Fused Partial Gromov-Wasserstein (FPGW)

• Fused Partial Gromov-Wasserstein is a framework that compares structured objects by jointly aligning node features and structural relations while allowing only a subset of mass to be matched.
• It extends FGW by integrating a partial mass mechanism which robustly handles outlier nodes, missing structures, and corrupted measurements.
• FPGW is supported by metric guarantees and efficient Frank–Wolfe optimization, showing effectiveness in subgraph matching and graph clustering under noise.

Searching arXiv for papers on Fused Partial Gromov-Wasserstein and closely related FGW work. Fused Partial Gromov-Wasserstein (FPGW) is a framework for comparing structured objects—especially attributed graphs and related metric-measure representations—when only part of the mass should be matched. It extends Fused Gromov-Wasserstein (FGW), which jointly accounts for feature similarity and structural similarity, by relaxing the equal-mass constraint inherited from classical optimal transport. The resulting formalism is designed for settings with outlier nodes, extra substructures, missing nodes, corrupted measurements, or unequal total mass, where forcing a full correspondence can distort both feature and structural alignment (Bai et al., 14 Feb 2025). In current arXiv literature, FPGW appears both as a general theory for structured objects and as a practical modeling principle for subgraph matching, where only a subset of a larger graph should align with a query graph (Bai et al., 14 Feb 2025, Pan et al., 2024).

1. Conceptual position within optimal transport on structured data

FGW compares two structured objects by combining a feature-matching term and a structure-matching term in a single transport objective. In the graph setting, this means simultaneously aligning node attributes and pairwise relations such as adjacency or shortest-path structure. Classical FGW, however, assumes balanced transport: all mass must be matched across the two objects. FPGW relaxes this requirement and allows some mass to remain unmatched, thereby introducing a partial matching mechanism into the fused structured-transport setting (Bai et al., 14 Feb 2025).

This partiality is the defining distinction. In the formulation of "Fused Partial Gromov-Wasserstein for Structured Objects" (Bai et al., 14 Feb 2025), the purpose is to retain FGW’s joint modeling of features and geometry while permitting mass deletion. The motivation is explicit: graphs may contain outlier nodes, extra substructures, missing nodes, or corrupted measurements, and a balanced FGW plan can be unduly influenced by such irregularities. FPGW therefore occupies the same conceptual niche relative to FGW that partial optimal transport occupies relative to classical OT.

A closely related but task-specific manifestation appears in "Subgraph Matching via Partial Optimal Transport" (Pan et al., 2024). There, subgraph matching is formulated as a partial fused Gromov-Wasserstein problem so that a smaller query graph can be matched inside a larger source graph while accounting for both node features and graph structure. This suggests that FPGW is not merely a robustness device for noisy balanced comparisons, but also a natural formalism whenever the semantics of the problem are inherently subset-to-subset rather than full-object-to-full-object.

By contrast, several recent FGW-based methods remain strictly in the full-matching regime. "Fused Gromov-Wasserstein Graph Mixup for Graph-level Classifications" (Ma et al., 2023) uses full FGW couplings between complete node distributions, and "A Fused Gromov-Wasserstein Approach to Subgraph Contrastive Learning" (Sangare et al., 28 Feb 2025) likewise employs standard couplings with uniform marginals over sampled subgraphs. These works are relevant because they clarify how fused structure-signal alignment behaves algorithmically, but they do not introduce partial transport.

2. Formal definitions and objective functions

The most general formulation in the current literature is given for structured metric-measure spaces $\mathbb{X}=(X,d_X,\mu)$ and $\mathbb{Y}=(Y,d_Y,\nu)$, with a feature cost $C:X\times Y\to\mathbb{R}_+$, a structure cost $L:\mathbb{R}^2\to\mathbb{R}_+$, and weights $\omega_1,\omega_2\ge 0$ satisfying $\omega_1+\omega_2=1$ (Bai et al., 14 Feb 2025). The penalized fused partial GW objective is defined as

$$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$

Here $\gamma$ is the coupling, $\gamma_1,\gamma_2$ are its marginals, and $|\cdot|_{TV}$ denotes total variation (Bai et al., 14 Feb 2025).

