Optimal Quantization for Nonuniform Densities on Spherical Curves

TL;DR

This paper analyzes optimal quantization of nonuniform probability densities on spherical curves, deriving conditions, asymptotic formulas, and applying them to von Mises distributions for improved geometric and quantization understanding.

Contribution

It introduces a centroid condition, high-resolution asymptotic analysis, and error quantification specific to nonuniform densities on spherical curves, including applications to von Mises distributions.

Findings

Derived the centroid condition for nonuniform densities.

Established a point-density formula for high-resolution asymptotics.

Quantified the asymptotic error for nonuniform densities.

Abstract

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.