Cascade of Phase Transitions for Multi-Scale Clustering

TL;DR

This paper introduces a new multi-scale clustering framework based on phase transitions during simulated annealing of the EM algorithm, enabling detection of cluster structures at various scales without prior knowledge.

Contribution

It develops a novel approach leveraging phase transition cascades and local covariance to identify multi-scale clusters and their sizes, advancing clustering techniques.

Findings

Identifies phase transition thresholds for clustering stability.

Extracts multi-scale cluster information without prior knowledge.

Combines simulated annealing with regularised Gaussian mixture models.

Abstract

We present a novel framework exploiting the cascade of phase transitions occurring during a simulated annealing of the Expectation-Maximisation algorithm to cluster datasets with multi-scale structures. Using the weighted local covariance, we can extract, a posteriori and without any prior knowledge, information on the number of clusters at different scales together with their size. We also study the linear stability of the iterative scheme to derive the threshold at which the first transition occurs and show how to approximate the next ones. Finally, we combine simulated annealing together with recent developments of regularised Gaussian mixture models to learn a principal graph from spatially structured datasets that can also exhibit many scales.