The Geometry of Anisotropic Dilation for Optimal Regularization

Abstract

How can structured transformations of data improve regularizer design for inverse problems? We address this question through anisotropic dilation, a direction-preserving map that rescales each point along its Euclidean ray by a positive profile on the sphere. Our analysis builds on a characterization of optimal regularizers within a broad Gibbs class, where the regularizer best matched to a distribution $P$ is determined by a direction-dependent radial summary statistic ${\rho }_P$. We show that anisotropic dilation acts explicitly and transitively on the space of radial summary statistics: any prescribed optimal regularizer in this class can be obtained by transforming an admissible source distribution using a single explicit profile. We characterize the resulting distributional orbits and establish conditions under which this control of regularizer geometry yields a weakly convex denoising formulation even when the original objective is not weakly convex. We also derive in closed form the profile that optimally adapts a fixed base regularizer to data. Its improvement over isotropic rescaling with respect to KL divergence is governed by a Jensen gap, for which we establish strict positivity under explicit conditions for several natural families of distributions. For jointly learning the base regularizer and anisotropic profile from samples, we establish finite-sample generalization bounds, with explicit rates when the logarithm of the profile is parameterized by linear feature models or sparse ReLU networks. Experiments on a controlled two-dimensional family and on MNIST denoising demonstrate that learning the anisotropic profile can improve reconstruction performance for both handcrafted and learned base regularizers.

arXiv comment58 pages, 3 figures, 3 tables