The equivalent mass-constrained version is

$\mathbb{Y}=(Y,d_Y,\nu)$0

with feasible set

$\mathbb{Y}=(Y,d_Y,\nu)$1

In discrete form, if $\mathbb{Y}=(Y,d_Y,\nu)$2 and $\mathbb{Y}=(Y,d_Y,\nu)$3, then

$\mathbb{Y}=(Y,d_Y,\nu)$4

where $\mathbb{Y}=(Y,d_Y,\nu)$5, $\mathbb{Y}=(Y,d_Y,\nu)$6, and $\mathbb{Y}=(Y,d_Y,\nu)$7 (Bai et al., 14 Feb 2025).

A key simplification established for the penalized formulation is that, when $\mathbb{Y}=(Y,d_Y,\nu)$8 and $\mathbb{Y}=(Y,d_Y,\nu)$9,

$C:X\times Y\to\mathbb{R}_+$0

This explicit mass-penalty form makes clear that unmatched mass is not handled through balanced marginal constraints, but through a penalty depending only on the transported mass $C:X\times Y\to\mathbb{R}_+$1 (Bai et al., 14 Feb 2025).

For subgraph matching, the formulation becomes concrete. Given a large source graph $C:X\times Y\to\mathbb{R}_+$2 with $C:X\times Y\to\mathbb{R}_+$3 nodes and a query graph $C:X\times Y\to\mathbb{R}_+$4 with $C:X\times Y\to\mathbb{R}_+$5 nodes, the source structure is represented by $C:X\times Y\to\mathbb{R}_+$6, the query structure by $C:X\times Y\to\mathbb{R}_+$7, and node-feature dissimilarities by $C:X\times Y\to\mathbb{R}_+$8. The structural tensor is

$C:X\times Y\to\mathbb{R}_+$9

and the tensor-matrix product is

$L:\mathbb{R}^2\to\mathbb{R}_+$0

The standard fused Gromov-Wasserstein problem is

$L:\mathbb{R}^2\to\mathbb{R}_+$1

whereas the partial version for subgraph matching is

$L:\mathbb{R}^2\to\mathbb{R}_+$2

Here

$L:\mathbb{R}^2\to\mathbb{R}_+$3

with $L:\mathbb{R}^2\to\mathbb{R}_+$4 under the uniform-mass choice used in that work (Pan et al., 2024).

3. Relation to FGW, partial GW, and unbalanced variants

FPGW is positioned explicitly as the fused counterpart of partial GW (Bai et al., 14 Feb 2025). The relation to FGW is structural: FGW jointly models feature similarity and relational geometry, while FPGW preserves this fused objective but allows only a fraction of mass to be transported. This means that the distinction from plain partial GW is not the partiality mechanism but the addition of a feature term.

Several equivalence and limiting-case statements are central in the theory (Bai et al., 14 Feb 2025). First, if unbalanced FGW uses total variation divergence rather than KL divergence, then FUGW and FPGW coincide. More precisely, with total variation marginal penalties, the search can be restricted to $L:\mathbb{R}^2\to\mathbb{R}_+$5, and the resulting objective is exactly the simplified FPGW expression above. Second, there is a threshold condition on the penalty parameter $L:\mathbb{R}^2\to\mathbb{R}_+$6 ensuring that the optimal plan uses full feasible mass: $L:\mathbb{R}^2\to\mathbb{R}_+$7 Under this condition, there exists an FPGW minimizer $L:\mathbb{R}^2\to\mathbb{R}_+$8 with

$L:\mathbb{R}^2\to\mathbb{R}_+$9

If the inequality is strict, every minimizer has full feasible mass. Under this regime, FPGW reduces to the mass-constrained problem at $\omega_1,\omega_2\ge 0$0, and when $\omega_1,\omega_2\ge 0$1, this further collapses to classical FGW (Bai et al., 14 Feb 2025).

The same reduction appears directly for the mass-constrained version: if $\omega_1,\omega_2\ge 0$2, then $\omega_1,\omega_2\ge 0$3 at $\omega_1,\omega_2\ge 0$4, so FMPGW becomes FGW (Bai et al., 14 Feb 2025). This establishes that FPGW is a genuine generalization rather than a different distance family.

