Year: 2023
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Patel, Dhruv, Shen, Hui, Bhamidi, Shankar, Liu, Yufeng, Pipiras, Vladas
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Dynamic network models: Entrance boundary and continuum scaling limits, condensation phenomena and probabilistic combinatorial optimization, G2103075710, Trustworthy Reinforcement Learning for Online Decision Making, G2290275552, Network Time Series: From Dynamics to Coevolution, Iterative testing procedures and high-dimensional scaling limits of extremal random structures, G2944537279, RTG: Networks: Foundations in Probability, Optimization, and Data Sciences, G4223711460, G4626621346, G6471324540, G7452299184, Excellence in Research: Statistical Network Modeling and Inference for Complex Data, G9000016791
OpenAlex ID: W4386436469