Computing the distance between unbalanced distributions: the flat metric.

Area of Science:

• Computational mathematics
• Machine learning
• Optimal transport theory

Background:

• The Wasserstein distance (W1) is a fundamental metric for comparing probability distributions.
• Existing methods often require distributions to have equal total mass, limiting their applicability.
• Unbalanced optimal transport and data analysis tasks frequently involve distributions with differing sample sizes or where normalization is not possible.

Purpose of the Study:

• To provide a robust implementation for computing the flat metric in arbitrary dimensions.
• To extend the utility of optimal transport metrics to scenarios with unequal total mass.
• To develop a method capable of distinguishing distributions based on mass differences.

Main Methods:

• The core methodology employs a neural network to identify an optimal test function.
• This test function is used to compute the flat metric, also known as the dual bounded Lipschitz distance.
• Emphasis was placed on ensuring comparability of distances computed from independently trained neural networks.

Main Results:

• The implementation successfully computes the flat metric across various dimensions.
• The method effectively adapts to and utilizes mass differences between distributions.
• Experimental validation with ground truth and simulated data demonstrated the quality of the computed distances.

Conclusions:

• The developed implementation offers a practical solution for computing the flat metric, generalizing the Wasserstein distance.
• This tool is particularly relevant for unbalanced optimal transport problems and the analysis of real-world data distributions.
• The approach provides a reliable way to compare distributions even when they have unequal total mass or cannot be normalized.