Skeletal Homology

What the math gives to ML

The paper provides a scale-aware homology framework in which Vietoris–Rips homology is canonically identified with skeletal homology for arbitrary metric spaces, together with explicit stability under small perturbations of points or maps. The transferable asset is the ultradiamond viewpoint: if two chains or cycles are close in a metric that makes the boundary map 1-Lipschitz, their homology classes agree after a controlled increase of scale. This suggests a topology-preserving consistency regularizer for neural embeddings, applied across stochastic augmentations or model perturbations, with the scale slack determined by the measured embedding displacement rather than by an arbitrary penalty weight. The first target should be point-cloud or graph encoders where persistent H_0/H_1 features can be computed on small batches and robustness of learned geometry is measurable.