunay Complexes

Manifold Reconstruction Using Tangential Delaunay Complexes

Manifold reconstruction using tangential Delaunay complexes

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

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