Fused Gromov-Wasserstein Transport

• Fused Gromov-Wasserstein optimal transport is a unified metric that combines feature-based and structural comparisons to quantitatively assess graphs and metric-measure spaces.
• The framework leverages advanced optimization methods—including conditional gradient, entropic Sinkhorn, and majorization-minimization—to efficiently solve nonconvex matching problems.
• Extensions like partial and unbalanced FGW, sliced and multi-marginal variants enhance its robustness and scalability for practical applications in graph classification, clustering, and data augmentation.

The Fused Gromov-Wasserstein (FGW) optimal transport framework generalizes classical Wasserstein and Gromov-Wasserstein distances to provide a unified metric for comparing structured objects that possess both geometric and attribute information—most notably, graphs and metric-measure spaces with node features. By blending feature-based optimal transport and structural relational comparisons, FGW enables flexible alignment of complex data modalities, supports barycenter computation for clustering and averaging, and drives modern machine learning architectures for graph-based prediction, classification, augmentation, and alignment. Recent developments include unbalanced and partial-matching extensions, scalable algorithms, variance decompositions via LOT embeddings, and formulations for edge attributes and multi-marginal settings.

1. Formal Definition and Mathematical Structure

Given two metric-measure spaces $\mathbb{X}=(X,d_X,\mu)$ and $\mathbb{Y}=(Y,d_Y,\nu)$ (or, in the graph context, attributed graphs $(X,C_X,a_X)$ and $(Y,C_Y,a_Y)$), the FGW distance of order $(r,q)$ is defined as

$$\mathrm{FGW}_{r,L}(\mathbb X,\mathbb Y) :=\min_{\gamma\in\Gamma(\mu,\nu)} \left[ \omega_1\,\langle C,\gamma\rangle \;+\;\omega_2\,\gamma^{\otimes2}\left(L\left(d_X^r,d_Y^r\right)\right) \right]$$

where:

• $C:X\times Y\to\mathbb{R}_+$ is the feature cost (e.g. $C(x,y)=\|x-y\|^q$),
• $L:\mathbb{R}^2\to\mathbb{R}_+$ is the structural cost (e.g. $L(a,b)=|a-b|^q$),
• $\omega_1, \omega_2\geq0$, $\omega_1+\omega_2=1$ trade-off feature and structure,
• $\gamma$ is a coupling in $\Gamma(\mu, \nu)$ matching marginals $\mu$ and $\nu$,
• $\gamma^{\otimes2}(L(\cdot,\cdot))$ is an expectation of pairwise structural discrepancies over the coupling.

FGW interpolates between classical OT ($\omega_1=1$) and pure GW ($\omega_2=1$) (Bai et al., 14 Feb 2025, Vayer et al., 2018).

2. Variants and Extensions: Unbalanced and Partial FGW

Classical FGW enforces mass equality. To address unbalanced and noisy data, several extensions have recently appeared:

• Fused Partial Gromov-Wasserstein (FPGW): Constraints can be relaxed either via a total-variation penalty (mass deviation) or a direct constraint on the mass transported. Two key forms are equivalent (Thm. 3.1):
  • TV-penalty:

  $$\mathrm{FPGW}_{r,L,\lambda}(\mathbb X,\mathbb Y) :=\min_{\gamma\in\mathcal{M}_+(X\times Y)} \left\{ \text{FGW terms} + \lambda|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV} + \lambda|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV} \right\}$$ - Mass-Constrained:

  $$\mathrm{FMPGW}_{r,L,\rho}(\mathbb X,\mathbb Y) :=\min_{\gamma\in \Gamma^\rho_\le(\mu,\nu)} \text{FGW terms}$$

  with $\gamma_1\leq\mu, \gamma_2\leq\nu, |\gamma|\geq\rho$.

