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

He Jiang, Ery Arias-Castro
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:2485-2493, 2021.

Abstract

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 Parna (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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-jiang21c, title = { On the Consistency of Metric and Non-Metric K-Medoids }, author = {Jiang, He and Arias-Castro, Ery}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {2485--2493}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/jiang21c/jiang21c.pdf}, url = {https://proceedings.mlr.press/v130/jiang21c.html}, abstract = { 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 Parna (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. } }
Endnote
%0 Conference Paper %T On the Consistency of Metric and Non-Metric K-Medoids %A He Jiang %A Ery Arias-Castro %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-jiang21c %I PMLR %P 2485--2493 %U https://proceedings.mlr.press/v130/jiang21c.html %V 130 %X 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 Parna (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.
APA
Jiang, H. & Arias-Castro, E.. (2021). On the Consistency of Metric and Non-Metric K-Medoids . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:2485-2493 Available from https://proceedings.mlr.press/v130/jiang21c.html.

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