Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs

What the math gives to ML

The paper defines a multi-color connectivity functional from the sublevel sets of a max-nearest-color distance field, rather than from ordinary pairwise distances. Its cost is determined by component birth and merge radii, providing a hierarchical signal that can encourage embeddings to become globally connected across colors such as classes, modalities, or augmentations. The asymptotic theorem is primarily geometric, but the filtration and its birth-minus-death cost can be approximated on minibatch graphs. The most practical transfer is a topology-aware metric-learning regularizer, with differentiable approximations for colorful lune radii and soft connectivity.