Title: An Efficient Orlicz-Sobolev Approach for Transporting Unbalanced Measures on a Graph
Venue: NeurIPS 2025
Accepted: yes
Keywords: Orlicz-Wasserstein

generalized Sobolev transport

Orlicz geometric structure

Lp geometric structure

measures on a graph

measures with different total mass
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 \emph{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's insights, we develop a novel variant of EPT endowed with Orlicz geometric structure, called \emph{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 \emph{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. Furthermore, we show preliminary evidence on the advantages of OST for measures on a graph in document classification and topological data analysis.

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

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 [12]'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. Furthermore, we show initial evidence on the advantages of OST for measures on a graph in document classification and topological data analysis.

1 Introduction

Orlicz-Wasserstein (OW) extends L p geometry by leveraging a specific class of convex functions for Orlicz geometric structure. Intuitively, OW is an instance of optimal transport (OT), which utilizes Orlicz metric as its ground cost [67, 32, 28, 3, 41]. Building on this foundation, OW has proven instrumental in advancing certain machine learning approaches. For example, recent works have leveraged OW to tackle challenging problems: Altschuler & Chewi [3] use OW as a metric

shift for Rényi divergence, enabling novel differential-privacy-inspired techniques to overcome longstanding challenges for fast convergence of hypocoercive differential equations, while Guha et al. [28] employ OW metric to significantly improve Bayesian contraction rates in hierarchical Bayesian nonparametric models by overcoming limitations raised from the usage of traditional OT with Euclidean ground cost. However, OW's high computational complexity, stemming from its two-level optimization formula, poses a significant limitation. To address this challenge, Le et al. [41] introduce generalized Sobolev transport (GST), a scalable variant of OW suitable for practical application domains, especially for large-scale settings. Moreover, Orlicz geometric structure has been successfully applied to various machine learning problems, including linear regression [4, 66], scalable approaches [21] for reinforcement learning, kernelized support vector machines, and clustering. Additionally, Orlicz metrics play a crucial role in deriving deviation bounds for polynomialgrowth functions to approximate kernel derivatives [13], and have been used as regularization in OT problems [46]. For in-depth studies on Orlicz functions, see [2, 60].

When dealing with input measures having different total masses, various approaches have been proposed in the literature to address this challenge [30, 29, 6, 12, 25, 44, 57, 58, 26, 33, 45, 18, 8, 27, 63, 64, 56, 62, 15, 5, 48, 36, 24, 16, 65, 40, 52, 7, 14, 72]. These approaches for unbalanced measures have proven effective in various domains, including color transfer [8], shape matching [8], imageto-image translation [72], multi-label learning [26], positive-unlabeled learning [15], point-cloud gradient flow [72], natural language processing [36, 40], topological data analysis (TDA) [36, 40], generative modeling [5, 72], domain adaptation [5], and robust approaches for handling noisy supports, outliers [26, 5, 48], or noisy ground cost [54, 42].

In this work, we focus on the OT problem with Orlicz geometric structure for unbalanced measures supported on a graph metric space. On one hand, OW naturally extends OT's flexibility to handle unbalanced measures by incorporating either mass constraints or marginal difference penalization, formulated as partial OT (POT) or unbalanced OT (UOT) respectively. However, these approaches result in a more complex two-level optimization problem, analogous to POT/UOT with Orlicz metric cost, which poses significant computational challenges. On the other hand, although GST provides a scalable alternative to the computationally intensive OW, it still assumes equal-mass input measures. Moreover, due to GST's definition as an optimization over the critic function, extending it to unbalanced measures is nontrivial. To address these limitations, we revisit the entropy partial transport (EPT) problem [36, 40, 72] and leverage insights from Caffarelli & McCann [12] to reformulate EPT as a standard complete OT problem. Then by carefully calibrating the corresponding ground cost for its nonnegativity, we propose Orlicz-EPT and establish a theoretical foundation for solving it by a binary search algorithmic approach. Furthermore, by exploiting the dual EPT and underlying graph structure, we introduce a novel regularization approach, leading to Orlicz-Sobolev transport (OST), which scales Orlicz-EPT for practical applications.

Contribution. In summary, our contributions are two-fold as follows:

We revisit the EPT problem and leverage Caffarelli & McCann [12]'s insights to reformulate EPT as a standard complete OT, leading to the development of the proposed Orlicz-EPT. We establish its theoretical foundations, enabling a binary search algorithmic approach for its computation. Additionally, we develop a novel regularization approach, resulting in the proposed OST. We show that OST can be efficiently computed by simply solving a univariate optimization problem, unlike the computationally intensive Orlicz-EPT.

We derive geometric structures for OST and establish its connections to other transport distances. Our empirical results demonstrate that OST is several-order faster than OrliczEPT. We also provide initial evidence on the advantages of OST for document classification and TDA.

Organization. In §2, we briefly review relevant background and notions. We revisit EPT problem and propose Orlicz-EPT in §3. In §4, we introduce the computationally efficient OST. Then we derive geometric structures for OST and draw its connections to other transport distances in §5. In §6, we discuss related work. Empirical results are presented in §7, followed by concluding remarks in §8. Proofs of key theoretical results and additional materials are deferred to the Appendices. Furthermore, we have released code for our proposed approaches. 2

2 Preliminaries

In this section, we introduce notations, and briefly review graph, and Orlicz functions.

Graph. We follow the graph setting as in [39]. Let V, E be the sets of nodes and edges respectively. We consider a connected, undirected, and physical 3 graph G = ( V, E ) with positive edge lengths { w e } e ∈ E . For continuous graph setting, we regard G as the set of all nodes in V and all points forming the edges in E . We equip G with graph metric d G ( x, y ) which equals to the length of the shortest path between x and y in G . Additionally, we assume that there exists a fixed root node z 0 ∈ V such that the shortest path connecting z 0 and x is unique for any x ∈ G , i.e., the uniqueness property of the shortest paths. We denote P ( G ) (resp. P ( G × G ) ) as the set of all nonnegative Borel measures on G (resp. G × G ) with a finite mass. Let [ x, z ] be the shortest path connecting x and z in G . For x ∈ G , edge e ∈ E , define the sets Λ( x ) and γ e as follows:

Functions on graph. By a continuous function f on G , we mean that f : G → R is continuous w.r.t. the topology on G induced by the Euclidean distance. Henceforth, C ( G ) denotes the set of all continuous functions on G . Similar notation is used for continuous functions on G × G . Given a positive scalar b > 0 , then a function f : G → R is called b -Lipschitz w.r.t. d G if | f ( x ) -f ( y ) | ≤ b d G ( x, y ) for every x and y in G .

A family of convex functions. We consider the collection of N -functions [2, §8.2] which are special convex functions on R + . Hereafter, a strictly increasing and convex function Φ : [0 , ∞ ) → [0 , ∞ ) is called an N -function if lim t → 0 Φ( t ) t = 0 and lim t → + ∞ Φ( t ) t = + ∞ .

Orlicz functional space. Given an N -function Φ and a nonnegative Borel measure ω on G , let L Φ ( G , ω ) be the linear hull of the set of all Borel measurable functions f : G → R satisfying ∫ G Φ( | f ( x ) | ) ω ( d x ) < ∞ . Then, L Φ ( G , ω ) is a normed space with the Luxemburg norm, defined as

3 Orlicz-EPT: Entropy Partial Transport with Orlicz Geometric Structure

In this section, we revisit the entropy partial transport (EPT) problem [36, 40, 72], then develop Orlicz-EPT as a variant of EPT endowed with Orlicz geometric structure.

3.1 Entropy Partial Transport (EPT)

Let γ 1 , γ 2 be the first and second marginals of γ ∈ P ( G × G ) respectively. For unbalanced measures µ, ν ∈ P ( G ) , we consider the set Π ≤ ( µ, ν ) := { γ : γ 1 ≤ µ, γ 2 ≤ ν } . 4 Additionally, let f 1 , f 2 be the Radon-Nikodym derivatives of γ 1 w.r.t. µ and of γ 2 w.r.t. ν respectively, i.e., γ 1 = f 1 µ ( 0 ≤ f 1 ≤ 1 , µ -a.e.) and γ 2 = f 2 ν ( 0 ≤ f 2 ≤ 1 , ν -a.e.).

For convex and lower semicontinuous entropy functions F 1 , F 2 : [0 , 1] → (0 , ∞ ) , and nonnegative weight functions w 1 , w 2 : G → [0 , ∞ ) , we consider the weighted relative entropies F 1 ( γ 1 | µ ) := ∫ G w 1 ( x ) F 1 ( f 1 ( x )) µ (d x ) , and F 2 ( γ 2 | ν ) := ∫ G w 2 ( x ) F 2 ( f 2 ( x )) ν (d x ) . For scalar b > 0 , scalar m ∈ [0 , ¯ m ] with ¯ m := min { µ ( G ) , ν ( G ) } , and graph metric d G as ground cost, the EPT problem is

Following [40, §3], by using entropy functions F 1 ( s ) = F 2 ( s ) := | s -1 | and considering a Lagrange multiplier λ ∈ R conjugate to the constraint γ ( G × G ) = m , we instead study the problem

where C λ ( γ ) = ∫ G w 1 µ (d x ) + ∫ G w 2 ν (d x ) -∫ G w 1 γ 1 (d x ) -∫ G w 2 γ 2 (d x ) + b ∫ G × G [ d G ( x, y ) -λ ] γ (d x, d y ) . 5

EPT as a standard OT. Following Caffarelli & McCann [12]'s insights, we can reformulate problem (4) as the standard complete OT problem. However, it is not guarantee that the corresponding standard OT has a nonnegative ground cost, e.g., see [12, 36, 40, 72]. Therefore, such OT reformulation may not be applicable to derive corresponding OW as in [67, 32, 28, 3, 41] since N -function is only defined for nonnegative domain (§2). Therefore, it is essential to carefully calibrate the ground cost of the corresponding standard OT of EPT to ensure its nonnegativity.

