Hölder regularity and Harnack inequality for the logarithmic Laplacian

What the math gives to ML

The paper introduces the logarithmic Laplacian, a nonlocal operator with a scale-free kernel proportional to |z|^{-n}, a local compensation term, and a dimension-dependent zeroth-order coefficient. This provides a principled multiscale regularizer for spatial feature maps or ordered token embeddings, potentially avoiding the strong local oversmoothing induced by an ordinary Laplacian. The most direct experiment is to penalize the squared logarithmic-Laplacian response of intermediate representations and compare it against standard nearest-neighbor smoothness penalties at matched computational and gradient budgets.