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

Gourab Ghatak

TL;DR

This work introduces Weighted K-Harmonic Means (WKHM), a regularized clustering approach with soft inverse-distance weighting that yields numerically stable updates and a direct interpretation as RSS-based user association in wireless networks. It provides the first stochastic convergence guarantees for harmonic-mean-based clustering: monotone descent to a local minimum with fixed initialization, convergence in probability under Binomial Point Process initialization, and almost-sure convergence under mild decay conditions. The method bridges clustering theory and wireless network design by linking the weights to signal-strength-based associations and demonstrates superior trade-offs between minimum signal strength and load fairness in simulations, outperforming classical and modern baselines in joint RN placement and UE association. These results offer a principled, practically relevant tool for wireless network optimization under uncertainty and randomness.

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.

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

TL;DR

This work introduces Weighted K-Harmonic Means (WKHM), a regularized clustering approach with soft inverse-distance weighting that yields numerically stable updates and a direct interpretation as RSS-based user association in wireless networks. It provides the first stochastic convergence guarantees for harmonic-mean-based clustering: monotone descent to a local minimum with fixed initialization, convergence in probability under Binomial Point Process initialization, and almost-sure convergence under mild decay conditions. The method bridges clustering theory and wireless network design by linking the weights to signal-strength-based associations and demonstrates superior trade-offs between minimum signal strength and load fairness in simulations, outperforming classical and modern baselines in joint RN placement and UE association. These results offer a principled, practically relevant tool for wireless network optimization under uncertainty and randomness.

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.

Paper Structure

This paper contains 6 sections, 4 theorems, 18 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Key Contributions
  3. Regularized WKHM Formulation
  4. Application to Wireless Communications: Node Placement and Signal-Fraction Association
  5. Application to Wireless Communications: Node Placement and Signal-Fraction Association
  6. Conclusion

Key Result

lemma 1

Let $\mathcal{U}=\{u_1,\dots,u_N\}$ be a finite set of UE positions in $\mathbb R^d$ and let $\mathcal{R}=\{r_1,\dots,r_K\}$ denote the positions of $K$RN or cluster centers produced by the KM or CKM algorithm. Assume a (single-tone) RSRP model for a UE at position $x$ served by an RN at position $r

Figures (4)

  • Figure 1: Comparison of cluster formations with $K = 3$ and $N = 240$.
  • Figure 2: Comparison of cluster formations with $K = 3$ and $N = 330$.
  • Figure 3: Variation of performance with respect to $p$, $q$, and $\epsilon$: (a) SINR as a function of $p$ and $q$, (b) Mean SINR with respect to $q$ for a fixed $p = 2$, and (c) Mean SINR with respect to $\epsilon$ for fixed $p = 2$ and $q = 4$.
  • Figure 4: Empirical CDFs (a) SNR, (b) SINR, and (c) Cell load for different clustering schemes. Here the number of base stations is 9 in a 100 m x 100 m area with 100 users.

Theorems & Definitions (9)

  • lemma 1
  • proof
  • Theorem 1: Convergence to a Local Minimum
  • proof
  • Theorem 2: Convergence in Probability
  • proof
  • Theorem 3: Almost Sure Convergence
  • proof
  • Remark 1