Precisely, following [40, Theorem 3.1], we henceforth consider λ ≥ 0 . 6 Then let ˆ s be a point outside graph G , i.e., ˆ s / ∈ G , and extend graph metric cost d G on G to a new nonnegative cost function ˆ c with bλ -deviation on ˆ G := G ∪ { ˆ s } as follows:

For unbalanced measures µ, ν , we construct corresponding probability (balanced) measures ˆ µ = µ + ν ( G ) δ ˆ s µ ( G )+ ν ( G ) and ˆ ν = ν + µ ( G ) δ ˆ s µ ( G )+ ν ( G ) . Let Π(ˆ µ, ˆ ν ) := { ˆ γ ∈ P ( ˆ G × ˆ G ) : ˆ µ ( U ) = ˆ γ ( U × ˆ G ) , ˆ ν ( U ) = ˆ γ ( ˆ G × U ) for all Borel sets U ⊂ ˆ G } , then one can recast EPT (4) as a standard OT with cost ˆ c .

Proposition 3.1. Consider the standard OT W ˆ c between probability measures ˆ µ, ˆ ν with cost ˆ c ,

then we have

The proof is placed in Appendix §A.2.1.

Thus, we have reformulated EPT (4) for unbalanced measures as a corresponding standard complete OT (7) with nonnegative ground cost. Consequently, we bypass the technical challenges inherent in unbalanced settings and can leverage abundant existing results and approaches in the standard balanced setting for OT problems with unbalanced measures on a graph.

Remark 3.2 (Nonnegativity) . Unlike existing approaches, e.g., as in [12, 36, 40, 72], the new ground cost ˆ c of the corresponding standard OT problem (7) for EPT is guaranteed to be nonnegative. Our calibration is essential for developing the associated OW from its standard OT as in [67, 32, 28, 17].

3.2 Orlicz-EPT

Following the approaches in [67, 32, 28, 17], we define Orlicz-EPT , which is EPT endowed with an Orlicz geometric structure, based on the standard OT problem (7) as follows:

It should be noted that, similar to OW, Orlicz-EPT (8) is derived from the standard OT problem (7), which circumvents all challenges coming from the setting of unbalanced measures.

We next show that the objective function of Orlicz-EPT is monotone non-increasing w.r.t. t .

Proposition 3.3 (Monotonicity) . Let Φ be an N -function, and let ˆ c be the cost given by (5) . For any probability measures ˆ µ and ˆ ν on ˆ G , define

Then the function t ∈ (0 , + ∞ ) ↦-→ A ( t ; ˆ µ, ˆ ν ) is monotone non-increasing.

The proof is placed in Appendix §A.2.2.

Computation. Observe that for a fixed t , A is a standard OT problem between ˆ µ and ˆ ν with the cost function Φ ( ˆ c ( · , · ) t ) . For computational efficiency, 7 we consider its corresponding entropic regularization [20], and show that the monotonicity is preserved.

Proposition 3.4 (Entropic regularization) . Define the entropic regularization of A as

where ε ≥ 0 and H is Shannon entropy defined by H (˜ γ ) := -∫ ˆ G × ˆ G (log ˜ γ ( x, y ) -1)d˜ γ ( x, y ) . Then the function t ∈ (0 , + ∞ ) ↦-→ A ε ( t ; ˆ µ, ˆ ν ) is monotone non-increasing.

The proof is placed in Appendix §A.2.3.

In addition, we obtain the following upper and lower bounds for A ε .

Proposition 3.5 (Bounds) . Let supp( · ) be a set of supports of a measure, then we have

where L ˆ µ, ˆ ν := max x ∈ supp(ˆ µ ) ,y ∈ supp(ˆ ν ) ˆ c ( x, y ) .

The proof is placed in Appendix §A.2.4.

Thanks to the monotonicity of A ε in Proposition 3.4 and the limits of A ε in Proposition 3.5, we can leverage the binary search approach to compute the entropic regularized Orlicz-EPT, which corresponds to the original Orlicz-EPT (8). Precisely, this entropic regularization is defined as

Discussions. Orlicz-EPT is a novel variant of EPT that incorporates Orlicz geometric structure. Leveraging Caffarelli & McCann [12]'s insights and carefully calibrating the ground cost of the corresponding standard OT to ensure its nonnegativity, we are able to bypass all challenges of unbalanced measures and derive the proposed Orlicz-EPT from the standard OT, similar to OW [67]. We note that OE Φ ,ε (11) performs binary search with quadratic complexity A ε (10), instead of dealing with super-cubic complexity A (9) in OE Φ (8). Unfortunately, the two-level optimization structure of OE Φ ,ε still retains significant complexity, severely limiting its practical applications, particularly in large-scale settings. To address this computational challenge, in the next section we exploit the dual EPT and graph structure to develop a novel regularization approach, resulting in the proposed Orlicz-Sobolev transport. This approach adopts the Orlicz geometric structure used in Orlicz-EPT, but offers a much more efficient computation.

4 Orlicz-Sobolev Transport: A Scalable Variant of Orlicz-EPT

In this section, we leverage the dual EPT and underlying graph structure to develop a novel regularization approach, resulting in the proposed Orlicz-Sobolev transport (OST).

Dual EPT. For b -Lipschitz w 1 , w 2 (w.r.t. d G ), from [40, Corollary 3.2], the dual EPT is

Let Ψ be the complement N -function of Φ and ω be a nonnegative Borel measure on G . Then let WL Ψ ( G , ω ) be the graph-based Orlicz-Sobolev space [41, Definition 3.1] associated to Ψ and ω . Inspired by the approach of GST [41], we consider the critic function f ∈ U within WL Ψ ( G , ω ) . Consequently, the b -Lipschitz constraint on the critic function f ∈ U is replaced by ∥ f ′ ∥ L Ψ ≤ b .

For f ∈ WL Ψ ( G , ω ) , we have f ( x ) = f ( z 0 ) + ∫ [ z 0 ,x ] f ′ ( y ) ω (d y ) , ∀ x ∈ G . Let 1 be the indicator function. Then by using the generalized Hölder inequality [2, §8.11] and ∥ f ′ ∥ L Ψ ≤ b , we can control the integral part over the generalized graph derivative f ′ for f ( x ) as follows:

where the last inequality is due to the increasing property of N -function Φ . Therefore, instead of the bounded constraint on the critic function f in U , we constraint only on f ( z 0 ) .

Definition 4.1 (Orlicz-Sobolev transport (OST)) . For α ∈ [0 , 1 2 ( bλ + w 1 ( z 0 ) + w 2 ( z 0 ))] , let I α := [ -w 2 ( z 0 ) -bλ 2 + α, w 1 ( z 0 ) + bλ 2 -α ] . The Orlicz-Sobolev transport for µ, ν ∈ P ( G ) is defined

where U Ψ ,α := { f ∈ WL Ψ ( G , ω ) : ∥ f ′ ∥ L Ψ ≤ b, f ( z 0 ) ∈ I α } .

Intuitively, U Ψ ,α is the collection of all functions f expressed by f ( x ) = s + ∫ [ z 0 ,x ] h ( y ) ω (d y ) , ∀ x ∈ G , where s ∈ I α , and ∥ h ∥ L Ψ ≤ b . The upper bound constraint on α is to ensure that I α is nonempty. When α = 0 , I α is the largest interval. Also, OST is an instance of the integral probability metric [49].

Computation. Given unbalanced measures µ, ν ∈ P ( G ) , for brevity let us define

Theorem 4.2 (Univariate optimization problem for OST) . OST can be computed as follows

The proof is placed in Appendix §A.2.5.

We derive the discrete case for OST which provides an explicit expression for the integral in (63).

Corollary 4.3 (Discrete case) . Let ω be the length measure of graph G , and assume that input measures µ, ν are supported on nodes in V of graph G . 8 Then, we have

The proof is placed in Appendix §A.2.6.

Therefore, OST can be efficiently computed by simply solving the univariate optimization problem (17), thanks to the proposed novel regularization for critic functions in U Ψ ,α .

Remark 4.4 (Non-physical graph) . In §2, G is assumed to be a physical graph. Corollary 4.3 implies that OST only depends on graph structure ( V, E ) and edge weights w e when input measures are supported on nodes in V of G . Hence, OST is applicable for non-physical graph G for such cases.

Remark 4.5 (Complementary pairs of N -functions) . Corollary 4.3 also implies that one can compute OST with N -function Φ without involving its complementary N -function Ψ (17), unlike its definition (14). The univariate optimization formula (17)) for OST requires that Ψ is finite-valued, which is satisfied for any N -function Φ as it grows faster than linear.

Preprocessing for γ e . Similar to the GST computation [41], we precompute set γ e (1) for all edge e in G . More concretely, we apply the Dijkstra algorithm to recompute the shortest paths from z 0 to all other vertices in V with complexity O ( | E | + | V | log | V | ) , where | · | denotes the set cardinality.

Sparsity. Observe that for every x ∈ supp ( µ ) , its mass is gathered into µ ( γ e ) if and only if e ⊂ [ z 0 , x ] [41]. Let E µ,ν := { e ∈ E | ∃ z ∈ ( supp ( µ ) ∪ supp ( ν )) , e ⊂ [ z 0 , z ] } ⊂ E . Then it suffices to compute the summation only over edges e ∈ E µ,ν in (17) for OST, i.e., screen out all edges e ∈ E \ E µ,ν .

5 Theoretical Properties

In this section, we leverage the computational efficiency of OST to derive its geometric structure and explore its connections to other transport distances.

Geometric structures of OST.

Proposition 5.1 (Geometric structure) . Let 0 ≤ α < bλ 2 +min { w 1 ( z 0 ) , w 2 ( z 0 ) } and µ, ν, σ ∈ P ( G ) .

iii) With an additional assumption w 1 ( z 0 ) = w 2 ( z 0 ) , then OS Φ ,α is a metric.

We next establish connections of OST with other transport distances, including GST [41], Sobolev transport (ST) [39], unbalanced Sobolev transport (UST) [40].

Connection of OST with GST. Denote GS Φ for the GST with N -function Φ .

Connection of OST with ST. Denote S p for the p -order ST, for 1 < p < ∞ .

Connection of OST with UST. Denote US p,α for UST, for 1 < p < ∞ .

Additionally, we investigate the limit case for N -function, i.e., Φ( t ) = t , 10 for OST and Orlicz-EPT.

Proposition 5.5 (Limit case for OST) . For Φ( t ) = t , and with the same assumptions as in Corollary 4.3, then OST yields a closed-form expression as follows:

Proposition 5.6 (Limit case for Orlicz-EPT) . For Φ( t ) = t , then we have OE Φ ( µ, ν ) = KT( µ, ν ) for every µ, ν ∈ P ( G ) .

