Weighted K-Harmonic Means Clustering: Convergence Analysis and Applications to Wireless Communications

2025-12-18

AI Summary

This paper addresses critical limitations of K-means clustering in wireless networks—numerical instability, absence of soft assignment capability, and incompatibility with RSS-based user association. To overcome these, we propose the Weighted K-Harmonic Means (WKHM) clustering algorithm. WKHM enables interpretable soft user association via inverse-distance weighting and establishes, for the first time, a rigorous stochastic convergence theory for harmonic-mean-based clustering—including probabilistic convergence under binomial-Poisson point process initialization and almost-sure convergence under decaying step-size conditions. By integrating weighted harmonic-mean optimization, nonconvex analysis, and stochastic geometry modeling, WKHM significantly improves the joint optimization of minimum signal strength and load fairness across diverse user spatial distributions, outperforming both classical and state-of-the-art clustering baselines.

Technology Category

• Machine Learning: Clustering
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Heuristic Search

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We propose the emph{weighted K-harmonic means} (WKHM) clustering algorithm, a regularized variant of K-harmonic means designed to ensure numerical stability while enabling soft assignments through inverse-distance weighting. Unlike classical K-means and constrained K-means, WKHM admits a direct interpretation in wireless networks: its weights are exactly equivalent to fractional user association based on received signal strength. We establish rigorous convergence guarantees under both deterministic and stochastic settings, addressing key technical challenges arising from non-convexity and random initialization. Specifically, we prove monotone descent to a local minimum under fixed initialization, convergence in probability under Binomial Point Process (BPP) initialization, and almost sure convergence under mild decay conditions. These results provide the first stochastic convergence guarantees for harmonic-mean-based clustering. Finally, through extensive simulations with diverse user distributions, we show that WKHM achieves a superior tradeoff between minimum signal strength and load fairness compared to classical and modern clustering baselines, making it a principled tool for joint radio node placement and user association in wireless networks.

Problem

• Ensures numerical stability and soft assignments in clustering
• Establishes convergence guarantees under deterministic and stochastic settings
• Optimizes signal strength and load fairness in wireless networks

Innovation

• Weighted K-harmonic means with soft assignments
• Convergence guarantees under deterministic and stochastic settings
• Superior tradeoff between signal strength and load fairness