Generalized Sobolev IPM for Graph-Based Measures

TL;DR

This work generalizes the Sobolev IPM to graph-based measures by introducing an Orlicz geometric framework, enabling priors beyond the conventional $L^p$ structure. It develops a Musielak regularization that equates the Orlicz-Sobolev norm of a critic with a univariate gradient-based norm, allowing the generalized Sobolev IPM to be computed via a simple univariate optimization. The approach yields a metric, connects to GST, ST, OW, and OT, and demonstrates substantial computational gains over Orlicz-Wasserstein in large graphs while maintaining competitive performance on document classification and topological data analysis tasks. This framework broadens the toolkit for comparing graph-supported probability measures with scalable, geometry-informed priors, facilitating practical deployment in ML pipelines that rely on graph-structured data and persistence diagrams.

Abstract

We study the Sobolev IPM problem for measures supported on a graph metric space, where critic function is constrained to lie within the unit ball defined by Sobolev norm. While Le et al. (2025) achieved scalable computation by relating Sobolev norm to weighted $L^p$-norm, the resulting framework remains intrinsically bound to $L^p$ geometric structure, limiting its ability to incorporate alternative structural priors beyond the $L^p$ geometry paradigm. To overcome this limitation, we propose to generalize Sobolev IPM through the lens of \emph{Orlicz geometric structure}, which employs convex functions to capture nuanced geometric relationships, building upon recent advances in optimal transport theory -- particularly Orlicz-Wasserstein (OW) and generalized Sobolev transport -- that have proven instrumental in advancing machine learning methodologies. This generalization encompasses classical Sobolev IPM as a special case while accommodating diverse geometric priors beyond traditional $L^p$ structure. It however brings up significant computational hurdles that compound those already inherent in Sobolev IPM. To address these challenges, we establish a novel theoretical connection between Orlicz-Sobolev norm and Musielak norm which facilitates a novel regularization for the generalized Sobolev IPM (GSI). By further exploiting the underlying graph structure, we show that GSI with Musielak regularization (GSI-M) reduces to a simple \emph{univariate optimization} problem, achieving remarkably computational efficiency. Empirically, GSI-M is several-order faster than the popular OW in computation, and demonstrates its practical advantages in comparing probability measures on a given graph for document classification and several tasks in topological data analysis.

Figures (7)

• Figure 1: Time consumption on ${\mathbb G}_{\text{Log}}$.
• Figure 2: SVM results and time consumption for kernel matrices with graph ${\mathbb G}_{\text{Log}}$. For each dataset, the numbers in the parenthesis are the number of classes; the number of documents; and the maximum number of unique words for each document respectively.
• Figure 3: SVM results and time consumption for kernel matrices with graph ${\mathbb G}_{\text{Log}}$. For each dataset, the numbers in the parenthesis are respectively the number of PD; and the maximum number of points in PD.
• Figure 4: Time consumption for GSI-M, GST and OW on ${\mathbb G}_{\text{Sqrt}}$ with 1K nodes and 32K edges.
• Figure 5: SVM results and time consumption for kernel matrices with graph ${\mathbb G}_{\text{Sqrt}}$.

Theorems & Definitions (46)

• Definition 3.1: Graph-based Orlicz-Sobolev space le2024generalized
• Theorem 3.2: Equivalence
• Definition 3.3: Generalized Sobolev IPM with Musielak regularization
• Theorem 3.4: GSI-M as univariate optimization problem
• Theorem 3.5: Discrete case
• Proposition 3.6: Closed-form discrete case
• Theorem 4.1: Metrization
• Proposition 4.2: GSI-M with different $N$-functions
• Theorem 4.3: Relation with original generalized Sobolev IPM
• Proposition 4.4: Connection between GSI-M and regularized Sobolev IPM