Proposition 5.7 (Relation of OST and Orlicz-EPT) . For Φ( t ) = t , length measure ω on G , b -Lipschitz w 1 , w 2 (w.r.t. d G ), α = 0 , and p = 1 , then OS Φ ,α ( µ, ν ) ≥ OE Φ ( µ, ν ) + bλ 2 ( µ ( G ) + ν ( G )) .

The proofs for these theoretical results (in §5) are respectively placed in §A.2.7-§A.2.13.

6 Related Works and Discussions

In this section, we discuss relations between our proposals with related works in the literature.

GST [41]. Proposition 5.2 shows that OST provably generalizes GST [41] for unbalanced measures. We emphasize that GST is restricted for balanced measures and is defined as an optimization over the critic function, making it nontrivial to directly extend it to accommodate unbalanced measures.

EPT [36, 40, 72]. Orlicz-EPT and OST are developed from the primal and dual EPT respectively. Notably, the corresponding standard OT following [12] is not guarantee nonnegativity for ground cost, see [12, 36, 40]. Additionally, Caffarelli & McCann [12]'s insights may not be applicable to certain other UOT formulations, such as those proposed in [6, 26, 18, 63, 64, 27, 5, 48, 52]. The calibration is essential to guarantee nonnegativity for ground cost of the corresponding standard OT for EPT, paving ways to develop Orlicz-EPT. Similar to OW, Orlicz-EPT is derived from standard OT, thereby circumventing the challenges associated with unbalanced measures. Furthermore, we derive a novel regularization, resulting in the proposed OST with an efficient computation.

UST [40] and ST [39] Proposition 5.4 shows that OST provably generalizes UST to a more general collection of N -functions. Consequently, OST also provably generalizes ST to unbalanced measures, and to a more general set of N -functions (see Proposition 5.3).

Measures on a graph. We study OT problem between two unbalanced measures supported on the same graph, a setting also explored in [40]. One should distinguish our considered problem with the research lines on computing either distances/discrepancies [55, 74, 73, 22, 11, 47] or kernels [10, 34, 53, 61] between two (different) input graphs .

7 Experiments

In this section, we illustrate that the computation of Orlicz-EPT is costly. Especially, OST is severalorder faster than Orlicz-EPT. Following the problem setups in [40], we evaluate OST for unbalanced measures supported a given graph , 11 and show initial evidences on its advantages for document classification and topological data analysis (TDA).

Document classification. We use 4 real-world document datasets: TWITTER , RECIPE , CLASSIC , and AMAZON as in [40], and summarize their characteristics in Figure 2. By regarding each word in a document as a support with a unit mass, we represent each document as a nonnegative measure. Consequently, the representations of documents with different lengths are measures with different total mass . We apply the same word embedding procedure in [40] to map words into vectors in R 300 .

TDA. We consider orbit recognition on Orbit dataset [1], and object shape classification on MPEG7 dataset [35] as in [40]. We summarize these dataset characteristics in Figure 3. We use persistence diagrams (PD), a multiset of 2 -dimensional data points summarized topological features, to represent objects of interest. We then consider each data point in PD as a support with a unit-mass, and represent PD as nonnegative measures. As a result, PD having different numbers of topological features are presented as measures with different total mass . 12

Graph. Following [40], we use the graphs G Log and G Sqrt [39, §5] for our experiments, 13 which empirically satisfy the assumptions in §2. Additionally, we set M = 10 4 for the number of nodes for these graphs, except experiments on MPEG7 dataset with M = 10 3 due to its small size.

N -function. Following [41], we consider two N -functions: Φ 1 ( t ) = exp( t ) -t -1 , and Φ 2 ( t ) = exp( t 2 ) -1 , and the limit case of N -functions, i.e., Φ 0 ( t ) = t .

Parameters. For simplicity, we follow the experimental setup in [40]. We set λ = b = 1 , α = 0 , and consider the weight functions w 1 ( x )= w 2 ( x )= a 1 d G ( z 0 , x ) + a 0 where a 1 = b and a 0 =1 . The entropic regularization ε is chosen from { 0 . 01 , 0 . 1 , 1 , 10 } , typically via cross validation.

Optimization algorithm. For OST, we use fmincon MATLAB solver with trust-region-reflective algorithm, for solving the univariate optimization problem (17).

SVM classification. For document classification and TDA, we use support vector machine (SVM) with kernels exp( -¯ t ¯ d ( · , · )) , where ¯ d is a distance/discrepancy (e.g., OST, Orlicz-EPT) for unbalanced measures on a graph, and ¯ t > 0 . We regularize Gram matrices by adding a sufficiently large diagonal term for indefinite kernels [20]. Additionally, we note that there are more than 29 Mpairs for AMAZON which we need to evaluate distances/discrepancies for SVM in each run to illustrate the experiment scale. 14

Set up. We randomly split each dataset into 70% / 30% for training and test, and use 10 repeats. We basically choose hyper-parameters via cross validation. More concretely, we choose kernel hyperparameter from { 1 /q s , 0 . 5 /q s , 0 . 2 /q s } with s = 10 , 20 , . . . , 90 , where q s is the s % quantile of a subset of distances observed on a training set; SVM regularization hyperparameter from { 0 . 01 , 0 . 1 , 1 , 10 } ; root node z 0 from a random 10 -root-node subset of V in graph G . Note that reported time consumption includes all preprocessing procedures, e.g., preprocessing for γ e for OST.

7.1 Computation

(t) = t

(t)=exp(t)-t-1

(t)=exp(t

2

)-1

10 0

10 1

10 2

10 3

10 4

10 5

Time consumption (s)

Orlicz-EPT

OST

We compare the time consumption of OST and Orlicz-EPT with Φ 1 , Φ 2 , and with the limit case Φ 0 .

Set up. We randomly sample 10 4 pairs of nonnegative measures on AMAZON dataset for evaluation. We consider M = 10 3 for graphs, and ε = 0 . 1 for Orlicz-EPT.

Results. We illustrate the time consumptions on G Sqrt in Figure 1. OST is several-order faster than Orlicz-EPT, i.e., at least 250 × , 13800 × , 11200 × for Φ 0 , Φ 1 , Φ 2 respectively. Notably, for N -functions Φ 1 , Φ 2 , Orlicz-EPT takes at least 2 . 6 days , while OST takes less than 21 seconds . Note that for the limit case Φ 0 , Orlicz-EPT is equal to EPT on a graph (Proposition 5.6), and OST admits a closed-form expression (Proposition 5.5) for a fast computation. Consequently, Orlicz-EPT and OST with Φ 0 is more computationally efficient than those with Φ 1 , Φ 2 .

7.2 Document Classification

0.67

0.68

0.69

0.7

0.71

0.72

0.73

Average Accuracy

TWITTER (3/3108/26)

10 2

10 3

10 4

Time Consumption (s)

0.4

0.45

0.5

0.55

RECIPE (15/4370/340)

10 2

10 4

0.86

0.88

0.9

0.92

0.94

0.96

0.98

CLASSIC (4/7093/197)

10 3

10 4

10 5

0.75

0.8

0.85

0.9

AMAZON (4/8000/884)

10 3

10 4

10 5

UOT

d

0

-Tree

regEPT-Tree

OST-

0

OST-

1

OST-

2

Set up. We evaluate OST with Φ 0 , Φ 1 , Φ 2 (§7.1), denote them as OSTΦ i for i = 0 , 1 , 2 . We exclude Orlicz-EPT due to their heavy computations (§7.1). Additionally, following [40], we consider

UOT [26, 63] with ground cost d G , 15 and special cases with tree-structure graph. More concretely, we randomly sample a tree from the given graph G , then consider the regularized EPT and d 0 , denoted as d 0 -Tree and regEPT-Tree [36, Proposition 3.8, Equation (9)].

Results. We show SVM results and time consumptions of kernels on G Sqrt in Figure 2. The performances of OST with all Φ functions are comparable to UOT, but the computation of UOT is more costly than OST. Additionally, OST outperforms d 0 -Tree and regEPT-Tree. However, the computations of OSTΦ 1 , OSTΦ 2 are more expensive while the computation of OSTΦ 0 is comparative to those fast-computational variants of UOT on tree (i.e., d 0 -Tree and regEPT-Tree). Moreover, OSTΦ 1 and OSTΦ 2 improve performances of OSTΦ 0 , but their computational time is several-order higher, which may imply that Orlicz geometric structure in OST may be helpful for document classification. The performances of UOT also agree with observations in [40].

7.3 Topological Data Analysis (TDA)

0

0.1

0.2

0.3

0.4

Average Accuracy

Orbit (5000/300)

10 2

10 4

10 6

Time Consumption (s)

0.3

0.4

0.5

0.6

0.7

MPEG7 (200/80)

10 0

UOT

d

0

-Tree

regEPT-Tree

OST-

0

OST-

1

OST-

2

Set up. Similarly, we evaluate OSTΦ 0 , OSTΦ 1 , OSTΦ 2 , UOT, d 0 -Tree, and regEPT-Tree for TDA.

Results. We illustrate SVM results and time consumptions of kernels on G Sqrt in Figure 3. The performances of OST with all Φ functions compare favorably with other transport distance approaches. Especially, the performances of OSTΦ 1 and OSTΦ 2 compare favorably with those of OSTΦ 0 , but it comes with higher computational cost (i.e., OSTΦ 0 has a closed-form expression (Proposition 5.5)), which may imply that Orlicz geometric structure may be also helpful for TDA tasks.

8 Conclusion

In this work, we propose novel approaches to extend OW/GST for unbalanced measures on a graph. Building on the EPT problem and leveraging Caffarelli & McCann [12]'s insights, we derive Orlicz-EPT by recasting it as a standard OT with a carefully calibrated ground cost, thereby bypassing challenges raised from unbalanced measures. Furthermore, by exploiting dual EPT and the underlying geometric structure, we formulate a novel regularization, resulting in the proposed OST, which is efficient in computation. It provably suffices to compute OST by simply solving a univariate optimization problem, unlike the computationally intensive Orlicz-EPT. Moreover, we illustrate empirical evidence on the advantages of OST in document classification and topological data analysis.

Acknowledgments and Disclosure of Funding

We thank the area chairs and anonymous reviewers for their comments. KF has been supported in part by Grant-in-Aid for Transformative Research Areas (A) 22H05106 and JST CREST JPMJCR2015. HH acknowledges the support of JSPS KAKENHI JP25H01494 and JP23K24909. TL gratefully acknowledges the support of JSPS KAKENHI Grant number 23K11243, and Mitsui Knowledge Industry Co., Ltd. grant.

