Selecting Interpretable Circular Coordinates from Data

What the math gives to ML

The paper provides a principled way to turn abstract first cohomology classes into a small set of interpretable, circle-valued coordinates drawn from a user-supplied dictionary. The transferable asset is the combination of integer-period differential forms, an empirical cochain inner product, and minimum-weight basis selection in a vector matroid: coordinates can be selected for topological coverage and low geometric energy rather than by correlation or reconstruction error. A practical neural-network use is to replace unconstrained periodic latent dimensions with selected dictionary angles, while using the projection residual to detect missing topology or regularize representation learning.