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Trajectory Inference with Smooth Schrödinger Bridges
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Introduces Smooth Schrödinger Bridges that replace Brownian reference with Gaussian autoregressive processes (e.g., Matérn), lifts the problem to phase space enabling belief propagation, and applies a wavelet-based approximation for trajectory inference.
Problem
Trajectory inference from snapshots of particles: classical Schrödinger bridges with Brownian (Markov) reference produce rough, noisy paths and fail to borrow strength across time; smooth priors seemed computationally intractable for multi-marginal SB.
Approach
Use Gaussian autoregressive processes (GAPs, incl. Matérn with half-integer smoothness) as reference; their phase-space lift (process plus derivatives) is Markov, so the multi-marginal SB becomes a tree-structured problem solvable by belief propagation (equivalent to Sinkhorn), approximated by orthonormal (Haar wavelet) basis expansion of continuous messages.
Key results
- On OBO tracking, smooth SB achieves 5p accuracy 0.999 on N-Body vs 0.641 for BMSB and 0.649 for W2M.
- On 2D Gaussian process data, smooth SB reaches mean l2 of 3.57e-4 vs 2.82e-1 for BMSB.
- On LOT tasks smooth SB is best or second-best on most metrics, e.g., Petal W1 2.70e-2 vs F&S 2.05e-2, and DMSB failed to converge on Dyg.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use smooth Schrödinger Bridges (SSB) instead of W2M | One-By-One tracking · N Body | 5p acc: 0.649 -> 0.999 | Helped |
| Use smooth Schrödinger Bridges (SSB) instead of W2M | One-By-One tracking · 2D Gaussian Process | Mean ℓ2: 2.32e-1 -> 3.57e-4 | Helped |
| Use smooth Schrödinger Bridges (SSB) instead of W2M | One-By-One tracking · Tri-stable Diffusion | 5p acc: 0.956 -> 0.798 | Hurt |
| Use SSB instead of F&S | Leave-One-Timestep-Out inference · Petal j=2 | W1: 2.05e-2 -> 2.70e-2 | Hurt |
| Use SSB instead of MIOFlow | Leave-One-Timestep-Out inference · EB j=2 | W1: 1.34e-1 -> 8.45e-2 | Helped |
| Use SSB instead of F&S | Leave-One-Timestep-Out inference · Dyg j=1 | M_G: 2.00e-3 -> 1.64e-3 | Helped |
Surprising findings
- W2M achieves the best results on the Tri-stable diffusion dataset, outperforming smooth SB there.
- DMSB failed to converge on the Dyngen data despite extensive tuning.
Limitations
- Curse of dimensionality: number of basis coefficients M scales exponentially with GAP order m and dimension d.
- No rigorous approximation guarantee for the wavelet-approximated belief propagation.
- Assumes strict marginal constraints; noisy observations left to future work.
Experimental setup
Datasets: Tri-stable diffusion N-body orbits 2D Gaussian process simulated data Petal EB Dyngen Tree (Dyg)
Benchmarks: Tri-stable diffusion N-body 2D Gaussian process Petal LOT EB LOT Dyngen Tree LOT
Metrics: JumpP 5p accuracy mean l2 W1 MG MI
Baselines: Schrödinger bridge (Brownian) W2M MIOFlow DMSB F&S
Builds on: Schrödinger bridge Sinkhorn algorithm momentum Schrödinger bridge MIOFlow DMSB
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