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

In this appendix, we provide further theoretical results and detailed proofs in §A. Additionally, we give brief reviews on related notions used in our work, together with further discussions, and empirical results in §B.

A Detailed Proofs and Further Theoretical Results

In this section, we provide further theoretical results, and detailed proofs for all the theoretical results.

A.1 Further Theoretical Results

We investigate special cases for OST, alternative upper limit for A ε , and the limit case of N -function for entropic regularized Orlicz-EPT.

A.1.1 Special Cases for OST

We exam the special cases of OST when graph G is a tree.

Proposition A.1 (Relation of OST and a variant of regularized EPT) . Under the same assumptions as in Proposition 5.5, and assume in addition that graph G is a tree, then

where d α is a variant of the regularized EPT in [36, Equation (9)].

Proposition A.2 (Relation of OST and standard OT) . Under the same assumptions as in Proposition A.1, and assume in addition that µ ( G ) = ν ( G ) and b = 1 , then

where W d G is the standard OT with graph metric ground cost d G .

The proof is placed in Appendix §A.2.15.

A.1.2 Upper Limit of A ε w.r.t. Entropic Regularized OT

With a technical assumption that entropic regularized input is nonnegative for N -function Φ , 16 we derive an alternative upper limit of A ε w.r.t. entropic regularized OT as summarized in the following proposition.

Proposition A.3 (Upper bound w.r.t. entropic regularized OT) . We have

where W ˆ c,ε (ˆ µ, ˆ ν ) is the entropic regularized OT between probability measures ˆ µ, ˆ ν with ground cost ˆ c , defined as

The proof is placed in Appendix §A.2.16.

A.1.3 Limit Case for Entropic Regularized Orlicz-EPT

We consider the limit case for N -function, i.e., Φ( t ) = t , for OE Φ ,ε , similar to Proposition 5.6 for the original Orlicz-EPT.

Proposition A.4 (Limit case for entropic regularized Orlicz-EPT) . For Φ( t ) = t , and µ, ν ∈ P ( G ) , we have

where W ˆ c,ε is the entropic regularized optimal transport (see Equation (19) ).

The proof is placed in Appendix §A.2.17.

A.2 Detailed Proofs

A.2.1 Proof for Proposition 3.1

Proof. Consider the cost function ˜ c on ˆ G as follows:

Additionally, for unbalanced measures µ, ν , we construct corresponding balanced measures ˜ µ := µ + ν ( G ) δ ˆ s and ˜ ν := ν + µ ( G ) δ ˆ s where measures ˜ µ, ˜ ν have the same total mass ( µ ( G )+ ν ( G )) . Let ˜ Π(˜ µ, ˜ ν ) := { ˜ γ ∈ P ( ˆ G × ˆ G ) : ˜ µ ( U ) = ˜ γ ( U × ˆ G ) , ˜ ν ( U ) = ˜ γ ( ˆ G × U ) for all Borel sets U ⊂ ˆ G } , then following [40, Lemma A.8], we have

where W ˜ c (˜ µ, ˜ ν ) is a standard complete OT between two balanced measures ˜ µ and ˜ ν (i.e., having the same total mass µ ( G ) + ν ( G ) ), with cost ˜ c , defined as

Moreover, from Equation (22), we have

where the equality in Equation (24) is due to ˜ c ( x, y ) = ˆ c ( x, y ) -bλ for all x, y ∈ ˆ G . Hence, the proof is completed.

A.2.2 Proof for Proposition 3.3

Proof. From the definition, we have

Let 0 < t 1 ≤ t 2 < ∞ , denote ˜ γ ∗ t 1 , ˜ γ ∗ t 2 as the optimal transport plans of A ( t 1 ; ˆ µ, ˆ ν ) , A ( t 2 ; ˆ µ, ˆ ν ) respectively. Then, we have

where the second inequality is due to the strictly increasing property of the N -function Φ . Hence, the proof is completed.

A.2.3 Proof for Proposition 3.4

Proof. The result is followed by the same reasoning as in the proof for Proposition 3.3 where we leverage the strictly increasing property of the N -function Φ and the optimal transport plans for A ε .

More concretely, let 0 < t 1 ≤ t 2 < ∞ , denote ˜ γ ∗ t 1 , ˜ γ ∗ t 2 as the optimal transport plans of A ε ( t 1 ; ˆ µ, ˆ ν ) , A ε ( t 2 ; ˆ µ, ˆ ν ) respectively. Then, we have

Hence, the proof is completed.

A.2.4 Proof for Proposition 3.5

Proof. We provide the proof for the lower and upper limits for A ε as follows:

For lower limit. From the definition in Equation (10), we have

Additionally, since N -function Φ is strictly increasing, we have

For convenience, given any ˜ γ ∈ Π(ˆ µ, ˆ ν ) , we define

From the definition of H in Proposition 3.4, for any ˜ γ ∈ Π(ˆ µ, ˆ ν ) , we have

where the inequality is followed by using [19, Lemma 2.1.1] (i.e., ¯ H ( γ ) ≥ 0 ).

Thus, we have

The proof for the lower limit is completed.

For upper limit. For any ˜ γ ∈ Π(ˆ µ, ˆ ν ) , we have

where we use the Jensen's inequality for the second row.

Additionally, for any ˜ γ ∈ Π(ˆ µ, ˆ ν ) , we have

where we apply [19, Theorem 2.2.1 and Theorem 2.6.5] for the inequality in the second row. Thus, we have

Taking the infimum of ˜ γ in Π(ˆ µ, ˆ ν ) , we obtain

Therefore, by choosing t = W ˆ c (ˆ µ, ˆ ν ) Φ -1 (1+ ε [ H (ˆ µ )+ H (ˆ ν ) -1]) , then we have

The proof for the upper limit is completed.

A.2.5 Proof for Theorem 4.2

Proof. For f ∈ WL Φ ( G , ω ) , as in Equation (63), we have

Thus, following the Definition 4.1, we have

Thus, we can rewrite OS Φ ,α ( µ, ν ) as follows:

where recall that Θ is defined in Equation (15).

Additionally, recall that the indicator function of the shortest path [ z 0 , x ] is as follows:

We rewrite the objective function for the first term of OS Φ ,α ( µ, ν ) in Equation (30) as follows:

where we apply the Fubini's theorem to interchange the order of integration for the second row, and use the definition of Λ in Equation (1) for the last row. Consequently, following [60, Proposition 10, pp.81] and notice that ∥ bf ′ ∥ L Ψ = b ∥ f ′ ∥ L Ψ for b > 0 , we have

where ˜ f ( x ) := b ( µ (Λ( x )) -ν (Λ( x ))) , ∀ x ∈ G , and we write ∥ ∥ ∥ ˜ f ∥ ∥ ∥ Φ for the Orlicz norm of ˜ f with N -function Φ [60, Definition 2, pp.58] (i.e., see a review in Equation (58) in §B.1.3).

Moreover, following [60, Theorem 13, pp.69], we also have

Hence, putting these Equations (30), (34), (35) together, we have

The proof is completed.

A.2.6 Proof for Corollary 4.3

Proof. Following Theorem 4.2, we have

We next follow the same reasoning as in [41, Corollary 3.4] to compute the integral in (37) by an explicit expression.

For an edge e between two nodes u, v ∈ V of graph G , then u, v are also two data points in R n as G is a physical graph. For convenience, denote ⟨ u, v ⟩ as the line segment in R n connecting the two data points u, v , and ( u, v ) as the same line segment but without its two end-points. Therefore, we have e = ⟨ u, v ⟩ .

Additionally, for any x ∈ ( u, v ) , we have y ∈ G \ ( u, v ) belongs to Λ( x ) if and only if y ∈ γ e (see Equation (1) for the definitions of Λ( x ) and γ e ). Thus, we have

Consider the case where ω is the length measure of graph G , we have ω ( { x } ) = 0 for every x ∈ G . Consequently,

Additionally, for measures µ, ν supported on nodes V of G , and using Equation (38), then we have

for every edge e = ⟨ u, v ⟩ ∈ E of graph G .

Therefore, we can rewrite the identity (39) as follows:

By combining it with (37), we obtain

Hence, the proof is completed.

A.2.7 Proof for Proposition 5.1

Proof. The proof for each property on geometric structure of OST is as follows:

i) The result is directly followed from Equation (14) in Theorem 4.2 with the observation that

and

ii) From Definition 4.1, choosing f = 0 , then f ∈ U Ψ ,α , and for any µ, ν ∈ P ( G ) , we have that OS Φ ,α ( µ, ν ) ≥ 0 .

Assume that OS Φ ,α ( µ, ν )( µ, ν ) = 0 . Then, from Theorem 4.2, we obtain

Additionally, for 0 ≤ α < bλ 2 +min { w 1 ( z 0 ) , w 2 ( z 0 ) } , we have Θ > 0 . Consequently, we must have

Thus, µ (Λ( x )) = ν (Λ( x )) , ∀ x ∈ G .

By applying [40, Lemma A.9], 17 it leads to µ = ν .

Moreover, from Definition 4.1, we also have OS Φ ,α ( µ, µ ) = 0 .

Furthermore, for any feasible function f ∈ U Ψ ,α , we have

Therefore, by taking the infimum for f ∈ U Ψ ,α , it implies that

Hence, OS Φ ,α satisfies the triangle inequality.

iii) With an additional assumption w 1 ( z 0 ) = w 2 ( z 0 ) , then for any function f ∈ U Ψ ,α , we also have ( -f ) ∈ U Ψ ,α .

Therefore, from Definition 4.1, we obtain OS Φ ,α ( µ, ν ) = OS Φ ,α ( ν, µ ) .

Thus, together with results in ii), we have OS Φ ,α is a metric.

A.2.8 Proof for Proposition 5.2

Proof. For µ ( G ) = ν ( G ) and b = 1 , then following Theorem 4.2 for OST and [41, Theorem 3.3] for GST, we have

The proof is completed.

A.2.9 Proof for Proposition 5.3

Proof. For µ ( G ) = ν ( G ) , b = 1 , by applying Proposition 5.2, we have

where we recall that GS Φ is the GST for balanced measures on a graph.

Additionally, for 1 < p < ∞ and N -function Φ( t ) = ( p -1) p -1 p p t p , by leveraging [41, Proposition 4.4] for the connection between GST and ST, we have

Therefore, by combining Equations (40) and (41), we obtain

The proof is completed.