A task-specific variant of this relationship appears in subgraph matching. There, the partial problem is reformulated as a standard FGW problem by augmenting the query graph with a dummy node that carries the leftover mass $\omega_1,\omega_2\ge 0$5. The augmented target distribution is

$\omega_1,\omega_2\ge 0$6

with augmented feature cost

$\omega_1,\omega_2\ge 0$7

and augmented structure tensor

$\omega_1,\omega_2\ge 0$8

Unmatched source mass can therefore be sent to the dummy node at zero cost (Pan et al., 2024). This is a specialized equivalence: it does not define the general FPGW framework, but it shows how partial fused matching can be embedded into a standard FGW solver for the subgraph-search setting.

Full FGW methods provide useful contrasts. In graph mixup, FGW is defined over complete attributed graphs $\omega_1,\omega_2\ge 0$9, and the coupling set is the balanced transport polytope

$\omega_1+\omega_2=1$0

which enforces full mass preservation (Ma et al., 2023). In subgraph contrastive learning, the same full-coupling principle is used with uniform node distributions on fixed-size sampled subgraphs (Sangare et al., 28 Feb 2025). These formulations underscore what FPGW changes: not the fused structure-feature cost, but the admissible set of couplings.

4. Metric properties and theoretical guarantees

A major theoretical result for FPGW is the establishment of metric and pseudo-metric properties for a normalized version of the distance (Bai et al., 14 Feb 2025). Under the assumptions that $\omega_1+\omega_2=1$1 are compact, $\omega_1+\omega_2=1$2 are metrics, and $\omega_1+\omega_2=1$3, the normalized quantity is defined by

$\omega_1+\omega_2=1$4

The cited theorem states that this defines a semi-metric on the quotient space of mm-spaces modulo measure-preserving isometry, is nonnegative and symmetric, and satisfies stronger properties when $\omega_1+\omega_2=1$5 (Bai et al., 14 Feb 2025).

Specifically, if $\omega_1+\omega_2=1$6, then $\omega_1+\omega_2=1$7 if and only if the two mm-spaces are equivalent under a measure-preserving isometry (Bai et al., 14 Feb 2025). If $\omega_1+\omega_2=1$8 and $\omega_1+\omega_2=1$9, then the generalized triangle inequality holds: $$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$0 and when $$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$1, the ordinary triangle inequality holds, so the distance is a metric (Bai et al., 14 Feb 2025).

The proof strategy uses an “FGW with auxiliary infinity points” construction. Additional points $$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$2 are adjoined so that partial couplings on the original spaces become full couplings on enlarged spaces. This allows the triangle inequality for FPGW to be reduced to a known FGW-type inequality on expanded spaces (Bai et al., 14 Feb 2025). This suggests a deep structural continuity between FGW and FPGW: the latter is not ad hoc partialization, but a transport geometry that can be embedded back into full-coupling theory on suitably augmented spaces.

The subgraph-matching literature does not emphasize metric axioms in the same generality, but it inherits the theoretical spirit of FGW and partial OT. The use of a dummy node with zero unmatched cost in the augmented FGW formulation is presented as a device that preserves computational tractability and allows the method to build on existing FGW theory and computational methods (Pan et al., 2024).

A plausible implication is that the metric formalization in (Bai et al., 14 Feb 2025) provides the more general foundation, while the subgraph-search construction in (Pan et al., 2024) demonstrates how partial fused geometry can be engineered for a concrete retrieval problem.

5. Algorithms and computational structure

The discrete FPGW and FMPGW objectives are nonconvex because of the quadratic structure term, and the principal optimization approach in the current general framework is Frank–Wolfe (conditional gradient) (Bai et al., 14 Feb 2025). For the mass-constrained problem, the discrete objective is

$$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$3

with gradient

$$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$4

At iteration $$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$5, the linearized subproblem is

$$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$6

A quadratic line search then determines the step size $$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$7, and the iterate is updated by

$$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$8

The step-size rule uses the coefficients

$$FPGW_{r,L,\lambda}(\mathbb{X},\mathbb{Y}) =\inf_{\gamma\in\mathcal{M}_+(X\times Y)} \omega_1 \gamma(C)+\omega_2 \gamma^{\otimes2}\!\big(L(d_X^r,d_Y^r)\big) +\lambda\big(|\mu^{\otimes2}-\gamma_1^{\otimes2}|_{TV}+|\nu^{\otimes2}-\gamma_2^{\otimes2}|_{TV}\big).$$9

where $\gamma$0 (Bai et al., 14 Feb 2025).

