Quantization of Probability Distributions via Divide-and-Conquer: Convergence and Error Propagation under Distributional Arithmetic Operations

TL;DR

This paper introduces a divide-and-conquer algorithm for approximating continuous probability distributions, providing error bounds and demonstrating improved stability during distributional arithmetic operations through numerical experiments.

Contribution

It offers a new approximation method with proven convergence bounds and enhanced stability compared to existing schemes for distributional arithmetic operations.

Findings

Error bound in Wasserstein-1 distance for all continuous distributions with finite mean

Method achieves optimal convergence rate in many cases

Numerical experiments confirm improved stability over existing schemes

Abstract

This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numerical study that compares pre-existing approximation schemes with a special focus on the stability of the discrete approximations when they undergo arithmetic operations. The main results are a simple upper bound of the approximation error in terms of the Wasserstein-1 distance that is valid for all continuous distributions with finite mean. In many use-cases, the studied method achieve optimal rate of convergence, and numerical experiments show that the algorithm is more stable than pre-existing approximation schemes in the context of arithmetic operations.