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Multi-Marginal Stochastic Flow Matching for High-Dimensional Snapshot Data at Irregular Time Points
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Proposes Multi-Marginal Stochastic Flow Matching (MMSFM), extending simulation-free score and flow matching to ordered multi-marginal snapshots at irregular time points using monotonic cubic Hermite transport splines over overlapping windows, evaluated on synthetic Gaussians, DynGen, CITEseq/Multiome, and Imagenette.
Problem
Modeling high-dimensional system dynamics from snapshot measurements taken at irregular time points, without reducing dimensionality, and without overfitting or expensive trajectory simulation during training.
Approach
Combine conditional flow matching and score matching with mini-flows over overlapping (k+1)-tuples of marginals, where conditioning paths are monotonic cubic Hermite transport splines built from first-order Markov OT couplings, and sample times stratified across intervals to handle uneven spacing.
Key results
- On S-shaped Gaussians with equidistant times (holding out t5), the Triplet model reduced W2^2 to 3.86 versus 66.91 for MIOFlow.
- On alpha-shaped Gaussians with arbitrary times T2, the Triplet model achieved W2^2 of 14.73 versus 76.82 for the Pairwise model.
- On CITEseq and Multiome (holding out t2=0.4), Triplet gave lower W2^2 than Pairwise in most PCA settings, e.g. 3942.08 vs 4036.41 on CITEseq PCA 100.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · S-shape, T1; hold t5=0.83 | W1: 2.36 -> 1.83 | Helped |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · α-shape, T1; hold t4=0.67 | W1: 3.78 -> 4.54 | Hurt |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · α-shape, T2; hold t4=0.54 | W1: 8.08 -> 3.79 | Helped |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · CITEseq PCA 50; hold t2=0.4 | W1: 54.18 -> 53.98 | Helped |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · CITEseq PCA 100; hold t2=0.4 | W1: 62.85 -> 62.08 | Helped |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · CITEseq Hi-Var 1000; hold t2=0.4 | W1: 50.64 -> 50.71 | Hurt |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · Multiome PCA 50; hold t2=0.4 | W1: 61.79 -> 60.92 | Helped |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · Multiome PCA 100; hold t2=0.4 | W1: 70.72 -> 70.39 | Helped |
| increase window size from Pairwise (k=1) to Triplet (k=2) | MMSFM · Multiome Hi-Var 1000; hold t2=0.4 | W1: 56.15 -> 56.10 | Helped |
| increase window size from Triplet (k=2) to All (k=M-1) (ablation) | MMSFM · S-shaped and α-shaped Gaussians, T1 and T2 held-out points | W1 | Mixed |
Surprising findings
- Pairwise model slightly outperformed Triplet on alpha-shaped Gaussians with equidistant times (T1), attributed to momentum from the prior interval.
- Neither model converged on alpha-shaped Gaussians with highly irregular times T3, and models struggled with bifurcating DynGen trajectories, possibly due to mini-batch OT allowing branch jumping.
Limitations
- Mini-batch OT does not enforce consistency, causing trajectories to jump between bifurcating branches.
- Performance degrades for highly irregular time intervals, with alpha-shaped T3 failing to converge.
- Evaluations use small MLPs on toy data and a 32x32 image task; the CITEseq/Multiome experiments use a single donor.
Experimental setup
Datasets: S-shaped Gaussians alpha-shaped Gaussians DynGen CITEseq Multiome Imagenette
Benchmarks: S-shaped Gaussians alpha-shaped Gaussians DynGen CITEseq Multiome Imagenette
Metrics: W1 W2^2 MMD (Gaussian kernel) MMD (mixture kernel)
Baselines: MIOFlow SF2M (pairwise k=1) All-marginal (k=M-1) model
Builds on: conditional flow matching simulation-free score and flow matching (SF2M) MIOFlow transport splines Schrödinger bridge
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