Clustering in Metric Spaces: Some Existence and Continuity Results for k-Centers

Pärna, Kalev

doi:10.1007/978-3-642-46757-8_10

Cited by 3 publications

(2 citation statements)

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“…Early in this line of work, Pollard (1981) established the consistency of K-means in Euclidean spaces. Pärna (1986) extended the result to separable metric spaces, while Pärna (1988Pärna ( , 1990Pärna ( , 1992 examined the particular situation of Hilbert and Banach spaces, where the existence of an optimal solution had been considered by Herrndorf (1983) and Cuesta and Matrán (1988).…”

Section: K-means and K-medoidsmentioning

confidence: 99%

“…Remark 2. As discussed in (Cuesta and Matrán, 1988;Pärna, 1990Pärna, , 1992, a K-means problem may not have a solution. In our situation, however, we are assuming that the space is a locally compact Polish space, and a solution can be shown to exist by a simple compactness argument together with our assumptions on φ (and the fact that the distance function is always continuous in any metric space it equips).…”

Section: Consistency Of K-medoidsmentioning

confidence: 99%

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On the Consistency of Metric and Non-Metric K-medoids

Arias-Castro,

Jiang

2020

Preprint

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We establish the consistency of K-medoids in the context of metric spaces. We start by proving that K-medoids is asymptotically equivalent to K-means restricted to the support of the underlying distribution under general conditions, including a wide selection of loss functions. This asymptotic equivalence, in turn, enables us to apply the work of Pärna (1986) on the consistency of K-means. This general approach applies also to non-metric settings where only an ordering of the dissimilarities is available. We consider two types of ordinal information: one where all quadruple comparisons are available; and one where only triple comparisons are available. We provide some numerical experiments to illustrate our theory.We establish the consistency of K-medoids by first showing that K-medoids is asymptotically equivalent to K-means restricted to the support of the underlying distribution, and then leveraging the work of Pärna (1986) on the consistency of K-means in metric spaces.

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