Research map / Learning theory and algorithms / Optimal transport
Improving Neural Optimal Transport via Displacement Interpolation
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Proposes DIOTM, a neural optimal transport method that trains forward and backward quadratic-cost transport maps with displacement interpolation and an HJB-based regularizer, improving stability over OTM on image-to-image translation.
Problem
Learning neural optimal transport maps via adversarial min-max training is unstable and sensitive to regularization, especially in high-dimensional image settings.
Approach
Parameterize time-dependent transport maps as displacement interpolants of forward and backward neural maps, and regularize the dual potential with a Hamilton-Jacobi-Bellman condition linking optimal potentials across time, so the whole intermediate dynamics is matched rather than only the endpoints.
Key results
- DIOTM achieves FID of 10.72 on Wild→Cat (64x64) with uniform time sampling, the best among the alternatives tested.
- DIOTM shows more robust FID across regularization strength λ than OTM, which is highly sensitive to λ (Fig. 4).
- DIOTM reaches 1-NFE generation and trains in 28h versus over 10 days for DSBM/ASBM on Male→Female 128x128.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DIOTM instead of OTM | Bidirectional transport maps · 8G→G→8G | reconstruction error: 1.06 -> 0.22 | Helped |
| Use DIOTM instead of OTM | Bidirectional transport maps · G→8G→G | reconstruction error: 0.040 -> 0.015 | Helped |
| Use DIOTM instead of OTM | Bidirectional transport maps · 25G→G→25G | reconstruction error: 4.13 -> 12.43 | Hurt |
| Use DIOTM instead of OTM | Bidirectional transport maps · G→25G→G | reconstruction error: 0.56 -> 0.68 | Hurt |
| Use DIOTM instead of OTM | Bidirectional transport maps · M→S→M | reconstruction error: 4.01 -> 1.11 | Helped |
| Use DIOTM instead of OTM | Bidirectional transport maps · S→M→S | reconstruction error: 1.12 -> 0.46 | Helped |
| Use Beta (2,2) time sampling instead of uniform (ablation) | DIOTM · Wild→Cat (64x64) | FID: 10.72 -> 12.46 | Hurt |
| Use Beta (0.5,0.5) time sampling instead of uniform (ablation) | DIOTM · Wild→Cat (64x64) | FID: 10.72 -> 11.85 | Hurt |
Surprising findings
- Baseline OTM collapses with a sudden loss spike after about 40K iterations, indicating instability even with regularization.
- DIOTM's Gaussian mixture high-dimensional results are reasonable but not state-of-the-art, attributed to not using ICNN-type constraints.
Limitations
- Restricted to quadratic cost; the displacement interpolation parametrization does not hold for general costs.
- Requires training two transport maps (forward and backward), making it less efficient than OTM.
- Theoretical guarantees gap because the potential network is not optimal during training; experiments limited to resolutions up to 128x128.
Experimental setup
Datasets: Wild→Cat (64x64) unpaired translation Male→Female (CelebA 128x128) Gaussian mixture benchmark
Benchmarks: AFHQ Wild→Cat (64x64) CelebA Male→Female (128x128) Gaussian mixture OT benchmark
Metrics: FID LPIPS reconstruction error training time
Baselines: OTM DSBM ASBM
Builds on: OTM (Neural optimal transport via min-max with quadratic cost) Wasserstein-2 barycenter duality Dynamic Schrödinger bridge matching
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