On the Consistency of Metric and Non-Metric K-Medoids
On the Consistency of Metric and Non-Metric K-Medoids
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发表时间：
2020-10
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通讯作者：
He Jiang;E. Arias-Castro
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作者：
He Jiang;E. Arias-Castro
我们在度量空间的背景下建立了 K 中心点的一致性。我们首先证明 K-medoids 渐近等价于一般条件下限制于底层分布支持的 K-means，包括广泛的损失函数选择。反过来，这种渐近等价使我们能够将 Parna (1986) 的工作应用于 K 均值的一致性。这种通用方法也适用于仅提供相异性排序的非度量设置。我们考虑两种类型的序数信息：一种是所有四重比较都可用；另一种是所有四重比较都可用。一种只能进行三重比较。我们提供一些数值实验来说明我们的理论。
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.