Approximation in Metric Sobolev Spaces: A General Framework

Abstract

In our recent work [FHS25], we introduced a numerical framework for approximating Sobolev functions on Wasserstein spaces from finite samples, leveraging structural properties established in [FSS23]. The present paper demonstrates that this methodology extends far beyond that specific setting. We identify a general class of metric measure spaces -- including weighted Riemannian manifolds and spaces of measures equipped with the Hellinger--Kantorovich distance -- for which the key hypotheses of Hilbertianity and the existence of a computable algebra of Lipschitz functions hold. Within this abstract framework, we recover and generalize the core approximation results of [FHS25] for recovering functions from random point evaluations. Our main contribution is to show that the combination of theoretical foundations from [FSS23] and algorithmic strategies from [FHS25] is robust enough to apply to a wide variety of infinite-dimensional spaces of current interest.