Consistency of Lloyd's Algorithm Under Perturbations

TL;DR

This paper proves that Lloyd's algorithm maintains an exponentially bounded mis-clustering rate under small perturbations, extending previous results to more realistic data scenarios involving pre-processing steps.

Contribution

It demonstrates that Lloyd's algorithm remains effective under data perturbations when combined with proper initialization, with implications for various clustering applications.

Findings

Mis-clustering rate remains exponentially bounded under small perturbations.

Proper initialization ensures the correctness of Lloyd's algorithm in perturbed settings.

Results apply to high-dimensional data, time series, and network community detection.

Abstract

In the context of unsupervised learning, Lloyd's algorithm is one of the most widely used clustering algorithms. It has inspired a plethora of work investigating the correctness of the algorithm under various settings with ground truth clusters. In particular, in 2016, Lu and Zhou have shown that the mis-clustering rate of Lloyd's algorithm on $n$ independent samples from a sub-Gaussian mixture is exponentially bounded after $O(\log(n))$ iterations, assuming proper initialization of the algorithm. However, in many applications, the true samples are unobserved and need to be learned from the data via pre-processing pipelines such as spectral methods on appropriate data matrices. We show that the mis-clustering rate of Lloyd's algorithm on perturbed samples from a sub-Gaussian mixture is also exponentially bounded after $O(\log(n))$ iterations under the assumptions of proper initialization and that the perturbation is small relative to the sub-Gaussian noise. In canonical settings with ground truth clusters, we derive bounds for algorithms such as $k$-means$++$ to find good initializations and thus leading to the correctness of clustering via the main result. We show the implications of the results for pipelines measuring the statistical significance of derived clusters from data such as SigClust. We use these general results to derive implications in providing theoretical guarantees on the misclustering rate for Lloyd's algorithm in a host of applications, including high-dimensional time series, multi-dimensional scaling, and community detection for sparse networks via spectral clustering.