Generalized Sobolev Transport for Probability Measures on a Graph

TL;DR

The paper addresses optimal transport for probability measures on graphs under non-L_p geometric priors. It introduces Generalized Sobolev Transport (GST), which fuses graph-based Sobolev duality with Orlicz function geometry to generalize ST and connect to OW, while enabling a univariate optimization for computation. GST recovers ST in the power-function (L_p) regime and aligns with OW on trees, providing substantial speedups over OW and extending OT-type geometry to broader priors. The approach yields practical benefits in document classification and topological data analysis, offering a scalable alternative to OW for graph-structured data with flexible geometry.

Abstract

We study the optimal transport (OT) problem for measures supported on a graph metric space. Recently, Le et al. (2022) leverage the graph structure and propose a variant of OT, namely Sobolev transport (ST), which yields a closed-form expression for a fast computation. However, ST is essentially coupled with the $L^p$ geometric structure within its definition which makes it nontrivial to utilize ST for other prior structures. In contrast, the classic OT has the flexibility to adapt to various geometric structures by modifying the underlying cost function. An important instance is the Orlicz-Wasserstein (OW) which moves beyond the $L^p$ structure by leveraging the \emph{Orlicz geometric structure}. Comparing to the usage of standard $p$-order Wasserstein, OW remarkably helps to advance certain machine learning approaches. Nevertheless, OW brings up a new challenge on its computation due to its two-level optimization formulation. In this work, we leverage a specific class of convex functions for Orlicz structure to propose the generalized Sobolev transport (GST). GST encompasses the ST as its special case, and can be utilized for prior structures beyond the $L^p$ geometry. In connection with the OW, we show that one only needs to simply solve a univariate optimization problem to compute the GST, unlike the complex two-level optimization problem in OW. We empirically illustrate that GST is several-order faster than the OW. Moreover, we provide preliminary evidences on the advantages of GST for document classification and for several tasks in topological data analysis.

Figures (14)

• Figure 1: Time consumption for GST and OW on ${\mathbb G}_{\text{Log}}$.
• Figure 2: Time consumption for GST and OW on ${\mathbb G}_{\text{Sqrt}}$.
• Figure 3: Document classification on graph ${\mathbb G}_{\text{Log}}$. For each dataset, the numbers in the parenthesis are respectively the number of classes; the number of documents; and the maximum number of unique words for each document.
• Figure 4: Document classification on graph ${\mathbb G}_{\text{Sqrt}}$.
• Figure 5: TDA on 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.

Theorems & Definitions (26)

• Definition 3.1: Graph-based Orlicz-Sobolev space
• Definition 3.2: Generalized Sobolev transport distance on graph
• Theorem 3.3: GST as univariate optimization problem
• Corollary 3.4: Discrete case
• Remark 3.5: GST for non-physical graph
• Remark 3.6: Complementary pairs of $N$-functions for GST.
• Remark 3.7: Sparsity in Problem \ref{['equ:GST_1d_optimization_discrete']}
• Theorem 4.1: Metrization
• Proposition 4.2: GST with different $N$-functions
• Proposition 4.3