Fused Gromov–Wasserstein: Theory & Applications

• FGW is an optimal transport metric that fuses feature similarity and structural consistency by jointly aligning data and internal geometries.
• It interpolates between Wasserstein and Gromov–Wasserstein distances via a parameter α, allowing flexible comparison of diverse structured objects.
• Advanced optimization techniques, such as barycenter computation and linear embeddings, improve efficiency and enable applications in graph analysis and beyond.

Searching arXiv for foundational and papers on Fused Gromov–Wasserstein. Fused Gromov–Wasserstein (FGW) is an optimal-transport distance for structured objects that have both feature information and relational structure. In its foundational formulation, a structured object is represented as a probability measure on a product of a structural space and a feature space, and FGW compares two such objects by optimizing a single coupling that simultaneously penalizes feature mismatch and distortion of intra-object relations. This construction interpolates between Wasserstein transport on features and Gromov–Wasserstein comparison of internal geometry, and it is used in the cited literature for labeled graphs, time series, images, meshes, and other structured data (Vayer et al., 2018).

1. Foundational formulation

The theoretical formulation models a structured object as a triplet $(X\times A,d_X,\mu)$, where $(X,d_X)$ represents structure, $A\subset \Omega$ is a common feature space, and $\mu$ is a fully supported probability measure on the product space. Given two structured objects $(X\times A,d_X,\mu)$ and $(Y\times B,d_Y,\nu)$, FGW is defined for $\alpha\in[0,1]$ and $p,q\ge 1$ by

$$d^{\Omega}_{FGW,\alpha,p,q}(\mu,\nu) = \left( \inf_{\pi\in \Pi(\mu,\nu)} \int \Big( (1-\alpha)\, d(a,b)^q + \alpha\, |d_X(x,x')-d_Y(y,y')|^q \Big)^p \, d\pi\, d\pi \right)^{1/p}.$$

The coupling $\pi$ is therefore asked to align feature values and structural relations with the same transport plan, rather than by optimizing two independent objectives (Vayer et al., 2018).

In the discrete graph setting, this becomes a transport problem on attributed nodes. A graph may be encoded as

$(X,d_X)$0

with node weights $(X,d_X)$1, node features $(X,d_X)$2, and a structural matrix $(X,d_X)$3 describing pairwise relations; a second graph is represented analogously by $(X,d_X)$4 and $(X,d_X)$5. The discrete FGW loss for $(X,d_X)$6 can be written as

$(X,d_X)$7

and FGW is the minimum of this quantity over admissible couplings $(X,d_X)$8 with prescribed marginals. The feature term compares nodes directly, whereas the structural term compares how matched nodes relate to other matched nodes inside their respective graphs (Vayer et al., 2018).

A persistent misconception is to regard FGW as a mere sum of a Wasserstein distance and a Gromov–Wasserstein distance. The cited formulations instead optimize a single probabilistic matching whose cost is quadratic in the coupling because the structural term couples pairs of matches. This joint coupling is what allows FGW to compare graphs of different sizes and objects whose features and structural coordinates live in different spaces (Vayer et al., 2018).

2. Interpolation, metric behavior, and geometry

The interpolation property is central. In the foundational theory, $(X,d_X)$9 suppresses the structural contribution and FGW converges to a Wasserstein-type comparison of feature marginals, whereas $A\subset \Omega$0 suppresses the feature contribution and yields Gromov–Wasserstein comparison of structural marginals. This is not only heuristic: the limit statements are proved in the theoretical treatment, and lower bounds explicitly relate FGW to both Wasserstein and Gromov–Wasserstein components (Vayer et al., 2018).

FGW also has nontrivial metric behavior. In the discrete graph formulation, for $A\subset \Omega$1, assuming the structure matrices are distance matrices, FGW defines a metric on structured data modulo measure-preserving isometries that also preserve features; for $A\subset \Omega$2, it is only a semi-metric because the triangle inequality is relaxed by a factor $A\subset \Omega$3 (Vayer et al., 2018). In the more general continuous treatment, the infimum is attained, FGW is nonnegative and symmetric, and the identity of indiscernibles is expressed through a measure-preserving map that is simultaneously feature preserving and structure preserving. The same treatment proves a true triangle inequality for $A\subset \Omega$4 and a relaxed triangle inequality for $A\subset \Omega$5 (Vayer et al., 2018).

These properties give FGW a genuine geometry rather than a task-specific loss only. In particular, the theoretical framework proves constant-speed geodesics when the feature space is $A\subset \Omega$6, with interpolation obtained by transporting along an optimal coupling, interpolating features linearly, and interpolating structure through the coupled product space. The same work also gives a finite-sample convergence result: if $A\subset \Omega$7 is the empirical measure of i.i.d. samples from $A\subset \Omega$8, then $A\subset \Omega$9, together with an expectation bound derived through comparison with Wasserstein transport on a product space (Vayer et al., 2018).

