A variational approach to the consistency of spectral clustering

TL;DR

This paper proves the consistency of spectral clustering methods by analyzing their convergence from discrete graph Laplacians to continuum operators, establishing conditions for reliable clustering as sample size grows.

Contribution

It introduces a variational framework to demonstrate spectral convergence and cluster consistency, providing sharp scaling conditions for the connectivity radius.

Findings

Spectral graph Laplacians converge to continuum operators under specific scaling.

Discrete spectral clusters approximate the true measure's partition as sample size increases.

The approach is general and adaptable to various spectral clustering scenarios.

Abstract

This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians towards the appropriate operators in the continuum domain. We obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the spectral convergence to hold. We also show that the discrete clusters obtained via spectral clustering converge towards a continuum partition of the ground truth measure. Such continuum partition minimizes a functional describing the continuum analogue of the graph-based spectral partitioning. Our approach, based on variational convergence, is general and flexible.