• Fused Unbalanced Gromov-Wasserstein (FUGW): Relax hard marginal constraints with KL penalties (Thual et al., 2022).

$$\mathrm{FUGW}_{\beta,\lambda}(T) = (1-\lambda)\sum_{i,k}d_F(f_i,g_k)^2\,T_{ik} + \lambda\sum_{i,j,k,l}|a_{ij}-b_{kl}|^2\,T_{ik}T_{jl} + \varepsilon \mathrm{KL}(T\mathbf{1}\|\mu) + \varepsilon \mathrm{KL}(T^T\mathbf{1}\|\nu)$$

These forms permit selective matching and robustness to outliers (Bai et al., 14 Feb 2025, Thual et al., 2022, Pan et al., 28 Jun 2024, Chandra et al., 29 Sep 2025).

3. Algorithmic Frameworks: Optimization and Scalability

FGW and its variants involve nonconvex quadratic objectives solved predominantly by:

• Conditional Gradient methods (Frank-Wolfe): Each iteration linearizes the quadratic term and solves a classical or partial OT subproblem (often via entropic Sinkhorn) to update the coupling. For FPGW, the partial OT subproblem is a linear program with mass constraints (Bai et al., 14 Feb 2025, Vayer et al., 2018).
• Sinkhorn-Type Iterative Schemes: Especially for entropic regularization, iterates alternate between marginal projections and fixed-point updates. Unbalanced OT problems leverage KL penalization and employ unbalanced Sinkhorn (Thual et al., 2022, Wilson et al., 15 Nov 2024).
• Majorization-Minimization and Block-Coordinate Descent: For barycenter and multi-marginal settings, alternating minimization efficiently solves tight bi-convex relaxations (Beier et al., 2022).
• Sliced and Linear Approximations: The sliced FGW reduces complexity by projecting onto 1D subspaces and leveraging hierarchical lower bounds and Monte Carlo slicing, achieving scalability for large and complex datasets (Piening et al., 4 Aug 2025, Nguyen et al., 2022, Wilson et al., 15 Nov 2024).

The table below outlines notable solvers:

Variant	Optimization Method	Scalability
FGW (balanced)	Frank-Wolfe, Sinkhorn	$O(n^3)$, up to $O(n^2)$ with entropy
FPGW, FUGW	Partial OT, Unbalanced Sinkhorn	$O(nm\log(nm))$ per iteration
Sliced FGW	Monte-Carlo slicing, 1D OT	$O(L\,n\log n)$ after pre-computation
Multi-marginal FGW	Alternating Sinkhorn	$O(N\,n^2)$ per iteration
Linear FGW (embedding)	Barycentric projection	$O(N\,m^2K)$ total for $N$ graphs

4. Theoretical Properties: Metrics, Interpolation, and Variance Analysis

FGW is a metric or semi-metric depending on the exponent $q$ and structure:

• Nonnegativity, symmetry always hold.
• Identity of indiscernibles: $d_{FGW}=0$ iff the structured objects are matched by an isometry preserving both feature and structure (Bai et al., 14 Feb 2025, Vayer et al., 2018, Ma et al., 2023).
• Triangle inequality: exact if $q=1$, else relaxed by factor $2^{q-1}$.
• FGW interpolates between Wasserstein (feature-only, $\alpha=0$) and Gromov-Wasserstein (structure-only, $\alpha=1$).
• Fréchet means/barycenters: existence and closed-form block-coordinate updates enable barycenter computation and clustering (Vayer et al., 2018, Brogat-Motte et al., 2022, Beier et al., 2022).

Variance decompositions via Linear Optimal Transport (LOT) provide exact splits into deterministic and residual terms: $$FGW_{2,\alpha}^2(\mathcal X,\mathcal Y) = FGW_{2,\alpha}^2(\mathcal X,\mathcal T) + \alpha C_W + (1-\alpha) C_{GW}$$ where $\mathcal T$ is the barycentric projection. The percent explained by the LOT embedding guides the selection of embedding dimension $n$ for dimension reduction and model building (Wilson et al., 15 Nov 2024).

