Computing the Distance between unbalanced Distributions -- The flat Metric

TL;DR

This paper introduces an implementation for computing the flat metric, a generalization of Wasserstein distance suitable for unbalanced distributions, using neural networks to efficiently measure differences in data distributions with varying total mass.

Contribution

The paper presents a neural network-based method to compute the flat metric for unbalanced distributions, enabling effective analysis of data with unequal total mass.

Findings

Successful implementation of the flat metric computation in any dimension.

Effective differentiation between distributions with different total masses.

Validation through experiments with ground truth and simulated data.

Abstract

We provide an implementation to compute the flat metric in any dimension. The flat metric, also called dual bounded Lipschitz distance, generalizes the well-known Wasserstein distance $W_1$ to the case that the distributions are of unequal total mass. Thus, our implementation adapts very well to mass differences and uses them to distinguish between different distributions. This is of particular interest for unbalanced optimal transport tasks and for the analysis of data distributions where the sample size is important or normalization is not possible. The core of the method is based on a neural network to determine an optimal test function realizing the distance between two given measures. Special focus was put on achieving comparability of pairwise computed distances from independently trained networks. We tested the quality of the output in several experiments where ground truth was available as well as with simulated data.