A.2.10 Proof for Proposition 5.4

Proof. For N -function Φ( t ) = ( p -1) p -1 p p t p with 1 < p < ∞ , from Theorem 4.2, we have

For convenience, for k > 0 , let

i.e., the objective function of the univariate optimization problem for OS Φ ,α .

We next consider two cases:

Case 1: ∫ G | µ (Λ( x )) -ν (Λ( x )) | p ω ( d x ) = 0 . Then, we have

Consequently, from Equation (42), we have

̸

Case 2: ∫ G | h ( x ) | p ω ( d x ) = 0 . Then, we have

Therefore, Equation (42), we have

for some finite number k 0 ∈ (0 , + ∞ ) satisfying T ′ ( k 0 ) = 0 . Additionally, we have

Consequently, by solving the equation T ′ ( k 0 ) = 0 w.r.t. k 0 , we obtain

Therefore, by plugging this value of k 0 into T , we have

Thus, by plugging this value of T ( k 0 ) into Equation (43), we obtain

Hence, we have shown that OS Φ ,α ( µ, ν ) = US p,α ( µ, ν ) in both cases. The proof is completed.

A.2.11 Proof for Proposition 5.5

Proof. Following Corollary 4.3, we have

For Φ( t ) = t , then we have

Hence, the proof is completed.

A.2.12 Proof for Proposition 5.6

Proof. From Equation (8), we have

For Φ( t ) = t , we further have

Then, the infimum ( t ∗ , ˜ γ ∗ ) satisfies

Therefore, we obtain t ∗ = ∫ ˆ G × ˆ G ˆ c ( x, y ) d ˜ γ ∗ ( x, y ) = W ˆ c (ˆ µ, ˆ ν ) .

Hence, we have

The proof is completed.

The proof is completed.

A.2.13 Proof for Proposition 5.7

Proof. For Φ( t ) = t , p = 1 , from Theorem 4.2, we have

Additionally, for Φ( t ) = t , from Proposition 5.6, we have

With additional assumptions that λ ≥ 0 and the nonnegative weight functions w 1 , w 2 are b -Lipschitz w.r.t. d G , then by applying [40, Corollary 3.2], we have

Consequently, for α = 0 , and the length measure ω on G , then following [40, Proposition 5.2], we have

The proof is completed.

A.2.14 Proof for Proposition A.1

Proof. From Proposition 5.5, we have

For the case when G is a tree, then following [40, Proposition 5.3 i)], we further have

Thus, from Equations (44) and (46), we have

A.2.15 Proof for Proposition A.2

Proof. From Equation (44) in the proof of Proposition A.1, we have

Additionally, when G is a tree, and with an additional assumption that µ ( G ) = ν ( G ) , by applying [40, Proposition 5.3 ii)], and notice that p = 1 and b = 1 , we obtain

where recall that W d G is the standard optimal transport with graph metric ground cost d G .

Hence, from Equations (47) and (48), we get

The proof is completed.

A.2.16 Proof for Proposition A.3

Proof. Following [19, Theorem 2.2.1] and definition of conditional entropy [19, Equation 2.10], for any ˜ γ ∈ Π(ˆ µ, ˆ ν ) , we have

where we recall that ¯ H and H are defined in Equation (26) and in Proposition 3.4 respectively. Therefore, as in the proof for Proposition 3.5 in §A.2.4, from Equation (28), we have

where we assume that the entropic regularized input of N -function Φ is nonnegative in the second row (Equation (53)), i.e.,

for any ˜ γ ∈ Π(ˆ µ, ˆ ν ) .

Taking the infimum of ˜ γ in Π(ˆ µ, ˆ ν ) , we obtain

Therefore, by choosing t = W ε (ˆ µ, ˆ ν )+ ε 2 ( H (ˆ µ )+ H (ˆ ν )) Φ -1 (1+ ε [ H (ˆ µ )+ H (ˆ ν ) -1]) , then we have

The proof is completed.

A.2.17 Proof for Proposition A.4

Proof. We use the same reason as in the proof for Proposition 5.6. From Equation (11), we have

For Φ( t ) = t , we further have

Then, let ˜ γ ∗ ε is the optimal solution for the entropic regularized OT

Thus, for the infimum ( t ∗ , ˜ γ ∗ ε ) , we have

Therefore, we obtain

Hence, we have

The proof is completed.

B Brief Reviews, Further Discussions and Empirical Results

In this section, we give brief reviews on important related notions to our proposed approaches. We next give further discussions on several aspects, and provide further empirical results.

B.1 Brief Reviews

We provide brief reviews on important related notions to our proposed approaches.

B.1.1 Sobolev Transport (ST)

We briefly review main notions for Sobolev transport (ST) [39] for probability measures on a graph.

L p functional space. For a nonnegative Borel measure ω on G , denote L p ( G , ω ) as the space of all Borel measurable functions f : G → R such that ∫ G | f ( y ) | p ω (d y ) < ∞ . For p = ∞ , we instead assume that f is bounded ω -a.e. Then, L p ( G , ω ) is a normed space with the norm defined by

and for p = ∞ ,

Functions f 1 , f 2 ∈ L p ( G , ω ) are considered to be the same if f 1 ( x ) = f 2 ( x ) for ω -a.e. x ∈ G .

Graph-based Sobolev space [39]. Let ω be a nonnegative Borel measure on G , and let 1 ≤ p ≤ ∞ . A continuous function f : G → R is said to belong to the Sobolev space W 1 ,p ( G , ω ) if there exists a function h ∈ L p ( G , ω ) satisfying

Such function h is unique in L p ( G , ω ) and is called the generalized graph derivative of f w.r.t. the measure ω . The generalized graph derivative of f ∈ W 1 ,p ( G , ω ) is denoted f ′ ∈ L p ( G , ω ) .

Sobolev transport [39]. Let ω be a nonnegative Borel measure on G . Given 1 ≤ p ≤ ∞ , and let p ′ be its conjugate, i.e., the number p ′ ∈ [1 , ∞ ] satisfying 1 p + 1 p ′ = 1 . For probability measures µ, ν supported on graph G , the p -order Sobolev transport (ST) [39, Definition 3.2] is defined as

where we write f ′ for the generalized graph derivative of f , W 1 ,p ′ ( G , ω ) for the graph-based Sobolev space on G , and L p ′ ( G , ω ) for the L p functional space on G .

B.1.2 Length measure

We briefly review the length measure on graph G in [39].

Definition B.1 (Length measure [39]) . Let ω ∗ be the unique Borel measure on G such that the restriction of ω ∗ on any edge is the length measure of that edge. That is, ω ∗ satisfies:

For any edge e connecting two nodes u and v , we have ω ∗ ( ⟨ x, y ⟩ ) = ( t -s ) w e whenever x = (1 -s ) u + sv and y = (1 -t ) u + tv for s, t ∈ [0 , 1) with s ≤ t , where recall that ⟨ x, y ⟩ denotes the line segment in e connecting x and y .

ii) For any Borel set G ⊂ G , we have

Lemma B.2 ( ω ∗ is the length measure on graph [39]) . Suppose that G has no short cuts, i.e., any edge e is a shortest path connecting its two end-points. Then, ω ∗ is a length measure in the sense that

for any shortest path [ x, y ] connecting x, y . Particularly, ω ∗ has no atom in the sense that ω ∗ ( { x } ) = 0 for every x ∈ G .

B.1.3 Orlicz functions

We describe a brief review on Orlicz functions as summarized in [41] for completeness. Please see [2, 60], for in-depth studies on Orlicz functions.

Afamily of convex functions. Weconsider the collection of N -functions [2, §8.2] which are special convex functions on R + . Hereafter, a strictly increasing and convex function Φ : [0 , ∞ ) → [0 , ∞ ) is called an N -function if lim t → 0 Φ( t ) t = 0 and lim t → + ∞ Φ( t ) t = + ∞ .

Examples of N -functions. Some popular examples for N -functions [2, §8.2] are

Φ( t ) = t p with 1 < p < ∞ .

Φ( t ) = exp( t ) -t -1 .

Φ( t ) = exp( t p ) -1 with 1 < p < ∞ .

For Luxemburg norm. For Luxemburg norm (see Equation (2)) for Orlicz functional space, the infimum in its definition is attained [2, §8.9].

Complementary function. For N -function Φ , its complementary function Ψ : R + → R + [2, §8.3] is the N -function, defined as follows

Examples of complementary pairs of N -functions. Some popular complementary pairs of N -functions [2, §8.3], [60, §2.2] are as follows:

Φ( t ) = t p p and Ψ( t ) = t q q where q is the conjugate of p , i.e., 1 p + 1 q = 1 and 1 < p < ∞ .

Φ( t ) = exp( t ) -t -1 and Ψ( t ) = (1 + t ) log(1 + t ) -t .

For the N -function Φ( t ) = exp( t p ) -1 with 1 < p < ∞ , its complementary N -function yields an explicit for, but not simple [60, §2.2], see [41, §A.8] for the detailed derivation of the complementary N -function.

Young inequality. Let Φ , Ψ be a pair of complementary N -functions, then we have

Orlicz norm. Besides the Luxemburg norm, the Orlicz norm [60, §3.3, Definition 2] is also a popular norm for L Φ ( G , ω ) , defined as

where Ψ is the complementary N -function of Φ .

Computation for Orlicz norm. By applying [60, §3.3, Theorem 13], we can rewrite the Orlicz norm as follows:

Therefore, one can use any second-order method, e.g., fmincon solver in MATLAB (with trust region reflective algorithm), for solving the univariate optimization problem.

Equivalence [2, §8.17] [50, §13.11]. The Luxemburg norm is equivalent to the Orlicz norm. In fact, we have

Connection between L p and L Φ functional spaces. When the convex function Φ( t ) = t p , for 1 < p < ∞ , we have

Generalized Hölder inequality. Let Φ , Ψ be a pair of complementary N -functions, then generalized Hölder inequality w.r.t. Luxemburg norm [2, §8.11] is as follows:

Additionally, we have the generalized Hölder inequality w.r.t. Luxemburg norm and Orlicz norm [50, §13.13] is as follows:

B.1.4 Wasserstein Distance and Orlicz Wasserstein (OW)

We briefly review p -Wasserstein distance with graph metric cost, and the Orlicz Wasserstein (OW) for probability measures on a graph.

