Uniform distributions on curves and quantization

Authors

Joseph Rosenblatt
Mrinal Kanti Roychowdhury, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

4-30-2023

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 \cite{BW}, which says that for a Borel probability measure P with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.

Comments

Original published version available at https://doi.org/10.4134/CKMS.c210434

Recommended Citation

Rosenblatt, Joseph, and Mrinal Kanti Roychowdhury. “Uniform Distributions on Curves and Quantization.” Communications of the Korean Mathematical Society, vol. 38, no. 2, Apr. 2023, pp. 431–50. https://doi.org/10.4134/CKMS.c210434.

Publication Title

Commun. Korean Math. Soc.

DOI

https://doi.org/10.4134/CKMS.c210434