Generalized Mode and Ridge Estimation

TL;DR

This paper introduces a method to analyze the geometric structure of generalized densities, including modes and ridges, using modified mean shift algorithms, with applications in clustering and astronomical data analysis.

Contribution

It develops a novel approach for finding modes and ridges of generalized densities, extending mean shift algorithms with proven consistency and convergence rates.

Findings

Effective identification of modes and ridges in generalized densities.

Application to astronomical data demonstrates practical utility.

Theoretical guarantees for estimator performance.

Abstract

The generalized density is a product of a density function and a weight function. For example, the average local brightness of an astronomical image is the probability of finding a galaxy times the mean brightness of the galaxy. We propose a method for studying the geometric structure of generalized densities. In particular, we show how to find the modes and ridges of a generalized density function using a modification of the mean shift algorithm and its variant, subspace constrained mean shift. Our method can be used to perform clustering and to calculate a measure of connectivity between clusters. We establish consistency and rates of convergence for our estimator and apply the methods to data from two astronomical problems.