An Efficient Orlicz-Sobolev Approach for Transporting Unbalanced Measures on a Graph

TL;DR

This work addresses transporting unbalanced measures on graph metric spaces by introducing Orlicz-EPT, which reformulates Entropy Partial Transport as a standard OT on an augmented graph with a calibrated nonnegative ground cost. Building on the dual EPT and a graph-based Orlicz-Sobolev framework, it then derives Orlicz-Sobolev transport (OST), a scalable regularization whose value can be computed via a single univariate optimization and which connects to GST, Sobolev transport, and unbalanced Sobolev transport in various limits. Theoretical results establish OST as a divergence and, under mild assumptions, a metric, while linking it to existing distances and showing limit cases that recover familiar graph-OT notions. Empirically, OST achieves several orders of magnitude faster computation than Orlicz-EPT and demonstrates competitive performance in document classification and topological data analysis, indicating strong practical potential for unbalanced transport on graphs.

Abstract

We investigate optimal transport (OT) for measures on graph metric spaces with different total masses. To mitigate the limitations of traditional $L^p$ geometry, Orlicz-Wasserstein (OW) and generalized Sobolev transport (GST) employ Orlicz geometric structure, leveraging convex functions to capture nuanced geometric relationships and remarkably contribute to advance certain machine learning approaches. However, both OW and GST are restricted to measures with equal total mass, limiting their applicability to real-world scenarios where mass variation is common, and input measures may have noisy supports, or outliers. To address unbalanced measures, OW can either incorporate mass constraints or marginal discrepancy penalization, but this leads to a more complex two-level optimization problem. Additionally, GST provides a scalable yet rigid framework, which poses significant challenges to extend GST to accommodate nonnegative measures. To tackle these challenges, in this work we revisit the entropy partial transport (EPT) problem. By exploiting Caffarelli & McCann (2010)'s insights, we develop a novel variant of EPT endowed with Orlicz geometric structure, called Orlicz-EPT. We establish theoretical background to solve Orlicz-EPT using a binary search algorithmic approach. Especially, by leveraging the dual EPT and the underlying graph structure, we formulate a novel regularization approach that leads to the proposed Orlicz-Sobolev transport (OST). Notably, we demonstrate that OST can be efficiently computed by simply solving a univariate optimization problem, in stark contrast to the intensive computation needed for Orlicz-EPT. Building on this, we derive geometric structures for OST and draw its connections to other transport distances. We empirically illustrate that OST is several-order faster than Orlicz-EPT.

Figures (7)

• Figure 1: Time consumption.
• Figure 2: Document classification on graph ${\mathbb G}_{\text{Sqrt}}$. 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 3: TDA on graph ${\mathbb G}_{\text{Sqrt}}$. For each dataset, the numbers in the parenthesis are respectively the number of PD; and the maximum number of points in PD.
• Figure 4: A geodetic graph illustration. The set of nodes $V$ has $10$ nodes, i.e., $V = \left\{x_1, x_2, \dotsc, x_{10}\right\}$. The set of edges $E$ has $15$ edges, i.e., $E = \left\{e_1, e_2, \dotsc, e_{15}\right\}$ where each edge weight/length is set to one, i.e., $w_{e_j}=1$, for $1 \le j \le 15$. For any $x_i, x_j$, there is a unique shortest path between them, with a length $2$. Let $x_1$ be the unique-path root node (i.e., $z_0 = x_1$) and $\widetilde{{\mathbb G}}$ be a subgraph containing 3 nodes $\left\{x_6, x_8, x_9\right\}$ and 2 edges $\left\{e_{12}, e_{15}\right\}$, then we have $\Lambda(x_6) = \gamma(e_6) = \widetilde{{\mathbb G}}$.
• Figure 5: Time consumption on graph ${\mathbb G}_{\text{Log}}$.

Theorems & Definitions (41)

• Proposition 3.1
• Remark 3.2: Nonnegativity
• Proposition 3.3: Monotonicity
• Proposition 3.4: Entropic regularization
• Proposition 3.5: Bounds
• Definition 4.1: Orlicz-Sobolev transport (OST)
• Theorem 4.2: Univariate optimization problem for OST
• Corollary 4.3: Discrete case
• Remark 4.4: Non-physical graph
• Remark 4.5: Complementary pairs of $N$-functions