Uniform Transformation of Non-Separable Probability Distributions

TL;DR

This paper introduces a theoretical framework and numerical method for transforming complex, non-separable probability distributions into uniform distributions in higher dimensions, extending beyond traditional cumulative approaches.

Contribution

It develops a potential function-based approach to generalize distribution transformation for non-separable, multi-dimensional probability densities.

Findings

Potential function links probability densities to transformations

Numerical method computes the potential in two dimensions

Examples demonstrate the approach's effectiveness

Abstract

A theoretical framework is developed to describe the transformation that distributes probability density functions uniformly over space. In one dimension, the cumulative distribution can be used, but does not generalize to higher dimensions, or non-separable distributions. A potential function is shown to link probability density functions to their transformation, and to generalize the cumulative. A numerical method is developed to compute the potential, and examples are shown in two dimensions.