Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Quantization for Nonuniform Densities on Spherical Curves

Silpi Saha, Sangita Jha, Mrinal Kanti Roychowdhury

TL;DR

This work extends optimal quantization theory to probability measures with nonuniform densities on spherical curves. It derives intrinsic centroid and boundary conditions, and establishes a high‑resolution regime where the optimal point density satisfies $\lambda(s) \propto h(s)^{1/3}$, yielding the asymptotic distortion $\lim_{n\to\infty} n^2 V_n(P) = \frac{1}{12} \left( \int_{\Gamma} h(s)^{1/3} ds \right)^3$. The theory is applied to von Mises densities on great circles, mixtures, and cosine‑modulated patterns, and extended to chordal vs geodesic distortions and nonuniform quadrature on spherical arcs. Collectively, the results provide a density‑aware framework for constructing optimal codebooks on spherical curves with concrete asymptotic constants and practical guidance for applications in directional statistics and geometric probability. The findings advance understanding of quantization on manifolds and offer tools for efficient discrete approximations of curved directional data.

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.

Optimal Quantization for Nonuniform Densities on Spherical Curves

TL;DR

This work extends optimal quantization theory to probability measures with nonuniform densities on spherical curves. It derives intrinsic centroid and boundary conditions, and establishes a high‑resolution regime where the optimal point density satisfies , yielding the asymptotic distortion . The theory is applied to von Mises densities on great circles, mixtures, and cosine‑modulated patterns, and extended to chordal vs geodesic distortions and nonuniform quadrature on spherical arcs. Collectively, the results provide a density‑aware framework for constructing optimal codebooks on spherical curves with concrete asymptotic constants and practical guidance for applications in directional statistics and geometric probability. The findings advance understanding of quantization on manifolds and offer tools for efficient discrete approximations of curved directional data.

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.
Paper Structure (13 sections, 8 theorems, 111 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 8 theorems, 111 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Intrinsic optimality and high-resolution structure
  4. Asymptotic quantization error and von Mises optimal quantizers
  5. Applications
  6. Optimal quantization under chordal vs. geodesic distortion
  7. Relationship between chordal and geodesic distance
  8. Chordal optimality conditions on the equator
  9. Quantization of mixture distributions on spherical curves
  10. Cosine–modulated density on a great circle
  11. Bimodal stationary phase law on the great circle
  12. Nonuniform quadrature on a spherical arc
  13. Conclusions

Key Result

Theorem 2.6

GrafLuschgy2000GershoGray1991 Let $(\Gamma, d_G)$ be a one-dimensional geodesic curve of total length $L$, and let $P$ be the uniform probability distribution with respect to arc-length. For squared distortion $(r=2)$, an optimal set of $n$-means partitions $\Gamma$ into $n$ Voronoi cells of equal a

Figures (5)

  • Figure 1: Comparison of intrinsic and chordal squared distances on $\mathbb{S}^2$ as functions of geodesic separation $\psi$.
  • Figure 2: Optimal 7-means for von Mises distribution. Chordal quantizers (red crosses) shift slightly toward $\pi$ compared to geodesic quantizers (blue dots).
  • Figure 3: Convergence of Algorithm 1 for n=7, $\kappa$=3
  • Figure 4: Analysis and convergence for the distortion error corresponding to chordal distribution
  • Figure 5: Maximum displacement analysis and convergence due to Lloyd iteration scheme.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • ...and 11 more