Generalized Sobolev IPM for Graph-Based Measures

2025-10-29

AI Summary

This work addresses the Sobolev-type Integral Probability Metric (IPM) problem on graph metric spaces, where critic functions are constrained by Sobolev norms. Existing approaches rely on $L^p$ geometry, limiting incorporation of non-$L^p$ structural priors. To overcome this, we introduce Orlicz geometry into the Sobolev IPM framework for the first time, establishing a theoretical connection between Orlicz–Sobolev norms and Musielak norms. We propose Generalized Sobolev IPM (GSI-M), which reduces high-dimensional optimization to univariate solving. GSI-M integrates Orlicz–Wasserstein theory, generalized Sobolev transport, and graph-structural analysis, employing Musielak regularization for computational efficiency. Experiments demonstrate that GSI-M significantly outperforms state-of-the-art methods in document classification and topological data analysis, achieving speedups of several orders of magnitude over Orlicz–Wasserstein (OW) while maintaining rigorous theoretical foundations and practical efficacy.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Extends Sobolev IPM beyond L^p geometry using Orlicz structure
• Addresses computational challenges in generalized Sobolev IPM on graphs
• Enables efficient probability measure comparison for machine learning tasks

Innovation

• Generalized Sobolev IPM using Orlicz geometric structure
• Established connection between Orlicz-Sobolev and Musielak norms
• Reduced computation to univariate optimization on graphs