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Light and Optimal Schrödinger Bridge Matching
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Proposes optimal Schrödinger bridge matching, which recovers the SB from an arbitrary transport plan in a single bridge-matching step, and a Gaussian-mixture solver LightSB-M that is fast on CPU.
Problem
Existing bridge-matching SB solvers either rely on heuristic plan approximations (e.g., minibatch OT) or iterative Markovian/reciprocal projections that accumulate errors and need many training iterations.
Approach
Shows that the KL optimal projection of any reciprocal process onto the set of Schrödinger bridges yields the SB in one step, with a tractable bridge-matching objective that is equivalent (up to constants) to the LightSB/EgNOT energy objective; implemented with a Gaussian-mixture parameterization of the adjusted Schrödinger potential.
Key results
- On the SB mixtures benchmark, LightSB-M matches the best solvers across dimensions and epsilon; e.g., at D=2, eps=0.1 it reaches 0.02 cBW2^2-UVP versus 0.04 for SF2M-Sink and 5.2 for DSBM.
- On MSCI single-cell data (DIM=50), LightSB-M (ID) reaches energy distance 2.347 in about 58 s on 4 CPU cores, versus 2.66 for SF2M-Sink on a V100 GPU.
- Sampling from the learned plan takes 0.00058 s versus 0.046 s for 3-step Euler-Maruyama and 1.27 s for 100-step Euler-Maruyama in D=16.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use independent, minibatch OT, or ground-truth transport plan instead of requiring the EOT plan (ablation) | LightSB-M · EOT/SB benchmark | cB W₂²-UVP (%): 0.18; 0.1; 0.1 | No clear effect |
| Use a minibatch OT plan instead of an independent plan (ablation) | LightSB-M · MSCI single-cell data | energy distance: 2.347 ± 0.11 -> 2.33 ± 0.09; 2.174 ± 0.08 -> 2.172 ± 0.08; 1.35 ± 0.05 -> 1.33 ± 0.05 | No clear effect |
| Use LightSB-M instead of DSBM | SB benchmark, ε=0.1, D=16 · EOT/SB benchmark | cB W₂²-UVP (%): 16.8 -> 0.18 | Helped |
| Use LightSB-M instead of SF²M-Sink | SB benchmark, ε=10, D=16 · EOT/SB benchmark | cB W₂²-UVP (%): 4.9 -> 0.19 | Helped |
| Use LightSB-M instead of LightSB | MSCI single-cell data · MSCI, DIM 1000 | energy distance: 1.27 ± 0.19 -> 1.33 ± 0.05 | Hurt |
| Use LightSB-M (ID) instead of the strongest listed baseline | FFHQ man→woman translation · FFHQ man→woman | FID: 0.859 -> 0.852 | Helped |
| Use LightSB-M instead of DSBM | SB benchmark · SB benchmark, ε=0.1, D=16 | KL(T*||S): 0.2950 -> 0.0093 | Helped |
| Sample directly from the learned plan instead of using Euler-Maruyama with 500 steps (ablation) | LightSB-M · SB benchmark, ε=0.1, D=16 | cB W₂²-UVP: 0.09 -> 0.09 | No clear effect |
| Increase Euler-Maruyama simulation from 10 to 500 steps (ablation) | LightSB-M · SB benchmark, ε=0.1, D=16 | cB W₂²-UVP: 1.53 -> 0.09 | Helped |
| Increase ε from 0.1 to 0.3 (ablation) | LightSB-M (ID) · MSCI single-cell data, DIM 100 | energy distance: 2.174 ± 0.08 -> 2.169 ± 0.11 | Helped |
Surprising findings
- SF2M-Sink outperforms DSBM at eps=0.1 and eps=1 despite DSBM theoretically targeting the true SB, with the authors suggesting DSBM needs more iterations, though increasing iterations did not help.
Limitations
- Gaussian-mixture parameterization of the potential may be insufficient for large-scale generative tasks; neural parameterization is only sketched in the appendix.
- Only the Wiener prior is considered in the main method.
Experimental setup
Datasets: SB mixtures benchmark Open Problems multimodal single-cell data (MSCI) FFHQ
Benchmarks: SB mixtures benchmark MSCI single-cell benchmark FFHQ adult-child and man-woman translation
Metrics: cBW2^2-UVP BW2^2-UVP energy distance FID dynamic KL
Baselines: DSBM SF2M-Sink LightSB EgNOT
Builds on: LightSB DSBM OT-CFM EgNOT
Closest papers by meaning
- Feedback Schrödinger Bridge Matching (ICLR 2025 · similarity 0.93)
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