Intrinsic dimension identification via graph-theoretic methods

Brito, M. R.; Quiróz, Adolfo J.; Yukich, J. E.

doi:10.1016/j.jmva.2012.12.007

Cited by 16 publications

(12 citation statements)

References 16 publications

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“…This permits evaluating both separation and contraction properties. It can be viewed as a weak estimation of the intrisic dimension [2], even if the manifold hypothesis might not hold. We compute an estimate of the efficiency of a k-NN to correctly find the label of a local support vector.…”

Section: Complexity Of the Classification Boundarymentioning

confidence: 99%

Building a Regular Decision Boundary with Deep Networks

Oyallon

2017

2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

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Section: Complexity Of the Classification Boundarymentioning

confidence: 99%

Building a Regular Decision Boundary with Deep Networks

Oyallon

2017

2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

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“…Manifold recovery from a sample of points, Genovese et al (2012b); Genovese et al (2012c). Inference on dimension, Fefferman et al (2016), Brito et al (2013). Estimation of measures (perimeter, surface area, curvatures), Cuevas et al (2007), Jiménez and Yukich (2011), Berrendero et al (2014).…”

Section: Introductionmentioning

confidence: 99%

Detection of low dimensionality and data denoising via set estimation techniques

Aaron¹,

Cholaquidis²,

Cuevas³

2017

Electron. J. Statist.

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This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set S ⊂ R d . The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of S, of geometric or topological character. The available information is just a random sample of points drawn on S. The term "to identify" means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: is S full dimensional? Is S "close to a lower dimensional set" M? If so, can we estimate M or some functionals of M (in particular, the Minkowski content of M)? As an important auxiliary tool in the answers of these questions, a denoising procedure is proposed in order to partially remove the noise in the original data. The theoretical results are complemented with some simulations and graphical illustrations. arXiv:1702.05193v2 [math.ST] 3 Nov 2017 Estimation of some other relevant quantities in a manifold, Niyogi, Smale and Weinberger (2008), Chen and Müller (2012). Dimensionality reduction, Genovese et al. (2012a), Tenebaum et al. (2000).The problems under study. The contents of the paper. We are interested in getting some information (in particular, regarding dimensionality and Minkowski content) about a compact set M ⊂ R d . While the set M is typically unknown, we are supposed to have a random sample of points X 1 , . . . , X n whose distribution P X has a support "close to M". To be more specific, we consider two different models:The noiseless model : the support of P X is M itself; Aamari and Levrard (2015), Amenta et al. (2002), Cholaquidis et al. (2014), Cuevas and Fraiman (1997). The parallel (noisy) model : the support of P X is the parallel set S of points within a distance to M smaller than R 1 , for some R 1 > 0, where M is a d -dimensional set and d ≤ d; Berrendero et al. (2014). Note that other different models "with noise" are considered in Genovese et al. (2012a), Genovese et al. (2012b) and Genovese et al (2012c).In Section 3 we first develop, under the noiseless model, an algorithmic procedure to identify, eventually, almost surely (a.s.), whether or not M has an empty interior; this is achieved in Theorems 1 and 2 below. A positive answer would essentially entail (under some conditions, see the beginning of Section 3) that M has a dimension smaller than that of the ambient space.Then, assuming the noisy model andM = ∅ ( whereM denotes the interior of M) Theorems 3 (i) and 4 (i) provide two methods for the estimation of the maximum level of noise R 1 , giving also the corresponding convergence rates. If R 1 is known in advance, the remaining results in Theorems 3 and 4 allow us also to decide whether or not the "inside set" M has an empty interior.The identification methods are "algorithmic" in the sense that they are based on automatic procedures to perform them with arbitrary precision. This will requi...

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“…On the other hand, k-nearest-neighbors methods have found many applications in different statistical contexts, from the two-sample-problem (Schilling 1986), to clustering (Brito et al 1997) and even in dimensionality reduction (Brito et al 2002(Brito et al , 2013.…”

Section: A Sub-sample and K-nearest-neighbor Approach To The Svm Problemmentioning

confidence: 99%

Nearest neighbors methods for support vector machines

Camelo¹,

González-Lima

Quiróz

2015

Ann Oper Res

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A key issue in the practical applicability of the support vector machine methodology is the identification of the support vectors in very large data sets, a problem to which a great deal of attention has been given in the literature. In the present article we propose methods based on sampling and nearest neighbors, that allow for an efficient implementation of an approximate solution to the classification problem and, at least in some problems, will help in identifying a significant fraction of the support vectors in large data sets at low cost. The performance of the proposed method is evaluated in different examples and some of its theoretical properties are discussed.

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Intrinsic dimension identification via graph-theoretic methods

Cited by 16 publications

References 16 publications

Building a Regular Decision Boundary with Deep Networks

Building a Regular Decision Boundary with Deep Networks

Detection of low dimensionality and data denoising via set estimation techniques

Nearest neighbors methods for support vector machines

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