On some spectral properties of stochastic similarity matrices for data clustering

TL;DR

This paper investigates spectral properties of similarity matrices in image clustering, establishing conditions for spectral gaps in Gaussian mixtures and proposing an algorithm leveraging these properties for efficient clustering.

Contribution

It provides new spectral analysis results for Gaussian mixture similarity matrices and introduces an algorithm that exploits spectral gaps for improved clustering performance.

Findings

Spectral gap exists under specific conditions for Gaussian mixtures.

The proposed algorithm effectively clusters image elements using spectral properties.

Conditions for spectral gap facilitate more accurate and efficient clustering.

Abstract

Clustering in image analysis is a central technique that allows to classify elements of an image. We describe a simple clustering technique that uses the method of similarity matrices. We expand upon recent results in spectral analysis for Gaussian mixture distributions, and in particular, provide conditions for the existence of a spectral gap between the leading and remaining eigenvalues for matrices with entries from a Gaussian mixture with two real univariate components. Furthermore, we describe an algorithm in which a collection of image elements is treated as a dynamical system in which the existence of the mentioned spectral gap results in an efficient clustering.