On Hölder fields clustering

Cadre, Benoı̂t; Paris, Quirino

doi:10.1007/s11749-011-0244-4

Cited by 6 publications

(5 citation statements)

References 31 publications

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“…For instance Biau et al (2008) prove that in a separable Hilbert space, and provided |X | ≤ L almost surely, then ER(ĉ) − R ≤ 12kL 2 / n, for all n ≥ 1. A similar result is established in Cadre and Paris (2012) relaxing the hypothesis of bounded support by supposing only the existence of an exponential moment for X . In the context of a separable Hilbert space, Levrard (2015) establishes a stronger result under some conditions involving the quantity p(t ) defined as follows.…”

Section: Risk Boundssupporting

confidence: 62%

A notion of stability for k-means clustering

Gouic¹,

Paris²

2018

Electron. J. Statist.

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Section: Risk Boundssupporting

confidence: 62%

A notion of stability for k-means clustering

Gouic¹,

Paris²

2018

Electron. J. Statist.

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“…By means of the Kappa coefficient of agreement, it is shown that if the data are sampled at regular time intervals the functional nature of the data can be ignored without damaging the performance of the clustering procedure. This is a numerical validation of the theoretical result of [5]. A numerical counter-example has been included, with a non-uniform discretization strategy, for raw-data clustering methods in which the performance of aKKm and the K-means algorithm is significantly deteriorated compared with the KK-means algorithm, which accounts for the functional nature.…”

Section: Introductionmentioning

confidence: 92%

“…The K-means algorithm works well for uniformly spaced functional data (see [5]). In our case, this is shown in Table 5 where the K-means algorithm gives κ values similar to those obtained with methods based on projections on RKHS in all the test problems with uniformly spaced data.…”

Section: P-kk-meansmentioning

confidence: 99%

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K-means algorithms for functional data

2015

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a b s t r a c tCluster analysis of functional data considers that the objects on which you want to perform a taxonomy are functions f : X & R p ↦R and the available information about each object is a sample in a finite set ofThe aim is to infer the meaningful groups by working explicitly with its infinite-dimensional nature.In this paper the use of K-means algorithms to solve this problem is analysed. A comparative study of three K-means algorithms has been conducted. The K-means algorithm for raw data, a kernel K-means algorithm for raw data and a K-means algorithm using two distances for functional data are tested. These distances, called d Vn and d ϕ , are based on projections onto Reproducing Kernel Hilbert Spaces (RKHS) and Tikhonov regularization theory. Although it is shown that both distances are equivalent, they lead to two different strategies to reduce the dimensionality of the data. In the case of d Vn distance the most suitable strategy is Johnson-Lindenstrauss random projections. The dimensionality reduction for d ϕ is based on spectral methods.A key aspect that has been analysed is the effect of the sampling fx i g n i ¼ 1 on the K-means algorithm performance. In the numerical study an ex professo example is given to show that if the sampling is not uniform in X, then a K-means algorithm that ignores the functional nature of the data can reduce its performance. It is numerically shown that the original K-means algorithm and that suggested here lead to similar performance in the examples when X is uniformly sampled, but the computational cost when working with the original set of observations is higher than the K-means algorithms based on d ϕ or d V n , as they use strategies to reduce the dimensionality of the data.The numerical tests are completed with a case study to analyse what kind of problem the K-means algorithm for functional data must face.

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“…The question of general unbounded distributions is more challenging and has been studied less. The case where the vectors X i have well behaved exponential moments was analyzed in (Cadre and Paris, 2012). Results under less restrictive assumptions include: the uniform deviation bounds in (Telgarsky and Dasgupta, 2013;Bachem, Lucic, Hassani and Krause, 2017); a sub-Gaussian excess distortion bound in (Brownlees, Joly and Lugosi, 2015) for the so-called k-medians problem; and the results for trimmed quantizers in (Brécheteau, Fischer and Levrard, 2018).…”

Section: Introductionmentioning

confidence: 99%

Robust $k$-means Clustering for Distributions with Two Moments

Klochkov¹,

Kroshnin²,

Zhivotovskiy³

2020

Preprint

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We consider the robust algorithms for the k-means clustering problem where a quantizer is constructed based on N independent observations. Our main results are median of means based non-asymptotic excess distortion bounds that hold under the two bounded moments assumption in a general separable Hilbert space. In particular, our results extend the renowned asymptotic result of Pollard (1981) who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer in R d . In a special case of clustering in R d , under two bounded moments, we prove matching (up to constant factors) non-asymptotic upper and lower bounds on the excess distortion, which depend on the probability mass of the lightest cluster of an optimal quantizer. Our bounds have the sub-Gaussian form, and the proofs are based on the versions of uniform bounds for robust mean estimators.

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On Hölder fields clustering

Cited by 6 publications

References 31 publications

A notion of stability for k-means clustering

A notion of stability for k-means clustering

K-means algorithms for functional data

Robust $k$-means Clustering for Distributions with Two Moments

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