A Geometric Analysis of Initialization Bias in Spherical K-means in the Weak Signal Regime
Abstract
We study initialization bias in spherical K-means for weakly informative directional mixtures. We model the observations by a K-component von Mises-Fisher mixture with a small concentration parameter κ, corresponding to a high-dispersion regime in which the data provide limited information about the underlying directions. Our analysis begins with the limiting case κ=0 (corresponding to a uniform distribution over the sphere), where one population spherical K-means update is governed entirely by the Voronoi tessellation induced by the initialized templates. For uniformly random initializations in fixed dimension d, the updated templates become asymptotically aligned with their initial values as K→∞: the average squared geodesic error scales as O(K−2/(d−1)), while the worst-case error is O((log K/K)2/(d−1)). We then show that, in the weak-signal regime of small positive κ, the population update remains an O(κ) perturbation of this limiting map. Thus, in the weak-signal regime, spherical K-means can preserve initialization-induced structure despite the presence of a genuine but highly dispersed directional signal.
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BibTeX
@misc{balanov2026geometric,
title = {{A Geometric Analysis of Initialization Bias in Spherical $K$-means in the Weak Signal Regime}},
author = {Amnon Balanov and Tamir Bendory},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02205},
eprint = {2609.02205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02205},
}
RIS
TY - UNPB
TI - A Geometric Analysis of Initialization Bias in Spherical K-means in the Weak Signal Regime
AU - Amnon Balanov
AU - Tamir Bendory
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02205
UR - https://arxiv.org/abs/2609.02205
AB - We study initialization bias in spherical K-means for weakly informative directional mixtures. We model the observations by a K-component von Mises-Fisher mixture with a small concentration parameter κ, corresponding to a high-dispersion regime in which the data provide limited information about the underlying directions. Our analysis begins with the limiting case κ=0 (corresponding to a uniform distribution over the sphere), where one population spherical K-means update is governed entirely by the Voronoi tessellation induced by the initialized templates. For uniformly random initializations in fixed dimension d, the updated templates become asymptotically aligned with their initial values as K→∞: the average squared geodesic error scales as O(K^−2/(d−1)), while the worst-case error is O((log K/K)^2/(d−1)). We then show that, in the weak-signal regime of small positive κ, the population update remains an O(κ) perturbation of this limiting map. Thus, in the weak-signal regime, spherical K-means can preserve initialization-induced structure despite the presence of a genuine but highly dispersed directional signal.
ER -