Pyramidal Compactification of Asymmetric Metric Measure Spaces via Adjoint Transport

What the math gives to ML

The paper develops a compactification for asymmetric metric-measure spaces using one-sided Lipschitz observables, domination by measure-preserving nonexpansive maps, and multiscale lower sets called pyramids. Its most transferable asset is that directed geometry can be represented and compared through families of asymmetric observables without collapsing directionality via early symmetrization. This suggests a directed-attention or graph-neural module whose logits are constrained to be one-sided Lipschitz with respect to a learned quasi-metric, while matching observable families across augmentations or resolutions. The paper's explicit Hausdorff-distortion bound provides a practical stability loss for this representation.