Weak elastic energy of rectifiable curves in Riemannian surfaces

What the math gives to ML

The paper constructs a relaxed curvature functional for nonsmooth curves on a Riemannian surface and proves that finite energy is equivalent to second-order Sobolev regularity of the arc-length parameterization. The transferable asset is a reparameterization-aware penalty on changes of tangent direction: unlike a first-derivative smoothness penalty, it penalizes bending while allowing a trajectory to move at nearly constant speed. A practical neural adaptation is to place normalized hidden states on a sphere and penalize their discrete geodesic curvature across network depth or diffusion time. This produces a geometrically defined second-order regularizer that can be tested against ordinary residual-state smoothing.