Rectifiable sets and the Traveling Salesman Problem

Jones, Peter W.

doi:10.1007/bf01233418

Cited by 233 publications

(227 citation statements)

References 11 publications

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“…His construction is a kind of variant of P. Jones' Traveling Salesman Theorem (see [5]) and its main drawback is that it is very difficult to extend it to dimensions higher than 1. The construction given here to prove Theorem 0.1 extends naturally to any dimension, the main problem being to find interesting analytic or geometric criteria for it to hold.…”

Section: Introductionmentioning

confidence: 99%

Menger Curvature and Rectifiability

Léger¹

1999

The Annals of Mathematics

133

146

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Section: Introductionmentioning

confidence: 99%

Menger Curvature and Rectifiability

Léger¹

1999

The Annals of Mathematics

133

146

View full text Add to dashboard Cite

“…Unilateral local approximation of sets by G(n, m) was introduced independently by Jones [11], and it is now an important tool in the theory of quantitative rectifiability (for example, see [8]). EXAMPLE 1.7.…”

Section: Metadefinitionmentioning

confidence: 99%

“…EXAMPLE 1.7. In [11], Jones introduced the idea of a unilateral approximation number, now called a Jones beta number β E (Q), which measures how closely the set E ⊆ R n is to lying on a straight line inside a cube Q in a scale-invariant fashion. Specifically,…”

Section: Metadefinitionmentioning

confidence: 99%

Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

Badger

Lewis

2015

Forum math. Sigma

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We investigate the interplay between the local and asymptotic geometry of a set A ⊆ R n and the geometry of model sets S ⊂ P(R n ), which approximate A locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an (n − 1)-dimensional asymptotically optimally doubling measure in R n (n 4) has upper Minkowski dimension at most n − 4.

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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%

“…In the seminal paper [33] 1 multiscale quantities that measure geometric quantities of k-dimensional sets in R D were introduced. These quantities could be used to characterized rectifiability and construct near-optimal solutions to the analyst's traveling salesman problem.…”

Section: Multiscale Geometric Analysis and Dimension Estimationmentioning

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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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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Rectifiable sets and the Traveling Salesman Problem

Cited by 233 publications

References 11 publications

Menger Curvature and Rectifiability

Menger Curvature and Rectifiability

Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

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

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