Definition and Properties of m-Dimensional n-Principal Points

Abstract

In this article, we introduce the notion of “m-dimensional n-principal points,” which is a generalization of the notion of n-principal points. A set of m-dimensional n-principal points of a distribution is defined as a set of n points that optimally represents the distribution in terms of mean squared distance subject to the condition that the dimension of the linear subspace spanned by the n points is at most m. Its properties and connections to principal components are investigated for elliptically symmetric distributions and a location mixture of spherically symmetric distributions.

Keywords:

• Elliptically symmetric distribution
• Location mixture
• Mean squared distance
• Principal points
• Spherically symmetric distribution

Mathematics Subject Classification:

• Primary 62E17
• Secondary 62H30

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Errata

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which led to significant improvements in this article. Kurata's portion of this work was supported by JSPS KAKENHI Grant Number 23700341. Kurata's portion of this work was supported by JSPS KAKENHI Grant Number 20243016, 21500272.