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Joint Metric Space Embedding by Unbalanced Optimal Transport with Gromov–Wasserstein Marginal Penalization
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Proposes an unbalanced optimal transport formulation with Gromov-Wasserstein marginal penalization that jointly embeds two heterogeneous metric measure spaces into a fixed metric space, solved by block-coordinate descent with multi-marginal Sinkhorn.
Problem
Unsupervised alignment and joint embedding of heterogeneous datasets lying in different metric spaces, where no ground cost between domains is available and exact isometric embeddings may not exist.
Approach
Relax Sturm's embedded Wasserstein distance by penalizing the marginals of an unbalanced OT plan with GW distances to the input spaces, yielding a quadratic multi-marginal unbalanced OT problem with a bi-convex relaxation; the limit as penalty goes to infinity recovers the embedded Wasserstein distance and the small-penalty limit gives a GW barycenter.
Key results
- On SNAREseq, the method achieves FOSCTTM 0.165 and KNN-Acc 0.943, versus 0.219 / 0.872 for JMDS and 0.402 / 0.447 for SCOT.
- On FAUST joint 2D embeddings with lambda=10, GW2 0.0048 and W2 0.027 versus JMDS 0.0057 and 0.046.
- On scGEM, the method's FOSCTTM 0.208 is better than JMDS 0.230, but its KNN-Acc 0.610 is slightly below JMDS 0.616.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use EWλ instead of JMDS at λ = 1 | Joint embedding method · FAUST | GW²: 0.0071±0.007 -> 0.0057±0.004 | Helped |
| Use EWλ instead of JMDS at λ = 1 | Joint embedding method · FAUST | W²: 0.029±0.010 -> 0.017±0.007 | Helped |
| Use EWλ instead of JMDS at λ = 10 | Joint embedding method · FAUST | GW²: 0.0057±0.005 -> 0.0048±0.004 | Helped |
| Use EWλ instead of JMDS at λ = 10 | Joint embedding method · FAUST | W²: 0.046±0.023 -> 0.027±0.016 | Helped |
| Use EWλ instead of JMDS | Joint feature-space alignment · SNAREseq and scGEM | FOSCTTM: SNAREseq: 0.219 -> 0.165; scGEM: 0.230 -> 0.208 | Helped |
| Use EWλ instead of JMDS | Joint feature-space alignment · SNAREseq and scGEM | KNN-Acc: SNAREseq: 0.872 -> 0.943; scGEM: 0.616 -> 0.610 | Mixed |
| Use EWλ instead of JMDS | Joint feature-space alignment · SNAREseq | Time: 8.7s -> 28.9s | Hurt |
| Use EWλ instead of JMDS | Joint feature-space alignment · scGEM | Time: 0.3s -> 1.7s | Hurt |
Surprising findings
- On scGEM, the proposed method has slightly lower KNN accuracy than JMDS (0.610 vs 0.616) despite better FOSCTTM.
Limitations
- Working on a grid in R^d restricts the method to small dimensions due to computational cost.
- Requires a pre-chosen target metric space (grid) and its appropriate definition.
- Not a free-support solver; extension to multiple input spaces is left open.
Experimental setup
Datasets: FAUST SNAREseq scGEM MNIST FashionMNIST Blobs (scikit-learn) Moons (scikit-learn)
Benchmarks: FAUST SNAREseq scGEM
Metrics: GW distance Wasserstein distance FOSCTTM KNN accuracy runtime
Baselines: JMDS SCOT UnionCom
Builds on: Gromov-Wasserstein distance embedded Wasserstein distance joint multidimensional scaling multi-marginal unbalanced Sinkhorn
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