Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$

What the math gives to ML

The paper introduces an affine Sobolev energy that aggregates directional derivatives through a negative-power average over the unit sphere, rather than using only the ordinary Euclidean gradient norm. This construction is sensitive to all directions and is designed to respect affine changes of coordinates, making it a plausible regularizer for neural functions whose input geometry is anisotropic or poorly conditioned. The most direct transfer is a stochastic directional-Jacobian penalty, tested against standard isotropic input-gradient regularization on small vision and tabular models.