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Neural Optimal Transport with General Cost Functionals
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Proposes a neural saddle-point method for continuous optimal transport with general convex cost functionals, with class-guided and pair-guided functionals, plus duality-gap error bounds.
Problem
Computing continuous (out-of-sample) optimal transport plans for general cost functionals, e.g., class-preserving or pair-guided costs, in high-dimensional spaces such as images; existing general-OT methods are discrete.
Approach
Reformulates general OT as a maximin problem over potentials v and stochastic maps T (implicit plan representation via latent noise), solved with neural networks by stochastic gradient ascent-descent; instantiates a class-guided functional based on energy distance per class and a pair-guided L2 functional, and gives duality-gap error bounds that need no convexity assumption on the potential.
Key results
- On FMNIST to MNIST, the class-guided functional reaches 83.22% accuracy vs 10.96% for Neural OT W2 and 10.28% for OTDD.
- On MNIST to MNIST-M, the class-guided functional reaches 94.62% accuracy (95.27% without noise) vs 38.77% for W2 Neural OT.
- On MNIST-M transfer the class-guided functional achieves FID 6.67 vs 11.68 for MUNIT (best baseline), and reaches FID 21.1 on CelebAMask-HQ paired translation.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the class-guided cost functional instead of the compared label-agnostic neural OT costs | Neural OT for dataset transfer · FMNIST → MNIST | accuracy: 10.96 -> 83.22 | Helped |
| use the class-guided cost functional instead of the compared label-agnostic neural OT costs | Neural OT for dataset transfer · MNIST → MNIST-M | accuracy: 38.77 -> 94.62 | Helped |
| use the class-guided cost functional instead of the compared label-agnostic neural OT costs | Neural OT for dataset transfer · FMNIST → MNIST | FID: 7.51 -> 5.26 | Helped |
| use the class-guided cost functional instead of the compared label-agnostic neural OT costs | Neural OT for dataset transfer · MNIST → MNIST-M | FID: 19.43 -> 6.67 | Helped |
| add latent noise z to the class-guided transport map (ablation) | Class-guided OT · FMNIST → MNIST | accuracy: 82.79 -> 83.22 | Helped |
| add latent noise z to the class-guided transport map (ablation) | Class-guided OT · MNIST → MNIST-M | accuracy: 95.27 -> 94.62 | Hurt |
| add latent noise z to the class-guided transport map (ablation) | Class-guided OT · FMNIST → MNIST | FID: 7.14 -> 5.26 | Helped |
| add latent noise z to the class-guided transport map (ablation) | Class-guided OT · MNIST → MNIST-M | FID: 18.56 -> 6.67 | Helped |
Surprising findings
- Neural OT with classic or weak quadratic costs yields only around 10% class accuracy, near random guessing, on class-preserving transfer.
- Discrete general OT (SinkhornLpL1) and OTDD fail to preserve class structure in high dimensions, producing FIDs above 100 on some transfers.
Limitations
- Requires an estimator of the general functional, which may be non-trivial to construct.
- Constructing task-specific cost functionals may not be straightforward.
- Error bounds require strong convexity of the functional in the chosen metric; the class-guided functional is only convex, not shown strongly convex in the main experiments.
Experimental setup
Datasets: MNIST FashionMNIST MNIST-M Comic-Faces-V1 Edges-to-Shoes CelebAMask-HQ
Benchmarks: FMNIST to MNIST class-preserving transfer MNIST to MNIST-M class-preserving transfer CelebAMask-HQ paired translation Comic-Faces-V1 paired translation Edges-to-Shoes paired translation
Metrics: accuracy via pretrained ResNet18 FID
Baselines: AugCycleGAN MUNIT Neural OT W2 weak quadratic Neural OT OTDD flow General discrete OT (SinkhornLpL1) Pix2Pix RMSE regression
Builds on: Neural OT weak optimal transport OTDD flow energy distance
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