For penalized FPGW, the simplified objective becomes

$\gamma$1

with gradient

$\gamma$2

The linear subproblem is then a standard partial OT problem over $\gamma$3 (Bai et al., 14 Feb 2025). The stopping criterion is based on the Frank–Wolfe gap

$\gamma$4

and the nonconvex FW convergence theorem yields

$\gamma$5

so an $\gamma$6-stationary point is reached in $\gamma$7 iterations (Bai et al., 14 Feb 2025).

The computational bottleneck is the tensor contraction $\gamma$8. Naively it is $\gamma$9, but under the separable loss form

$\gamma_1,\gamma_2$0

the contraction can be reduced to

$\gamma_1,\gamma_2$1

bringing the cost down to $\gamma_1,\gamma_2$2 (Bai et al., 14 Feb 2025). This is the standard efficient FGW trick adapted to the partial setting.

The subgraph-matching formulation (Pan et al., 2024) also uses Frank–Wolfe, but on the dummy-augmented FGW problem. With objective

$\gamma_1,\gamma_2$3

the gradient is

$\gamma_1,\gamma_2$4

The initial feasible plan is $\gamma_1,\gamma_2$5, and the stopping condition is

$\gamma_1,\gamma_2$6

with $\gamma_1,\gamma_2$7 in experiments (Pan et al., 2024).

That work also derives a reduced-complexity formula for the structural contraction: $\gamma_1,\gamma_2$8 which lowers the cost from $\gamma_1,\gamma_2$9 to $|\cdot|_{TV}$0 (Pan et al., 2024).

The comparison with full FGW is instructive. In FGW graph mixup, a relaxed solver is introduced by splitting the transport polytope into row- and column-simplex constraints,

$|\cdot|_{TV}$1

and alternating projected exponentiated-gradient updates,

$|\cdot|_{TV}$2

with

$|\cdot|_{TV}$3

The reported convergence-rate improvement is from $|\cdot|_{TV}$4 to $|\cdot|_{TV}$5 for the marginal-constraint relaxation (Ma et al., 2023). This is not an FPGW solver, but it indicates the broader algorithmic trend in fused structured OT: efficiency often comes from reparameterizing or relaxing coupling constraints rather than changing the fused cost itself.

6. Applications and empirical behavior

The clearest general-purpose application evidence for FPGW comes from graph classification and clustering under outlier corruption (Bai et al., 14 Feb 2025). The graph classification experiments consider four benchmark datasets—MUTAG, MSRC-9, Proteins, and Synthetic—with 50% of graphs corrupted by added outlier nodes at noise levels $|\cdot|_{TV}$6. The reported pattern is that on clean data FGW and FPGW are comparable, but as outlier noise increases, FGW degrades substantially on several datasets, whereas FPGW remains stable and often retains near-clean accuracy even at 20–30% outlier levels (Bai et al., 14 Feb 2025). The stated explanation is direct: outlier nodes can be left unmatched, so they do not contaminate the correspondence plan.

The clustering experiments in the same work compare FGW-kmeans and FPGW-kmeans on synthetic SBM-style graphs with 30% outliers inserted into half of the graphs. FPGW yields clustering assignments closer to ground truth, and its centroids visibly exclude outlier structure, while FGW barycenters absorb outlier noise and become distorted (Bai et al., 14 Feb 2025). This is an important empirical distinction because it shows that partiality affects not only pairwise distances but also barycentric estimation.

The main computational tradeoff is that FPGW is slower than FGW but faster and more stable than FUGW in the reported settings. In the clustering experiment, FGW took about $|\cdot|_{TV}$7 seconds versus $|\cdot|_{TV}$8 seconds for FPGW, while FUGW’s Sinkhorn-based solver was slower and less stable (Bai et al., 14 Feb 2025). This suggests that the robustness benefit of partial matching comes with a moderate but nontrivial computational overhead.

For subgraph matching, the application is structurally different. "Subgraph Matching via Partial Optimal Transport" (Pan et al., 2024) introduces Subgraph Optimal Transport (SOT) and Sliding Subgraph Optimal Transport (SSOT). SOT solves one large partial FGW problem over the whole source graph, whereas SSOT improves efficiency by comparing the query only to candidate $|\cdot|_{TV}$9-hop neighborhoods: $\mathbb{Y}=(Y,d_Y,\nu)$00 Candidate neighborhoods are filtered first by requiring at least as many nodes as the query and then by a cheap feature-only partial OT test

$\mathbb{Y}=(Y,d_Y,\nu)$01

Only candidates passing the filter are evaluated with full partial FGW (Pan et al., 2024).