Wasserstein distance with graph metric cost. Given 1 ≤ p < ∞ , and probability measures µ and ν supported on graph G , then the p -order Wasserstein distance is defined as follows:

where Π( µ, ν ) := { γ ∈ P ( G × G ) : γ 1 = µ, γ 2 = ν } , and γ 1 , γ 2 are the first and second marginals of γ respectively.

Orlicz Wasserstein (OW). Following Guha et al. [28, Definition 3.2], the OW with the N -function Φ for probability measures µ, ν supported on graph G is defined as follows:

where recall that Π( µ, ν ) is the set of all couplings between µ and ν .

B.1.5 Generalized Sobolev transport (GST)

We briefly review main results on generalized Sobolev transport (GST) [41] for probability measures on a graph.

Graph-based Orlicz-Sobolev space [41]. Let Φ be an N -function and ω be a nonnegative Borel measure on graph G . A continuous function f : G → R is said to belong to the graph-based Orlicz-Sobolev space WL Φ ( G , ω ) if there exists a function h ∈ L Φ ( G , ω ) satisfying

Such function h is unique in L Φ ( G , ω ) and is called the generalized graph derivative of f w.r.t. the measure ω . This generalized graph derivative of f is denoted as f ′ .

Generalized Sobolev transport (GST) [41]. Let Φ be an N -function and ω be a nonnegative Borel measure on G . For probability measures µ, ν on a graph G , the generalized Sobolev transport (GST) is defined as follows:

where Ψ is the complementary function of Φ (see (57)).

B.1.6 Unbalanced Sobolev transport (UST)

We give a brief review on main results of unbalanced Sobolev transport (UST) [40] for measures on a graph, possibly having different total masses.

The regularized set U α p ′ for critic function [40]. For 1 ≤ p ≤ ∞ and 0 ≤ α ≤ 1 2 [ bλ + w 1 ( z 0 ) + w 2 ( z 0 )] , let U α p ′ be the collection of all functions f ∈ W 1 ,p ′ ( G , ω ) satisfying

and

Equivalently, U α p ′ is the collection of all functions f of the form

with s ∈ I α and with h : G → R being some function satisfying

Unbalanced Sobolev transport (UST) [40]. Let ω be a nonnegative Borel measure on graph G . Given 1 ≤ p ≤ ∞ and 0 ≤ α ≤ 1 2 [ bλ + w 1 ( z 0 ) + w 2 ( z 0 )] . For unbalanced measures µ, ν ∈ P ( G ) , the unbalanced Sobolev transport (UST) is defined as follows

For simplicity, we also use US p for the p -order UST when the context for α is clear.

Equal measures on a graph [40].

Lemma B.3 (Lemma A.9 in [40]) . For unbalanced measures µ, ν ∈ P ( G ) , then µ = ν if and only if µ (Λ( x )) = ν (Λ( x )) for every x in G .

B.1.7 Regularized EPT and Distance d α

We briefly review main results for the regularized EPT and distance d α in [36] for probability measures on tree T .

Regularized set of critic functions [36]. Let L α be a collection of all functions f of the form

where r is the tree root, and s is a constant in the interval [ -w 2 ( r ) -bλ 2 + α, w 1 ( r ) + bλ 2 -α ] and with ∥ g ∥ L ∞ ( T ) ≤ b .

Regularized EPT [36]. For unbalanced measures µ, ν supported on tree T , the regularized EPT is defined as follows:

Following [36, Proposition 3.8], we have

with i := 1 if µ ( T ) ≥ ν ( T ) and i := 2 if µ ( T ) < ν ( T ) .

Distance d α [36]. We briefly review the definition of distance d α in [36]

Following [36, Proposition 3.10], the distance d α is a metric.

B.2 Further Discussions

We give further discussions and details for various aspects in our work. For completeness, we recall important discussions on the graph in [39] (for Sobolev transport for probability measures), since these discussions and results are also applied and/or easily adapted for our proposed approaches.

Further details for the computation of Orlicz-EPT and OST. We describe further details for the computation for Orlicz-EPT and OST.

For Orlicz-EPT. Following the theoretical ground derived for the computation of the entropic regularized Orlicz-EPT, i.e., the objective function is monotone non-increasing (Proposition 3.4), and the lower and upper limits for the objective functions (Proposition 3.5), one can compute the entropic regularized Orlicz-EPT by a binary search algorithmic approach. For completeness, we straightforwardly describe the pseudo-code for it in Algorithm 1. Additionally, notice that we can leverage Proposition A.3 in §A.1.2 to alternatively set the initial value for t ℓ in Algorithm 1 (line 3).

For OST. For popular N -function, it is easy to derive its gradient and Hessian for the objective function of the univariate optimization problem. Therefore, in our experiments, we leverage the fmincon MATLAB solver with the trust-region-reflective algorithm to solve the univariate optimization problem for OST computation.

Algorithm 1 Compute entropic regularized Orlicz-EPT OE

Φ ,ε Input: Input measures µ, ν , function Φ , graph G , parameters b, λ, ε , and stopping threshold ¯ ε Output: entropic regularized Orlicz-EPT OE Φ ,ε ( µ, ν ) 1 Construct ˆ G = G ∪ { ˆ s } and corresponding nonnegative cost function ˆ c (§3) 2 Construct corresponding probability measures ˆ µ = µ + ν ( G ) δ ˆ s µ ( G )+ ν ( G ) and ˆ ν = ν + µ ( G ) δ ˆ s µ ( G )+ ν ( G ) . 3 Set t r = L ˆ µ, ˆ ν Φ -1 (1+ ε ) and t ℓ = W ˆ c (ˆ µ, ˆ ν ) Φ -1 (1+ ε [ H (ˆ µ )+ H (ˆ ν ) -1]) 4 while t r -t ℓ > ¯ ε do 5 Set t m = t ℓ + t r 2 6 Compute f m = A ε ( t m ; ˆ µ, ˆ ν ) 7 if f m ≤ 1 then 8 Set t r = t m 9 if f m == 1 then 10 Break else 11 Set t ℓ = t m 12 Return OE Φ ,ε ( µ, ν ) = ( µ ( G ) + ν ( G )) ( t r -bλ )

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Figure 4: A geodetic graph illustration. The set of nodes V has 10 nodes, i.e., V = { x 1 , x 2 , . . . , x 10 } . The set of edges E has 15 edges, i.e., E = { e 1 , e 2 , . . . , e 15 } where each edge weight/length is set to one, i.e., w e j = 1 , for 1 ≤ j ≤ 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 ˜ G be a subgraph containing 3 nodes { x 6 , x 8 , x 9 } and 2 edges { e 12 , e 15 } , then we have Λ( x 6 ) = γ ( e 6 ) = ˜ G .

Further discussion on Orlicz-EPT and OST. Following Proposition 5.2, we can view OST as an extension of GST [41] for handling unbalanced measures, and Orlicz-EPT as an extension of OW [67] for unbalanced measures. Notably, when the input measures have equal mass (i.e., µ ( G ) = ν ( G ) ), and b = 1 , Orlicz-EPT reduces to OW. Furthermore, Orlicz-EPT shares the same computational complexity as OW, as shown in Equation (8). It is worth noting that GST is a scalable variant of OW for balanced measures, in the same sense, OST can be seen as a scalable variant of Orlicz-EPT.

As a result, Orlicz-EPT is applicable for all applications of OW and extends OW to handle unbalanced input measures, whereas OST may not reserve all the properties of OW.

Like OW, Orlicz-EPT involves a two-level optimization problem, limiting its applicability to smallscale domains. In contrast, OST, similar to GST, is scalable and can be applied to large-scale domains.

Illustrations for notations on a graph. We illustrate notions on a graph in our work in Figure 4.

Geodetic graph. Recall that for a graph, for every pairs of nodes, the shortest path between them is unique, then it is called geodetic graph [39]. Therefore, geodetic graphs are special examples

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Path length. As discussed in [39], we can compute a path length connecting any two points x, y ∈ G , where the points x, y are not necessary to be nodes in V (but even for any point on an edge as well). Indeed, for the points x, y ∈ R n belonging to the same edge e = ⟨ u, v ⟩ , connecting two nodes u, v ∈ V . We have

for some scalars t, s ∈ [0 , 1] . Thus, the length of the path connecting x, y along the edge e (i.e., the line segment ⟨ x, y ⟩ ) can be computed by | t -s | w e .

Consequently, the length for an arbitrary path in G can be computed by breaking down into pieces over edges, and then summing over their corresponding lengths [39].

Extension to measures supported on G . For the discrete case, similar to ST [39], OST (17) is extendable for measures with finite supports on G , i.e., measures which may have supports on edges, by following the strategy to compute a path length for points in G . Precisely, we break down edges containing supports into pieces, and then sum over their corresponding values, instead of the sum over edges as in (17) for OST.

About the uniqueness property of the shortest paths on G . As discussed in [39], note that edge length is a real nonnegative scale, w e ∈ R + for any edge e ∈ E in G ., it is almost surely that every node in V can be regarded as unique-path root node since with a high probability, lengths of paths connecting any two nodes in graph G are different.

Moreover, for some special graph, e.g., a grid of nodes, there is no unique-path root node for such graphs. However, by perturbing each node (or also perturbing edge lengths in case G is a non-physical graph) with a small deviation, the perturbed graph will satisfy the unique-path root node assumption.

For continuous case, when measures are extended to support on G , and input measures are fully supported, then for some finite special nodes where there are multiple shortest paths, we fix one of the shortest paths to those points, and treat it as the chosen shortest path for those special points. However, for practical applications, the number of supports are finite, which bypasses the mentioned issue.

For the given graph setting. As in [39], we assume that the graph metric space is given. The question to learn an optimal graph metric structure from data is not considered in this work and leave for future investigation.

Measures on a graph. In this work, we consider OT problem for two input unbalanced measures supported on the same graph. See [40] for the same problem setting.

The proposed approaches, Orlicz-EPT and OST, are for input unbalanced measures , i.e., to compute the distance between two unbalanced measures, on the same graph. We distinguish our considered problem to the following related problems:

Compute distances/discrepancies between two (different) input graphs. For examples, [55, 22] compute distance/discrepancy between two (different) input graphs . They are essentially different to our considered problem which computes distance between two input probability measures supported on the same graph. In particular, Le et al. [38] consider a variant of Gromov-Wasserstein problem for two input probability measures , but possibly supported on different tree metric spaces.

