Uniform distributions on curves and quantization

TL;DR

This paper investigates the quantization of uniform distributions on various curves, providing exact formulas for optimal quantization sets and errors, and confirming the quantization dimension equals the curve's dimension.

Contribution

It derives explicit formulas for optimal quantization on curves like lines, circles, and triangles, extending quantization theory to these geometric distributions.

Findings

Quantization dimension equals the curve's dimension.

Quantization coefficient exists as a finite positive number.

Formulas for optimal n-means and errors are provided for each curve.

Abstract

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise (1982), which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number