Experiments compare SOT and SSOT with NeMa and G-Finder on synthetic Erdős–Rényi graphs and real-world datasets BZR, FIRSTMM_DB, LastFM, and Deezer. The reported metrics are success rate and average query time. In noisy settings, SOT and especially SSOT outperform NeMa, SSOT exhibits the best robustness under high feature noise, and on large graphs SSOT scales much better, with query time growing roughly linearly with source size (Pan et al., 2024). On Deezer, SSOT and G-Finder achieve perfect success, while SOT cannot run due to memory; SSOT is reported to be about $\mathbb{Y}=(Y,d_Y,\nu)$02 faster than G-Finder and $\mathbb{Y}=(Y,d_Y,\nu)$03 faster than NeMa in the noise-free setting (Pan et al., 2024).

These results support two distinct empirical roles for FPGW-style formulations. In whole-graph comparison, they act primarily as an outlier-robust distance. In subgraph retrieval, they act as a principled subset-matching mechanism.

7. Related directions, misconceptions, and open technical context

A common misconception is to treat any FGW-based graph method as an instance of FPGW. Recent literature makes the distinction sharp. FGWMixup formulates graph augmentation as a full FGW midpoint problem between two complete graphs and updates a synthetic graph by transporting both structure and features through learned full couplings (Ma et al., 2023). FOSSIL uses FGWD in subgraph-level contrastive learning with full transport coupling $\mathbb{Y}=(Y,d_Y,\nu)$04 under uniform marginals over fixed-size subgraphs (Sangare et al., 28 Feb 2025). Neither work models partial transport, unmatched nodes, graph completion, or subset-to-subset matching.

Another misconception is that partiality is simply a heuristic to drop noisy nodes. The theory in (Bai et al., 14 Feb 2025) indicates a more formal role: FPGW is a structured partial-transport geometry with metric properties, explicit reductions to FGW in appropriate limits, and a variational interpretation via mass penalties or minimum transported mass. In this sense, robustness to outliers is a consequence of the admissible coupling class rather than an auxiliary regularization effect.

A related line of work concerns computational surrogates rather than partiality. "A Novel Sliced Fused Gromov-Wasserstein Distance" (Piening et al., 4 Aug 2025) introduces a sliced fused third lower bound (SFTLB), which is a pseudo-metric and lower bound to FGW that avoids the quadratic program through local 1D OT, quadrature, and sliced Wasserstein. It does not impose partial couplings, but it is relevant because it addresses the same structural challenge: how to preserve fused feature-structure comparison while improving numerical robustness and scalability. This suggests that future FPGW research may combine partial transport with lower-bound hierarchies or slicing techniques.

Another adjacent direction is private synthetic graph generation, where FGW is used as the utility metric for comparing attributed graphs under vertex-level differential privacy (Wirth et al., 17 Feb 2025). That work remains in the full-mass regime, but it highlights the broader significance of fused transport distances as graph-distribution metrics. A plausible implication is that partial fused variants may become relevant in privacy-preserving settings when synthetic graphs do not preserve all nodes or when the goal is to retain only a structurally meaningful subset.

The present arXiv landscape therefore supports a coherent picture. FPGW is the partial-transport extension of FGW for structured objects (Bai et al., 14 Feb 2025). It has a general theoretical formulation with metric guarantees, specialized formulations for subgraph matching (Pan et al., 2024), Frank–Wolfe optimization schemes for both penalized and mass-constrained variants (Bai et al., 14 Feb 2025), and empirical evidence that partial matching improves robustness under outlier corruption and enables subset-aware graph retrieval (Bai et al., 14 Feb 2025, Pan et al., 2024). Full FGW methods on graph mixup, contrastive learning, privacy, and sliced lower bounds provide important neighboring context, but they should not be conflated with FPGW unless unmatched mass is explicitly part of the model (Ma et al., 2023, Sangare et al., 28 Feb 2025, Wirth et al., 17 Feb 2025, Piening et al., 4 Aug 2025).