Homotopic distances and group-like spaces

What the math gives to ML

The paper gives an exact reduction of multi-map homotopic disagreement to the Lusternik–Schnirelmann category of a single relative-difference map whenever the target has a group-like multiplication. This suggests replacing many pairwise representation-consistency terms by an O(m) gauge-invariant relative-difference layer, especially when latent codes are constrained to a compact Lie group such as SO(2), SO(3), or unit complex numbers. The LS-category interpretation additionally motivates a local-chart regularizer: learn a small collection of contractible latent charts and penalize relative differences that cannot be represented within one chart. The method is most plausible for multi-view learning, augmentation consistency, and equivariant or group-valued latent architectures.