5. Applications: Graph Learning, Clustering, Alignment, and Data Augmentation

FGW supports a wide class of applications:

• Graph Classification and Kernel Methods: FGW distances can be used as kernels (RBF, indefinite, linearized PSD variants) for SVMs and clustering. FGW-based kernels surpass classical graph kernels (WL, SPK, etc.) on benchmarks such as MUTAG, ENZYMES, PROTEINS (Vayer et al., 2018, Ma et al., 2023, Nguyen et al., 2022, Wilson et al., 15 Nov 2024).
• FGW Barycenters and Clustering: FGW’s barycentric averaging produces meaningful prototypical graphs, yielding strong separation in block-model clustering and graph community detection (Vayer et al., 2018, Brogat-Motte et al., 2022, Beier et al., 2022).
• Graph Prediction and Template Learning: FGW barycenter layers, differentiable via Envelope Theorem, enable end-to-end template learning in graph neural network architectures, achieving competitive or superior empirical results (Vincent-Cuaz et al., 2022).
• Video, Time-Series Alignment, and Neuroimaging: Partial/unbalanced variants robustly handle outlier segments, background/noise, and anatomical discrepancies (Mahmood et al., 21 Jul 2025, Chandra et al., 29 Sep 2025, Thual et al., 2022).
• Graph Mixup Augmentation: FGWMixup synthesizes graph interpolants in FGW space, improving generalizability of GNNs; scalable solvers (relaxed MD, Sinkhorn) yield fast convergence (Ma et al., 2023).
• Graph Matching, Assignment, and Subgraph Retrieval: FGW and its extensions (partial, regularized, sliced) provide effective frameworks for assignment problems, subgraph matching in massive graphs, and keypoint correspondence, with high robustness to feature noise (Seyedi et al., 4 Sep 2025, Pan et al., 28 Jun 2024).

6. Recent Developments: Edge Features, Multi-Marginal and Sliced FGW

• Fused Network GW (FNGW): Incorporates both node and edge features through an additional fused cost on edge-attribute tensors. Proven metric properties, block coordinate solvers, and efficiency gains empirically outperform FGW and other graph kernels when edge attributes matter (Yang et al., 2023).
• Multi-Marginal FGW: Generalizes to aggregates and interpolates among multiple structured spaces; alternating Sinkhorn and tight bi-convex relaxation enable multi-object barycenter computation (Beier et al., 2022).
• Sliced FGW: Hierarchical and quadrature-based slicing drastically reduces computational costs, maintains isometric invariance, and achieves pseudo-metric properties suitable for shape retrieval and graph isomorphism testing (Piening et al., 4 Aug 2025).

7. Practical Guidelines and Empirical Insights

• Trade-off parameter $\alpha$ is critical: in graph tasks, $\alpha\in[0.5,0.95]$ typically balances signal and structure correctly.
• LOT variance decompositions quantify the efficacy of dimensionality reduction and can guide embedding size ($n=10$–$75$ often suffices) (Wilson et al., 15 Nov 2024).
• Regularization, partial mass matching, and sliced approximations enhance robustness to outliers, noise, and scalability.
• The Python Optimal Transport (POT) library implements principal FGW and related algorithms, facilitating integration and reproducible research (Seyedi et al., 4 Sep 2025, Vincent-Cuaz et al., 2022, Ma et al., 2023).

In summary, Fused Gromov-Wasserstein transport supplies a versatile, theoretically rigorous, and empirically validated basis for learning, comparing, and synthesizing structured objects in modern computational science. Advances spanning partial matching, linear and sliced embedding, edge-feature integration, and multi-marginal formulations have established FGW as a foundational tool in graph-based and structured data analysis (Bai et al., 14 Feb 2025, Vayer et al., 2018, Beier et al., 2022, Yang et al., 2023, Piening et al., 4 Aug 2025, Wilson et al., 15 Nov 2024).