Random Normed k-Means: A Paradigm-Shift in Clustering within Probabilistic Metric Spaces

2025-04-04

AI Summary

Traditional clustering methods, such as k-means, suffer from limited robustness on stochastic data due to their reliance on deterministic distances. To address this, this paper introduces the first generalization of k-means to probabilistic metric spaces: it replaces pointwise deterministic distances with distance distribution functions and establishes a probabilistic metric framework grounded in random normed distances. We propose a distribution-aware clustering objective and an iterative optimization algorithm, accompanied by rigorous theoretical guarantees on convergence and statistical consistency. The method unifies treatment of both deterministic and stochastic data and can uncover nonlinearly separable structures. Extensive experiments on diverse real-world and synthetic multivariate datasets demonstrate that our approach significantly outperforms k-means++, fuzzy c-means, and kernel-based probabilistic k-means across multiple evaluation metrics—including Silhouette score and Adjusted Rand Index—validating its robustness and generalization capability.

Technology Category

Machine Learning: ClusteringReasoning under Uncertainty: Stochastic OptimizationData Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

Web Mining and Content Analysis: Robustness and generalizability of Web mining methodsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Large-scale security measurements

Abstract

Existing approaches remain largely constrained by traditional distance metrics, limiting their effectiveness in handling random data. In this work, we introduce the first k-means variant in the literature that operates within a probabilistic metric space, replacing conventional distance measures with a well-defined distance distribution function. This pioneering approach enables more flexible and robust clustering in both deterministic and random datasets, establishing a new foundation for clustering in stochastic environments. By adopting a probabilistic perspective, our method not only introduces a fresh paradigm but also establishes a rigorous theoretical framework that is expected to serve as a key reference for future clustering research involving random data. Extensive experiments on diverse real and synthetic datasets assess our model's effectiveness using widely recognized evaluation metrics, including Silhouette, Davies-Bouldin, Calinski Harabasz, the adjusted Rand index, and distortion. Comparative analyses against established methods such as k-means++, fuzzy c-means, and kernel probabilistic k-means demonstrate the superior performance of our proposed random normed k-means (RNKM) algorithm. Notably, RNKM exhibits a remarkable ability to identify nonlinearly separable structures, making it highly effective in complex clustering scenarios. These findings position RNKM as a groundbreaking advancement in clustering research, offering a powerful alternative to traditional techniques while addressing a long-standing gap in the literature. By bridging probabilistic metrics with clustering, this study provides a foundational reference for future developments and opens new avenues for advanced data analysis in dynamic, data-driven applications.

Problem

• Introduces k-means for probabilistic metric spaces
• Enables robust clustering in random datasets
• Addresses limitations of traditional distance metrics

Innovation

• k-means variant in probabilistic metric space
• distance distribution function replaces metrics
• robust clustering for random datasets