Flat Metric

Flat metrics, a generalization of Wasserstein distance to distributions with unequal mass, are a focus of current research, particularly in the context of unbalanced optimal transport problems and data analysis where normalization is unsuitable. Researchers are employing neural networks, specifically focusing on achieving consistent distance computations across independently trained models, to efficiently calculate these metrics. This work has applications in machine learning, enabling advancements in techniques like tropical principal component analysis and tropical logistic regression, and also finds relevance in theoretical geometry, where machine learning is being used to approximate Calabi-Yau metrics, a long-standing challenge with implications for string theory. The development of robust and efficient methods for computing flat metrics is thus crucial for both theoretical and applied advancements.