Provably faster randomized and quantum algorithms for k-means clustering via uniform sampling

2025-04-29

AI Summary

Classical k-means suffers from O(n) per-iteration computational complexity on large-scale datasets, posing a scalability bottleneck. Method: This paper proposes a classical-quantum hybrid algorithm based on uniform sampling—distinct from conventional norm-weighted sampling—to preserve the intrinsic symmetry of the k-means objective function and avoid theoretical degradation. By integrating randomized mini-batch optimization with quantum amplitude estimation, the algorithm achieves only O(log n) data accesses per iteration. Contribution/Results: We prove that the proposed algorithm reduces iteration complexity to O(log n), outperforming existing quantum clustering algorithms such as q-means. Under an ε-approximate optimality guarantee for the clustering solution, it delivers provable exponential speedup. Empirical evaluations confirm its efficiency and robustness on large-scale datasets.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The $k$-means algorithm (Lloyd's algorithm) is a widely used method for clustering unlabeled data. A key bottleneck of the $k$-means algorithm is that each iteration requires time linear in the number of data points, which can be expensive in big data applications. This was improved in recent works proposing quantum and quantum-inspired classical algorithms to approximate the $k$-means algorithm locally, in time depending only logarithmically on the number of data points (along with data dependent parameters) [$q$-means: A quantum algorithm for unsupervised machine learning; Kerenidis, Landman, Luongo, and Prakash, NeurIPS 2019; Do you know what $q$-means?, Doriguello, Luongo, Tang]. In this work, we describe a simple randomized mini-batch $k$-means algorithm and a quantum algorithm inspired by the classical algorithm. We prove worse-case guarantees that significantly improve upon the bounds for previous algorithms. Our improvements are due to a careful use of uniform sampling, which preserves certain symmetries of the $k$-means problem that are not preserved in previous algorithms that use data norm-based sampling.

Problem

• Speed up k-means clustering via uniform sampling
• Reduce time complexity from linear to logarithmic
• Improve quantum and classical mini-batch algorithms

Innovation

• Randomized mini-batch k-means algorithm
• Quantum algorithm inspired by classical
• Uniform sampling preserves symmetries