An optimal approximation of discrete random variables with respect to the Kolmogorov distance

TL;DR

This paper introduces an algorithm to approximate discrete random variables with smaller support sets while minimizing the Kolmogorov distance, supported by theoretical analysis and empirical evaluation.

Contribution

The paper presents a novel algorithm for optimally approximating discrete random variables with limited support, with proven correctness and complexity analysis.

Findings

Algorithm achieves minimal Kolmogorov distance in practice

Theoretical guarantees of correctness and efficiency

Empirical results demonstrate practical effectiveness

Abstract

We present an algorithm that takes a discrete random variable $X$ and a number $m$ and computes a random variable whose support (set of possible outcomes) is of size at most $m$ and whose Kolmogorov distance from $X$ is minimal. In addition to a formal theoretical analysis of the correctness and of the computational complexity of the algorithm, we present a detailed empirical evaluation that shows how the proposed approach performs in practice in different applications and domains.