M-decomposability, elliptical unimodal densities, and applications to clustering and kernel density estimation

TL;DR

This paper extends the concept of M-decomposability to multiple dimensions, demonstrating its utility in identifying modes and improving clustering and density estimation methods.

Contribution

It generalizes M-decomposability to higher dimensions and applies it as a non-parametric criterion for mode detection in clustering and density estimation.

Findings

Elliptical unimodal densities are M-undecomposable.

Mixture representation improves density modeling.

M-decomposability aids in mode detection.

Abstract

Chia and Nakano (2009) introduced the concept of M-decomposability of probability densities in one-dimension. In this paper, we generalize M-decomposability to any dimension. We prove that all elliptical unimodal densities are M-undecomposable. We also derive an inequality to show that it is better to represent an M-decomposable density via a mixture of unimodal densities. Finally, we demonstrate the application of M-decomposability to clustering and kernel density estimation, using real and simulated data. Our results show that M-decomposability can be used as a non-parametric criterion to locate modes in probability densities.