Orlicz–Sobolev with Musielak: An Efficient Regularization Approach for Graph-based IPM

We study the Sobolev IPM problem for probability measures supported on a graph metric space, where critic function is constrained to lie within the unit ball defined by Sobolev norm. Sobolev IPM is intrinsically coupled with $L^p$ geometric structure within its definition, limiting its ability to incorporate other prior geometry beyond the $L^p$ paradigm. Conversely, the classic optimal transport is flexible, easy to adapt to various geometric structures by simply changing its ground cost. An important example is Orlicz-Wasserstein (OW) which utilizes \emph{Orlicz} geometric structure to generalize the $L^p$ within the standard $p$-order Wasserstein, and remarkably play a vital role to advance machine learning methodologies. Inspired by recent advantages of OW, in this work, we leverage a specific class of convex functions for Orlicz geometry to mitigate such limitation for Sobolev IPM, and propose the generalized Sobolev IPM (GSI). Our GSI approach encompasses Sobolev IPM as a special case while accommodating diverse geometric priors beyond $L^p$. It however brings up significant computational hurdles that compound those already notoriously inherent in Sobolev IPM. To address these challenges, we theoretically establish a novel connection between \emph{Orlicz-Sobolev} norm and \emph{Musielak} norm which facilitates a novel efficient regularization for GSI. By further exploiting the underlying graph structure, we show that the regularized GSI reduces to a simple univariate optimization problem, achieving notably computational efficiency, enabling its usage in practical applications. We empirically illustrate that the regularized GSI is several-order faster than the popular OW in computation, and its performances compare favorably to other transport baselines for comparing graph-based measures in document classification and topological data analysis.