Compute kernels between two (different) input graphs. Graph kernels are kernel functions between two input (different) graphs to assess their similarity. See [10, 34, 53] for comprehensive reviews on graph kernels. Essentially, it is different to our proposed approaches to compute distances between two input unbalanced measures on the same graph.

For (variational) OT problems on a graph. OT problems for measures supported on a graph metric space have been explored in previous studies [39-41]. Additionally, Le et al. [43] have recently studied a variational OT problem, e.g., Sobolev IPM where the critic function is constrained within

a unit ball of Sobolev norm involving both the critic function and its gradient, for graph-based measures. Notably, graph metrics extend tree metrics, which are utilized in scalable optimal transport methods like tree-sliced Wasserstein (TSW) [37, 70, 68, 71, 69]. TSW, in turn, generalizes the popular sliced-Wasserstein (SW) approach [59, 9, 51]. The graph structure offers greater flexibility and degrees of freedom compared to the tree structure in TSW and the line structure in SW. For a more in-depth discussion on the motivation for OT on a graph, please refer to [39].

Persistence diagrams for TDA. Persistence diagrams (PD) are multisets of data points in R 2 , containing the birth and death time respectively of topological features (e.g., connected component, ring, or cavity), extracted by algebraic topology methods (e.g., persistence homology) [23].

Graphs G Log and G Sqrt [39]. For completeness, we review the construction for graphs G Log and G Sqrt in [39]. We utilize the farthest-point clustering method to cluster supports of measures into at most M clusters. Then, let the vertex set V be the collection of centroids of these clusters, i.e., graph vertices. For edges, we randomly select ( M log M ) edges, and M 3 / 2 edges for graphs G Log, and G Sqrt respectively. Let ˜ E be the set of those randomly sampled edges. For each edge e , its edge length/weight w e is computed by Euclidean distance between the two corresponding end points (i.e., corresponding nodes of edge e ). Let n c be the number of connected components in ˜ G ( V, ˜ E ) . We randomly add ( n c -1) more edges between these n c connected components to form a connected graph G from ˜ G . Let E c be the set of these ( n c -1) added edges, and denote set E = ˜ E ∪ E c , then G ( V, E ) is the constructed graph.

Further discussion on graphs in experiments. For all datasets, except MPEG7 dataset, G Sqrt consists of 10K nodes and 1 million edges, while G Log comprises 10K nodes and 100K edges. Due to the smaller size of the MPEG7 dataset, we constructed G Sqrt with 1K nodes and 32K edges, and G Log with 1K nodes and 7K edges.

Datasets and Computational Devices. For the datasets in our experiments, one can contact the authors of Sobolev transport [39] to access to them. Additionally, all of our experiments are run on commodity hardware.

In our experiments, we demonstrate the effectiveness of our approach in document classification on four real-world datasets and topological data analysis (TDA), including orbit recognition for linked twist maps, i.e., a discrete dynamical system modeling flows in DNA microarrays [31], and object shape recognition in MPEG7 . These evaluations on document classification and TDA are often used for tasks involving comparing measures on a graph, see [39, 41]. We believe that such experimental coverage is rich and diverse enough.

Hyperparameter validation. We use the same validation as in [40]. Precisely, we further randomly split the training set into 70% / 30% for validation-training and validation with 10 repeats to choose hyper-parameters for the experiments.

Further discussion on hyperparameters. The performance of OST/Orlicz-EPT typically depends on the choice of the N -function Φ , similar to how kernel functions impact performance in kerneldependent machine learning frameworks. In our experiments with N-functions Φ 1 and Φ 2 (§7), we observed slightly different performances. Similar findings have been reported for GST [41].

Determining the optimal N -function Φ for OST/Orlicz-EPT in a given task is an open problem that warrants further investigation. We leave it for future work. As an interim solution, cross-validation can be used to select Φ from a set of candidate functions.

Regarding the regularization weight b , we used b = 1 in our experiments based on the results in Propositions 5.2 and 5.3, as well as for simplicity. This choice of b is supported by theoretical results on EPT on a graph [40, Lemma A.6, Remark 4.8] and has been used in previous experiments for EPT [36, 40].

For α , from the result in Proposition 5.7 and for simplicity, we use α = 0 for experiments. Such value for α is also supported by theoretical results of EPT on a graph [40, Lemma 4.4, Proposition 5.2] and used in experiments for EPT [36, 40]. Additionally, recall that I α is the largest interval when α = 0 .

The number of pairs for kernel SVM [41]. Denote N tr , N te for the number of measures used for training and test respectively. For the kernel SVM training, the number of pairs for computing the distances is ( N tr -1) × N tr 2 . For the test, the number of pairs for computing the distances is N tr × N te . Thus, for each run, the number of pairs for computing the distances for both training and test is totally N tr × ( N tr -1 2 + N te ) . In Table 1, we summarize the number of pairs which we need to evaluate distances/discrepancies for SVM in each run to illustrate the experimental scale, e.g., more than 29 Mpairs for AMAZON .

Debiased approaches for (Sinkhorn-based) UOT [63, 65]. We review the discussion and evaluation in [40] for the debiased approaches for (Sinkhorn-based) UOT in [63, 65], with the same setup as in our experiments. As noted in the main text, the debiased version for Sinkhorn-based approach for UOT [26, 63] which may be helpful for applications, and the debiased version is empirically indefinite. Both the UOT and its debiased version have the same computational complexity.

Empirical results for the debiased version [63, 65] are given in [40, Figures 41-44]. As in [40], the debiased version improve performances of UOT in some datasets, especially for datasets in TDA tasks ( Orbit and MPEG7 ). For document datasets, performances of the debiased version and UOT are comparative, i.e., the role of debias property is not clear for advantages in applications.

Broader impacts. In this work, we propose novel approaches to extend OW/GST for unbalanced measures on a graph. The proposed Orlicz-EPT is directly derived from standard OT, bypassing challenges raised from unbalanced measures, as in OW. Additionally, we formulate a novel regularization approach, resulting in the proposed OST, which is efficient in computation. Therefore, our proposals pave the ways to use OT approach endowed with Orlicz geometric structure for applications with unbalanced measures, which is common in real-world scenarios. To our knowledge, there is no foresee negative social impacts for our research.

B.3 Further Empirical Results

Further empirical results on graph G Log. We provide corresponding results as in §7 for graph G Log.

In Figure 5, we compare the time consumption of OST and Orlicz-EPT with Φ 0 , Φ 1 , Φ 2 on graph G Log.

In Figure 6, we illustrate the SVM results and time consumptions of kernels on document classification with graph G Log.

In Figure 7, we illustrate the SVM results and time consumptions of kernels on TDA with graph G Log.

Further empirical results for different hyperparameters b and α . We carry out more additional experiments for different b, α . The results for average accuracy, and the standard deviation (in the parentheses) are as follows:

· For document classification.

We present further SVM results for document classification on TWITTER dataset with graph G Sqrt and different hyperparameters b and α in Tables 2 and 3 respectively.

We present further SVM results for document classification on TWITTER dataset with graph G Log and different hyperparameters b and α in Tables 4 and 5 respectively.

· For TDA.

We present further SVM results for TDA on MPEG7 dataset with graph G Sqrt and different hyperparameters b and α in Tables 6 and 7 respectively.

We present further SVM results for TDA on MPEG7 dataset with graph G Log and different hyperparameters b and α in Tables 8 and 9 respectively.

We observe that turning these hyperparameters b and α , e.g., via cross-validation, may help to improve the performances further.

(t) = t

(t)=exp(t)-t-1

(t)=exp(t

2

)-1

10 0

10 1

10 2

10 3

10 4

10 5

Time consumption (s)

Orlicz-EPT

OST

0.67

0.68

0.69

0.7

0.71

0.72

0.73

Average Accuracy

TWITTER (3/3108/26)

10 2

10 3

10 4

Time Consumption (s)

0.4

0.45

0.5

0.55

RECIPE (15/4370/340)

10 3

10 4

10 5

0.86

0.88

0.9

0.92

0.94

0.96

CLASSIC (4/7093/197)

10 3

10 4

10 5

0.75

0.8

0.85

0.9

AMAZON (4/8000/884)

10 3

10 4

10 5

UOT

d

0

-Tree

regEPT-Tree

OST-

0

OST-

1

OST-

2

0

0.1

0.2

0.3

Average Accuracy

Orbit (5000/300)

10 2

10 4

10 6

Time Consumption (s)

0.2

0.3

0.4

0.5

0.6

0.7

MPEG7 (200/80)

10 0

UOT

d

0

-Tree

regEPT-Tree

OST-

0

OST-

1

OST-

2

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The proposed Orlicz-EPT is presented in §3, where we revisit the EPT problem, leverage Caffarelli & McCann [12]'s insights to reformulate EPT as a corresponding standard OT. By carefully calibrating the ground cost of the corresponding standard OT, we guarantee that its ground cost is nonnegative, which is essential to develop Orlicz-EPT, a variant of EPT with Orlicz geometric structure, since the N -function is only defined on the nonnegative domain. Additionally, we provide theoretical background to solve Orlicz-EPT by a binary search approach.

Additionally, we present the proposed Orlicz-Sobolev transport (OST) in §4. By leveraging the dual EPT and the underlying graph structure, we derive a novel regularization for the critic function, to develop the proposed OST. We prove that OST can be computed by simply solving a univariate optimization problem.

Furthermore, in §5, we derive geometric structures for OST, and show its connections to other transport distances. In §7, we empirically illustrate that OST is several-order faster than Orlicz-EPT. We also show evidences on the advantages of OST for unbalanced measures on a graph for document classification and TDA.

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The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

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2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We consider OT problem for unbalanced measures supported on a graph metric space.

For Orlicz-EPT, we show that it is directly derived from a standard OT, with a guarantee on the nonnegativity for its ground cost via calibration, similar to OW approach (derived from corresponding standard OT with nonnegative ground cost [67]). Therefore, we can bypass all challenges raised from unbalanced measures. We develop theoretical backgrounds to solve it by a binary search approach. However, Orlicz-EPT is still a two-level optimization problem, it has a high computational cost, illustrated in §7.

For OST, it is efficient in computation by simply solving a univariate optimization problem (§4). Although OST is a regularized approach, we theoretically show that OST generalize GST for unbalanced measures on a graph (Proposition 5.2).

