Thomas Gravier$^{\mathbf{}12}$, Thomas Boyer$^{\mathbf{}1}$, Auguste Genovesio$^{1}$
$^\mathbf{*}$Equal contribution $^1$IBENS, ENS Ulm, PSL, Paris, France $^2$ENS Paris-Saclay, Paris, France
This website serves as supplementary material for the MMtSBM paper, published at ICML 2026.
Multi-marginal temporal Schrödinger Bridge Matching from unpaired data
ICML Poster Multi-marginal temporal Schrödinger Bridge Matching from unpaired data
https://github.com/tgravier/MMDSBM-pytorch
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Many natural dynamic processes -such as in vivo cellular differentiation or disease progression- can only be observed through the lens of static sample snapshots. While challenging, reconstructing their temporal evolution to decipher underlying dynamic properties is of major interest to scientific research. Existing approaches enable data transport along a temporal axis but are poorly scalable in high dimension and require restrictive assumptions to be met. To address these issues, we propose Multi-Marginal temporal Schrödinger Bridge Matching (MMtSBM) from unpaired data, extending the theoretical guarantees and empirical efficiency of https://arxiv.org/abs/2303.16852 by deriving the Iterative Markovian Fitting algorithm to multiple marginals in a novel factorized fashion. Experiments show that MMtSBM retains theoretical properties on toy examples, achieves state-of-the-art performance on real world datasets such as transcriptomic trajectory inference in 100 dimensions, and for the first time recovers couplings and dynamics in very high dimensional image settings. Our work establishes multi-marginal Schrödinger bridges as a practical and principled approach for recovering hidden dynamics from static data.
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In this 2D experiment akin to https://arxiv.org/abs/2209.03003, we used $N=3$ mixtures of two standard Gaussian as marginals. In this configuration the optimal transport between each pair of marginals can be computed exactly: it is a pure translation of each Gaussian components inside the mixtures:
The ground-truth exact OT transport plan between each pair of mixture (the true multi-marginal transport plan is unknown), computed with POT.
Using this plan as a good approximation of the true multi-marginal plan for mental reference, we can observe the effect of MMtSBM training over simple noisy flow matching and verify visually that it indeed yields a much more optimal plan:
Epoch $0$ (only noisy flow matching).
Epoch $0$ (only noisy flow matching).
Epoch $5$ (after MMtSBM training).
Epoch $5$ (after MMtSBM training).
Above videos: True marginal times $(t_0, t_1, t_2)=(0, 1, 2)$. The order of the $3$ true marginals is: $t_0=$ dark blue; $t_1=$ red; $t_2=$ light blue. Generated samples are in green. In the background is the quiver plot of the learned score network.
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After only the warm-up phase, we can see that the learned transport maps mix the Gaussian components of the mixtures, resulting in intersecting trajectories as can be seen in the left video. However, after the SB learning phase of MMtSBM, we can see in the right video that the learned trajectories do not intersect each other anymore and that MMtSBM yields the expected exact optimal transport map: pure translations between Gaussians.
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