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Schrodinger Bridge Flow for Unpaired Data Translation
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Proposes alpha-IMF / alpha-DSBM, a sampling-based online flow-of-path-measures method that approximates the Schrödinger Bridge by taking gradient steps on a bridge-matching loss instead of solving a Markovian projection each iteration.
Problem
Computing the Schrödinger Bridge (entropic optimal transport in dynamic form) for high-dimensional unpaired data translation; existing DSBM/IMF methods require training a full diffusion-type model per iteration, and minibatch OT approximations introduce large errors in high dimensions.
Approach
Define a flow of path measures whose time discretisation (alpha-IMF) converges to the SB for any step size alpha in (0,1]; implement it as an online procedure where each iteration is a single gradient step on the bridge matching loss, with a bidirectional forward/backward network sharing parameters via a time-direction input s, removing the need for alternating full Markovian projections.
Key results
- On EMNIST to MNIST, online finetuning of the bidirectional model reaches FID 4.39 (MSD 0.387) versus 6.33 FID for the pretrained bidirectional model; the two-network online variant reaches FID 4.28.
- On AFHQ-64 Wild to Cat, the bidirectional online finetune reaches FID 26.58 and LPIPS 0.482 versus 29.44 FID and 0.584 LPIPS before finetuning.
- In a Gaussian setting, online alpha-DSBM approaches the true EOT covariance faster than iterative DSBM finetuning.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| online finetune the pretrained two-networks model | α-DSBM two-networks model · EMNIST → MNIST | FID: 6.02 -> 4.28\std{0.07} | Helped |
| online finetune the pretrained two-networks model | α-DSBM two-networks model · EMNIST → MNIST | MSD: 0.564 -> 0.368\std{0.001} | Helped |
| online finetune the pretrained two-networks model | α-DSBM two-networks model · AFHQ-64 Wild → Cat | FID: 25.97 -> 28.752\std{1.191} | Hurt |
| online finetune the pretrained two-networks model | α-DSBM two-networks model · AFHQ-64 Wild → Cat | LPIPS: 0.589 -> 0.487\std{0.003} | Helped |
| online finetune instead of iterative finetuning | Two-networks α-DSBM · EMNIST → MNIST | FID: 5.25\std{0.15} -> 4.28\std{0.07} | Helped |
| online finetune instead of iterative finetuning | Two-networks α-DSBM · EMNIST → MNIST | MSD: 0.345\std{0.001} -> 0.368\std{0.001} | Hurt |
| add EMA during online finetuning (ablation) | Two-networks α-DSBM · AFHQ-64 Wild → Cat | FID: 32.665\std{0.647} -> 28.752\std{1.191} | Helped |
| add EMA during online finetuning (ablation) | Two-networks α-DSBM · AFHQ-64 Wild → Cat | LPIPS: 0.445\std{0.002} -> 0.487\std{0.003} | Hurt |
| use a bidirectional model instead of two networks (ablation) | Pretrained model · EMNIST → MNIST | FID: 6.02 -> 6.33 | Hurt |
| online finetune the pretrained bidirectional model | Bidirectional α-DSBM · EMNIST → MNIST | FID: 6.33 -> 4.39\std{0.09} | Helped |
| online finetune the pretrained bidirectional model | Bidirectional α-DSBM · EMNIST → MNIST | MSD: 0.572 -> 0.387\std{0.003} | Helped |
| online finetune the pretrained bidirectional model | Bidirectional α-DSBM · AFHQ-64 Wild → Cat | FID: 29.44 -> 26.579\std{0.434} | Helped |
Surprising findings
- Disabling EMA at test time worsened AFHQ FID (e.g. 32.67 vs 28.75 for two-network online), though LPIPS improved, showing an FID/alignment trade-off.
Limitations
- Not sampling-free: each iteration needs samples from the previous model, so it still depends on sampling.
- FID is unreliable for MNIST and small AFHQ test sets, so quantitative results are to be interpreted cautiously.
- Entropic regularisation epsilon requires careful tuning to trade off visual quality and alignment.
Experimental setup
Datasets: MNIST EMNIST AFHQ (Wild and Cat, 64x64 and 256x256)
Benchmarks: EMNIST to MNIST translation AFHQ-64 Wild to Cat
Metrics: FID MSD LPIPS
Baselines: DSBM Bridge Matching pretrained model Iterative DSBM finetuning DSB
Builds on: Diffusion Schrödinger Bridge Matching Iterative Markovian Fitting Bridge Matching Adam
Closest papers by meaning
- Adversarial Schrödinger Bridge Matching (NeurIPS 2024 · similarity 0.92)
- Unpaired Image-to-Image Translation via Neural Schrödinger Bridge (ICLR 2024 · similarity 0.90)
- Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.89)
- Schrödinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres (NeurIPS 2025 · similarity 0.89)
- Feedback Schrödinger Bridge Matching (ICLR 2025 · similarity 0.88)
- Go With the Flow: Fast Diffusion for Gaussian Mixture Models (NeurIPS 2025 · similarity 0.88)
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