A Geometric Analysis of Initialization Bias in Spherical $K$-means in the Weak Signal Regime

What the math gives to ML

The paper identifies a concrete failure mode of spherical prototype updates: when directional data are weakly informative, the population Voronoi-centroid map largely preserves the geometry of the initial prototypes rather than discovering latent directions. Its transferable asset is the quantitative separation between initialization-induced quantization error, which follows a sphere-covering rate, and the genuine signal contribution, which is only an O(κ) perturbation for small concentration κ. This suggests initializing spherical prototype layers with an explicit low-covering-radius design and testing initialization sensitivity as a function of signal strength rather than relying on one random seed.