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the challenge of computing optimal transport (OT) between unbalanced measures on a graph, which is common in tasks like comparing documents with different word counts. Traditional OT methods assume equal mass and often rely on $\\ell_p$-based cost functions, which limits their flexibility and scalability. The standard approaches that tackle unbalanced cases also have a higher complexity.\n\nTo overcome this, the authors introduce two contributions:\n\n* Orlicz-EPT: A new variant of entropy partial transport (EPT) that incorporates Orlicz geometry, which is a generalization of OT that uses convex functions instead of fixed powers like in $\\ell_p$. By reformulating EPT as a standard OT problem with a carefully calibrated, nonnegative ground cost, they enable the use of Orlicz-Wasserstein-like metrics even in unbalanced settings. This adds modeling flexibility, especially for data with noise or outliers, but remains computationally expensive due to its two-level optimization structure.\n\n* Orlicz-Sobolev Transport (OST): To overcome the higher complexity of Orlicz-EPT, the authors explore a dual formulation that is significantly more efficient. It defines a new transport distance using Orlicz-Sobolev spaces on graphs and simplifies the computation to a univariate optimization problem. The resulting distance has several desirable properties, such as divergence (nonnegative and zero iff measures match), being a metric under mild conditions, and generalizing several known distances like GST (Generalized Sobolev Transport), ST (Sobolev Transport), and UST (Unbalanced Sobolev Transport).\n\nThe authors demonstrate the effectiveness of OST on large-scale experiments in document classification and topological data analysis (TDA).\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n* The paper introduces a novel variant of optimal transport (OT) that extends existing formulations to the Orlicz geometry, providing a more flexible alternative to traditional $\\ell_p$-based distances for unbalanced measures.\n* It establishes strong theoretical foundations, including connections to several known OT variants (e.g., GST, ST, UST), demonstrating that the proposed distance is a meaningful generalization.\n* The effectiveness of the proposed distance is supported by empirical results on document classification and topological data analysis tasks.\n\n**Weaknesses**\n* Although the paper is mathematically rigorous, it is quite dense in places. Some sections (e.g., Section 3) would benefit from additional intuition or illustrative examples.\n\nQUESTIONS:\n* Can the authors explain the reason behind the choices in your experiments? While some of the settings are said to follow prior work, it’s not entirely clear that they are optimal for the current setup. In particular, could the authors provide more intuition behind the selection of the Orlicz functions $\\phi_0, \\phi_1$ , and $\\phi_2$ ? Additionally, I would appreciate an explanation for the specific form of the weight functions $w_1$ and $w_2$. Have the authors performed any ablation or sensitivity analysis to study the impact of these parameter choices on performance or efficiency?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of optimal transport for unbalanced measures on graph metric spaces. The key idea is to leverage insights from Caffarelli and McCann to reformulate the unbalanced optimal transport problem as a corresponding complete transport problem with a nonnegative ground cost. To further enhance this formulation, the Orlicz geometric structure is introduced. By defining measures on a graph, the solution to the reformulated transport problem can be reduced to solving a univariate optimization problem. Experimental results demonstrate that the proposed method outperforms existing approaches in both accuracy and computational efficiency.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is well written and clearly structured.\n- The proposed method builds upon and integrates several well-established techniques. In particular:\n1. It reformulates problem (4), which incorporates Lagrangian multipliers for unbalanced measures, into a corresponding complete optimal transport (OT) problem (problem 6) with a nonnegative ground cost—an approach inspired by the insights of Caffarelli and McCann.\n2. Problem (6) is further enhanced using the Orlicz geometric structure, following the concept of Orlicz-Wasserstein distance.\n3. While the proposed Orlicz-Wasserstein formulation involves a two-level optimization—typically computationally expensive—this is efficiently addressed by defining measures on a graph (as in [35]), allowing the solution to be reduced to a univariate optimization problem (involving the variable t of the N-function).\n\nWeaknesses:\n- The method presents an incremental development of several well-established ideas. As such, the novelty of the approach may appear limited. However, I think that, this is the first work to explore Orlicz-Wasserstein distance for unbalanced measures specifically on graph structures, which is a meaningful contribution.\n- it remains unclear how to recover the solution to problem (3) (for a fixed m) from the solution to problem (4) (for a fixed \\lambda).\n\nQUESTIONS:\nSee the weakness part.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new framework for optimal transport (OT) between unbalanced measures on a graph based on two new distances: Orlicz-EPT and Orlicz-Sobolev Transport (OST). Orlicz-EPT extends entropy-regularized partial OT by adding Orlicz geometric structure and recasting it as a standard OT problem with a calibrated nonnegative cost, enabling computation via binary search. Then, OST leverages the dual of EPT and graph structure to present a regularized OT distance that can be computed efficiently using a univariate optimization problem. The paper shows that OST generalizes several known distances, including generalized Sobolev transport, standard Sobolev transport, and unbalanced Sobolev transport, and maintains desirable geometric properties such as being a divergence or a metric. Empirical results demonstrate that OST is significantly faster than Orlicz-EPT and is competitively running in document classification and topological data analysis, suggesting its scalability and practicality.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe paper is a technically sound and well-presented contribution to the theory of optimal transport (OT) with a new pair of distances—Orlicz-EPT and Orlicz-Sobolev Transport (OST). The theoretical method is sound, basing itself on Caffarelli &amp; McCann&#39;s result to redefine entropy partial transport as a regular OT with calibrated nonnegative cost, making it possible to construct Orlicz-EPT. Subsequent to this, OST formulation via duality and a new regularization leads to an efficient computationally distance that can be calculated by way of a simple univariate optimization, much quicker than existing methods like Orlicz-EPT and UOT. The paper is nicely organized and clear in its exposition, with full definitions, proofs, and relation to existing work. OST generalizes various earlier transport distances (e.g., GST, ST, UST), and its strong theoretical guarantees are complemented with empirical behavior on document classification and topological data analysis, hinting at its viability for large-scale structured data.\n\nWeaknesses:\nDespite the theory being strong, empirical consideration is relatively narrow in scope and supported by document and topological data instead of more diverse or realistic contexts (e.g., biological or dynamic networks). Orlicz-EPT, while new, remains computationally expensive and less convenient despite theoretical innovation, and the paper fails to convincingly describe when it must be used instead of OST. Also, OST&#39;s behavior is still hyperparameter-sensitive (e.g., with respect to $\\alpha$, the choice of Orlicz function), though the paper has minimal suggestion or sensitivity analysis for making these selections. There are some technical subsections, notably the dual forms and cost calibration, that may benefit from a little more intuition or figures. Finally, certain parts of the work, particularly Orlicz-EPT, may arguably be seen as a modest extension of available forms like EPT, and experimental comparisons pretty much emphasize improvements over closely related previous work that may be extended.\n\nQUESTIONS:\n1. While Orlicz-EPT is presented as a theoretically rich generalization, its application in practice is circumscribed by prohibitive computational cost. Can the authors say anything about operations or instances for which Orlicz-EPT would be preferable to use than OST, even with this penalty in terms of computation? Adding either empirical examples or a theoretical account with an explicit statement of under what conditions the additional expressiveness of Orlicz-EPT is valuable and worthwhile and justifies its use over OST\nwould be advantageous. If authors are able to detail compelling practical applications in which Orlicz-EPT is superior to OST (in quality or quantity), this would further strengthen the importance and applicability of the contribution.\n\n2. The experiments are largely limited to text files and persistence diagrams. Can the authors supplement, or at least mention, performance in other domains like biological graphs, social networks, or dynamic/temporal graphs? Include a single additional domain or application (even in a limited form) that tests OST in a very different setting in order to more evidently demonstrate its generality. A broader empirical range would offer evidence towards arguments of practical use and increase confidence that OST is a generally applicable tool for structured data.\n\n3. How sensitive is OST to the parameters such as the Orlicz function $\\phi$, the regularization weight $b$, and the $\\alpha$ range for critic functions? Did the author test this? Add a little sensitivity analysis or some pragmatic guidance on how to select such parameters in different settings. Even a graph of accuracy against $\\alpha$ or different $\\phi$ functions would be helpful. More interpretability and usability of OST would help in ease of use by practitioners and allow practitioners to adopt and experiment with the approach in their setting.\n\n4. The paper discusses scalability based on the number of transport calculations but would be useful to also describe how the performance grows if graph size and complexity are increased. The authors can provide a brief runtime or memory analysis that alters the size of the graph (nodes/edges) or density, especially since there are graphs of thousands of nodes in practice.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors investigate numerical methods to perform unbalanced optimal transport where the support of the measures is a graph. This paper proposes Orlicz-EPT and Orlicz-Sobolev Transport. Orlicz-EPT builds on the entropy partial transport (EPT) framework, extending it with Orlicz geometry by calibrating the ground cost to ensure nonnegativity—allowing standard optimal transport techniques to be applied. To accomplish this, the authors use insights once used by Caffarelli and McCann (2010) based on the obstacle problem. However, due to its two-level optimization structure, Orlicz-EPT remains computationally intensive. The authors then introduce Orlicz-Sobolev Transport (OST) as a means to address these computational hurdles, which is a dual formulation that regularizes the critic function using Orlicz-Sobolev spaces and reduces the problem to a single-variable optimization. The authors prove several theoretical properties about OST. Empirical results show that OST achieves competitive performance in document classification and topological data analysis while being several orders of magnitude faster than Orlicz-EPT.\n\nSTRENGTHS AND WEAKNESSES:\nA main strength of the paper is the development of a novel optimal transport algorithm on graphs for unbalanced measures, which is a common situation that does not appear to lie in the scope of many existing approaches. The paper contains many theoretical results and is the first instance I have seen Orlicz geometries used in this context, which is quite nice (though, there do appear related works cited by the authors here). The Orlicz structure leads to an expensive two-layer optimization problem, which the authors find a work-around for. Their numerical speedups are impressive as well as the newfound accuracy on several tasks.\n\nA main weakness of the paper is the writing and lack of clarity. From even the abstract, there are many repetitive concepts and sentence structures that make reading this paper incredibly difficult. There are simply too many equations (in-display equations) crammed into the text. The same is true for the tables and figures---everything is so small! I think the authors are trying to write too much instead of giving a more high-level insights into the problem and their proposed solution.\n\nQUESTIONS:\n- The topic of OT on a graph (why this would happen, for example) is not appropriately motivated, and some more context would be appreciated.\n\n- It seems like the computations leading up to equation (6) is relevant enough to be a Theorem or proposition --- why is this not the case?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-05"}