---

name: gromov description: "Use POT Gromov-Wasserstein and Fused GW solvers for graphs, structured data, heterogeneous spaces, GW barycenters, dictionary learning, quantized approximations, and validation." disable-model-invocation: true metadata: disco-role: operating license: MIT

POT Gromov-Wasserstein and Fused GW

Use this sub-skill when the task involves comparing or aligning objects whose samples live in different metric spaces or have graph/relational structure: Gromov-Wasserstein (GW), Fused Gromov-Wasserstein (FGW), structured graph matching, node-feature-aware graph comparison, GW/FGW barycenters, dictionary learning/unmixing, semirelaxed/partial/unbalanced variants, quantized approximations, and GW validation.

Read references/api-reference.md when choosing a POT GW API, checking verified signatures/defaults, understanding ot.solve_gromov versus ot.gromov.*, or validating expected plan/log/result fields.

Read references/workflows.md when building an end-to-end GW or FGW workflow for graphs, node features, heterogeneous spaces, barycenters, dictionary learning, or approximate/quantized GW.

Read references/troubleshooting.md when cost matrices, weights, loss_fun/loss, alpha, marginal constraints, symmetric, partial/semirelaxed/unbalanced semantics, convergence, quantized helpers, or optional GNN dependencies fail.

Run scripts/gromov_smoke.py after installing POT to exercise deterministic NumPy GW and FGW fixtures with no plotting, downloads, original checkout, or optional graph dependencies.

Route quickly

• Use GW when each object is represented only by an internal structure matrix (C1, C2), such as pairwise graph distances, adjacency-derived dissimilarities, shape geodesics, or relational similarities.
• Use FGW when structures also have cross-space attributes or node features. Provide a feature cost matrix M of shape (n_source, n_target) and tune alpha: 0 emphasizes features/linear OT and 1 emphasizes pure GW structure.
• Use ot.solve_gromov for a unified OTResult with .plan, .value, .value_linear, .value_quad, and marginal helpers. Use ot.gromov.* functions when you need a specific classical, entropic, semirelaxed, partial, quantized, barycenter, dictionary, or estimator API.
• Use entropic GW/FGW when exact conditional-gradient plans are too slow or too sparse and a smoother approximation is acceptable.
• Use semirelaxed GW/FGW when the source marginal is fixed but the target-side mass may be reweighted, for example matching a graph to a prototype or community structure.
• Use partial GW/FGW when only a known amount of mass should match, such as subgraph matching with outliers.
• Use fused unbalanced GW when marginal masses should be penalized rather than enforced. Route general unbalanced/partial OT fundamentals to unbalanced-partial.
• Use quantized, sampled, pointwise, or low-rank GW when graph/sample sizes make full GW too expensive; validate the approximation on a reduced problem before trusting ranking or downstream decisions.
• Use GW/FGW barycenters when the output is itself a structure matrix or an attributed graph prototype. Route ordinary Wasserstein barycenters to barycenters.
• Use dictionary learning/unmixing when many structured samples should be represented as convex combinations of learned GW/FGW atoms.
• Treat POT GNN layers (ot.gnn) as optional reference-only functionality unless the user's environment has PyTorch and PyTorch Geometric installed and verified. Route backend installation and batch-array issues to backend-and-batch.

Stay in scope

This sub-skill owns structured GW/FGW modeling, solver selection, validation, and workflow-specific recovery. It does not cover ordinary vector-space OT except as the M term inside FGW; route those tasks to core-solvers. It does not teach general mass-relaxation theory outside GW/FGW; route that to unbalanced-partial. It does not install optional backend stacks, PyTorch Geometric, plotting, NetworkX, or scikit-learn; route setup questions to backend-and-batch or keep them explicitly optional.

Minimal operating checklist

1. Build finite floating-point square structure matrices C1 and C2; normalize scales when comparing multiple costs.
2. If using FGW, build M with shape (C1.shape[0], C2.shape[0]) from comparable node-feature distances.
3. Set one-dimensional nonnegative weights p and q whose lengths match the structure matrices, or let POT use uniform weights.
4. Choose the mass model: balanced GW/FGW, semirelaxed target reweighting, partial transported mass, or unbalanced marginal penalty.
5. Run the solver with explicit loss_fun/loss, alpha, reg, symmetric, max_iter, and tol when reproducibility matters.
6. Validate the returned plan: finite nonnegative values, expected shape, total mass, row/column marginal behavior for the selected mass model, and a finite GW/FGW value.
7. For larger graphs, compare an approximate/quantized result against a tiny exact subset or the bundled smoke script before scaling up.