Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds

What the math gives to ML

The paper provides a principled small-scale expansion for averaging over metric balls in equiregular sub-Riemannian geometry. Its transferable asset is the separation between a first-moment term, which creates drift and can destroy a clean Laplacian limit, and a normalized second-moment tensor, which determines anisotropic diffusion. This suggests a geometry-aware graph or point-cloud layer that estimates local first and second moments, removes spurious drift, and applies the resulting positive-semidefinite diffusion tensor instead of using isotropic neighborhoods. The most direct test is whether this moment-corrected layer improves stability and sample efficiency on data lying near anisotropic or nonholonomic manifolds.