Uniform rectifiability and quasiminimizing sets of arbitrary codimension

David, Guy; Semmes, Stephen

doi:10.1090/memo/0687

Cited by 50 publications

(173 citation statements)

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“…When the data are modeled as a manifold, possibly corrupted by noise, we can define the intrinsic dimension, at least locally, as the dimension of the manifold. There is a large body of literature at the intersection between harmonic analysis and geometric measure theory ( [77,26,27,28] and references therein) that explores and connect the behavior of multiscale quantities, such as Jones' β-numbers [77], with quantitative notions of rectifiability. This body of work has been our major inspiration.…”

Section: Overview Of Previous Work On Dimension Estimationmentioning

confidence: 99%

“…The inspiration for the current work originates from ideas in classical statistics (principal component analysis), dimension estimation of point clouds (see Section 2.1, 7 and references therein) and attractors of dynamical systems [23,24,25], and geometric measure theory [26,27,28], especially at its intersection with harmonic analysis. The ability of these tools to quantify and characterize geometric properties of rough sets of interest in harmonic analysis, suggests that they may be successfully adapted to the analysis of sampled noisy point clouds, where sampling and noise may be thought of as new types of (stochastic) perturbations not considered in the classical theory.…”

Section: Introductionmentioning

confidence: 99%

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

Little

Maggioni

Rosasco

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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Section: Overview Of Previous Work On Dimension Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Little

Maggioni

Rosasco

2017

Applied and Computational Harmonic Analysis

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“…But G = G × 0x, then by [3, 8.3], G is a minimal cone in H, centered at x. Since H is a 3-plane, we must have that G is a 2-dimensional minimal cone of type P, Y or T and then G is also a 3-dimensional minimal cone of type P, Y or T. Next, as G is a blow-up limit of F at x, by [3,7.31], we have θ…”

Section: 43)mentioning

confidence: 99%

“…Where the last line is obtained from the fact that E is Alhfors-regular (see [7]). Now (3.1.7) contradicts the hypothesis that E is MS-minimal, we thus obtain the lemma.…”

Section: 43)mentioning

confidence: 99%

On some properties of three-dimensional minimal sets in ℝ 4

Luu¹

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…The general idea, as in [Se2], [DS2], and Section 12 of [DS3], is to require some property of topological nature (i.e., for instance, invariant under deformations) that would automatically imply that E is at least d-dimensional. Connectedness (in dimension d = 1), as in the result of Bishop and Jones, would be an example.…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of uniformly non flat sets with topology

David¹

2004

Publicacions Matemàtiques

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Let d be an integer, and let E be a nonempty closed subset of R n . Assume that E is locally uniformly non flat, in the sense that for x ∈ E and r > 0 small, E∩B(x, r) never stays ε 0 r-close to an affine d-plane. Also suppose that E satisfies locally uniformly some appropriate d-dimensional topological nondegeneracy condition, like Semmes' Condition B. Then the Hausdorff dimension of E is strictly larger than d. We see this as an application of uniform rectifiability results on Almgren quasiminimal (restricted) sets.

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Uniform rectifiability and quasiminimizing sets of arbitrary codimension

Cited by 50 publications

References 0 publications

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

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

On some properties of three-dimensional minimal sets in ℝ 4

Hausdorff dimension of uniformly non flat sets with topology

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