Optimal Quantization for Nonuniform Densities on Spherical Curves

We present an analysis of optimal quantization of probability measures with nonuniform densities on spherical curves. We begin by deriving the centroid condition, followed by a high-resolution asymptotic analysis to establish the point-density formula. We further quantify the asymptotic error formula for the nonuniform densities. We apply these theorems to the von Mises distributions and characterize the optimal condition. We also provide applications using the high-resolution asymptotic and its corresponding error formula. Our results can be used in geometric probability theory and quantization theory of spherical curves.

💡 Research Summary

This paper investigates optimal quantization of probability measures supported on one‑dimensional curves embedded in the unit sphere S², with particular emphasis on non‑uniform density functions. The authors consider a smooth curve Γ ⊂ S² parameterized by arc‑length s ∈