For graph prediction and comparison, an important corollary is invariance to graph isomorphism. In the graph-regression formulation based on FGW loss, isomorphic graphs have zero distance, and graphs of different sizes can still be compared within a common relaxed graph space $\mu$0 (Brogat-Motte et al., 2022).

3. Optimization and computational regimes

Classical FGW is computationally challenging because the structural term is quadratic in the coupling and the resulting optimization problem is nonconvex. In the common $\mu$1 case, the graph formulation can be rewritten in quadratic form, and the standard solver is a conditional-gradient or Frank–Wolfe scheme. Each iteration computes the gradient

$\mu$2

solves a classical optimal-transport problem with ground cost $\mu$3, performs a line search, and updates the coupling. The tensor form can be evaluated in $\mu$4 for $\mu$5, improving on a naive $\mu$6 implementation, but the procedure still converges only to a local stationary point because the objective is nonconvex (Vayer et al., 2018).

Several later works modify this computational core. One graph-linearization paper computes the FGW plans to a fixed reference graph with a proximal point algorithm with entropic regularization,

$\mu$7

and solves the resulting subproblems approximately with Sinkhorn–Knopp updates (Nguyen et al., 2022). In a different direction, the orthogonal Gromov–Wasserstein program extends directly to FGW by adding a linear feature-discrepancy term; the resulting orthogonal fused surrogate admits a closed-form lower bound with $\mu$8 complexity in the equal-size setting (Jin et al., 2022).

A further computational line replaces the underlying quadratic program by hierarchical transport and slicing. The sliced FGW construction starts from a fused third lower bound, approximates local 1D Wasserstein distances by quadrature, packages the result as a Euclidean transport problem, and finally applies sliced Wasserstein to obtain a pseudo-metric that bounds FGW from below in the exact empirical setting. This construction is designed to remain invariant to isometric transformations while allowing arbitrary geometries, in contrast with earlier sliced GW formulations restricted to Euclidean geometry (Piening et al., 4 Aug 2025).

The recurring practical difficulty is therefore not definitional but algorithmic. Classical FGW is expressive precisely because it matches structure and features jointly, yet that same coupling makes exact or large-scale optimization difficult. Many later approximations preserve the fused objective only approximately, or preserve it analytically while changing the computational regime (Nguyen et al., 2022).

4. Barycenters, regression, and linearized embeddings

One of the main geometric consequences of FGW is the existence of barycenters. In the original graph treatment, an FGW barycenter is a Fréchet mean of structured objects,

$\mu$9

and, for fixed barycenter support, the optimization alternates over couplings, barycenter structure, and barycenter features. For $(X\times A,d_X,\mu)$0 with Euclidean features, the barycenter updates have closed forms: $(X\times A,d_X,\mu)$1 This makes graph centroids and graph $(X\times A,d_X,\mu)$2-means possible without a pre-image problem (Vayer et al., 2018).

The barycentric viewpoint was later turned into a predictive model. In supervised graph prediction, the output graph is defined as an input-conditioned FGW barycenter,

$(X\times A,d_X,\mu)$3

where $(X\times A,d_X,\mu)$4 are template graphs and $(X\times A,d_X,\mu)$5 are nonnegative weights depending on the input. The same work introduces a non-parametric estimator based on kernel ridge regression, proves universal consistency and an excess-risk bound, and then develops a parametric neural model in which both template graphs and barycenter weights are learned (Brogat-Motte et al., 2022).

A complementary development is linearization. The linearFGW construction embeds each graph into a tangent-like Euclidean space around a fixed reference graph. Given optimal FGW plans from the reference to two graphs, linearFGW is defined as the squared Euclidean distance between the corresponding embeddings built from barycentric projections of nodes and edges. This yields a kernel-friendly approximation of FGW: for $(X\times A,d_X,\mu)$6 graphs, pairwise FGW requires $(X\times A,d_X,\mu)$7 optimal-transport computations, whereas linearFGW requires only $(X\times A,d_X,\mu)$8 computations to the reference graph. Because the resulting representations are Euclidean, the Gaussian kernel

$(X\times A,d_X,\mu)$9

is positive semidefinite, addressing the indefiniteness often encountered when one kernelizes pairwise OT distances directly (Nguyen et al., 2022).

The LOT perspective has also been pushed further to analyze how much FGW geometry is retained after linearization. In the finite-network setting, an FGW barycentric projection maps a measure network to projected node coordinates and a projected edge matrix, and the squared $(Y\times B,d_Y,\nu)$0-Fused 2-Gromov–Wasserstein distance decomposes into a deterministic term explained by the barycentric projection and a residual probabilistic term. This leads to a notion of percentage of variance explained by FGW LOT embeddings (Wilson et al., 2024).

