AISTATS 2021poster1 citations
On the Consistency of Metric and Non-Metric K-Medoids
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.
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. }
}