Computing the Distance between unbalanced Distributions -- The flat Metric
This work addresses unbalanced optimal transport tasks and data analysis where sample size matters or normalization is not possible, but it is incremental as it builds on existing flat metric theory with a neural network implementation.
The authors tackled the problem of computing distances between distributions with unequal total mass by implementing the flat metric, which generalizes the Wasserstein distance W1, and demonstrated its effectiveness in handling mass differences and distinguishing distributions through experiments with ground truth and simulated data.
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.