5. Variants and structural generalizations

Several variants modify FGW to address modeling assumptions that are restrictive in the classical balanced, node-feature-only setting. A direct extension is Fused Network Gromov–Wasserstein (FNGW), which compares graphs with both node features and edge features. Its objective fuses three terms inside the same pairwise integral: node-feature mismatch, edge-feature mismatch, and structural mismatch. In the discrete setting, the corresponding gradient augments the classical FGW gradient with an additional tensor contraction for edge attributes, and the framework also supports barycenters for graphs with attributed edges (Yang et al., 2023).

Another major extension relaxes the equal-mass constraint. Classical FGW assumes full couplings $(Y\times B,d_Y,\nu)$1, but Fused Partial Gromov–Wasserstein (FPGW) allows sub-couplings and adds a mass penalty

$(Y\times B,d_Y,\nu)$2

This makes the method robust to outlier noise because unmatched mass need not be forced into spurious correspondences. The same work proves existence of minimizers, establishes metric-like behavior on a quotient space, shows that FGW is recovered when the optimal plan uses all mass or when $(Y\times B,d_Y,\nu)$3 is large enough, and proposes Frank–Wolfe solvers with partial optimal-transport linear oracles (Bai et al., 14 Feb 2025).

Feature treatment has also been made adaptive. FGW with feature selection introduces suppression weights $(Y\times B,d_Y,\nu)$4 into the feature term,

$(Y\times B,d_Y,\nu)$5

together with either Lasso or Ridge penalties, or simplex-constrained variants. The resulting alternating scheme keeps the transport update in the FGW family while learning which features to suppress. The paper proves bounds

$(Y\times B,d_Y,\nu)$6

and shows that the Lasso and Ridge versions retain metric or semimetric behavior analogous to the classical case (Lee et al., 12 May 2026).

These variants clarify what the basic FGW model does and does not assume. Classical FGW is balanced, feature-uniform, and centered on node attributes plus internal structure; later work relaxes each of these choices separately rather than replacing the fused transport principle itself (Bai et al., 14 Feb 2025).

6. Applications and methodological roles

FGW has been used in the cited literature as a distance, a loss, a barycentric geometry, a regularizer, and an evaluation metric. In Hawkes-process alignment, fused Gromov–Wasserstein discrepancy regularizes the joint maximum-likelihood estimation of two Hawkes models so that base intensities and infectivity matrices are aligned simultaneously, and the resulting transport plan is interpreted as a soft correspondence matrix between event types (Luo et al., 2019). In template-based graph neural networks, vectors of FGW distances to learnable template graphs serve as graph embeddings, with end-to-end differentiation through the FGW layer via the envelope theorem (Vincent-Cuaz et al., 2022). In graph data augmentation, FGWMixup constructs a synthetic graph as a midpoint in FGW space, jointly mixing node features and structure rather than perturbing them independently (Ma et al., 2023).

The same fused principle has been adapted beyond ordinary graph classification. In unsupervised knowledge-graph entity alignment, FGW motivates a three-stage progressive optimization scheme combining semantic matching and global structural comparison across knowledge graphs (Tang et al., 2023). In private synthetic graph generation, FGW is the main utility notion used to compare true and synthetic attributed random graphs; in that work it is primarily an analytical evaluation metric rather than the optimization objective of the generator (Wirth et al., 17 Feb 2025). For node-attributed graphs viewed as structured objects with finite support and uniform measure, $(Y\times B,d_Y,\nu)$7-nearest neighbors with fGW distance is proved universally consistent on the corresponding space of weak isomorphism classes (Hohmeier et al., 9 Jun 2026).

More recent work uses FGW geometry itself as the organizing object. In graph gradual domain adaptation, the domain discrepancy is taken to be FGW, an error bound is derived in terms of the path length $(Y\times B,d_Y,\nu)$8, and the FGW geodesic is identified as the optimal path of intermediate domains (2505.12709). In semantic correspondence for images, pseudo-label generation is reformulated as an FGW problem that fuses semantic similarity with 3D geometric consistency; because the full quadratic program is prohibitive, the method approximates FGW by anchor-based linearization and then trains with a soft-target loss robust to noisy transport plans (Im et al., 12 Mar 2026).

Across these uses, the same pattern recurs. FGW is chosen when a task requires simultaneous control of feature similarity and structural consistency, especially when the compared objects differ in size, label space, or internal geometry. The principal limitations are also stable across the literature: classical FGW is nonconvex, computationally expensive, not naturally kernel-friendly in its pairwise form, and balanced by construction. Much of the subsequent research therefore consists of preserving the fused coupling idea while modifying the solver, the geometry, or the admissible couplings (Nguyen et al., 2022).