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Multimarginal Generative Modeling with Stochastic Interpolants
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Extends stochastic interpolants to a barycentric simplex parameterization so one velocity/score model transports among K+1 marginals, enabling multi-way correspondences, path optimization, and all-to-all image translation.
Problem
Generative modeling of a joint distribution whose K+1 marginals are prescribed, with transport between any pair of marginals and recovery of multi-way correspondences; also reducing dynamical transport cost in the two-marginal case.
Approach
Lift the interpolation time to a point on the simplex (barycentric stochastic interpolant x(alpha)=sum alpha_k x_k), derive continuity equations on the simplex whose conditional-expectation fields g_k are minimizers of square-loss regressions, and define transport along any simplex curve alpha(t); the path itself can be optimized to minimize Wasserstein-2 transport cost.
Key results
- Learning an optimized simplex path for 2D Gaussian-to-checkerboard reduces transport cost, yielding more accurate probability flows at fixed 5 midpoint steps (qualitative).
- A single shared multimarginal model over 4 marginals (AFHQ, Flowers, CelebA, Gaussian) at 64x64 produces all-to-all translations with emergent style transfer (qualitative).
- Training on the whole simplex rather than only edges better preserves image characteristics in MNIST digit translation (qualitative).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Optimize the interpolation path instead of using the linear path | Stochastic interpolant for Gaussian-to-checkerboard transport · Gaussian-to-checkerboard | transport cost | Helped |
| Optimize the interpolation path instead of using the linear path | Stochastic interpolant for Gaussian-to-checkerboard transport · Gaussian-to-checkerboard | accuracy at fixed function evaluations: 5 steps | Helped |
| Add training across the entire simplex rather than only on simplex edges (ablation) | Multimarginal image translation model · MNIST digits 0–5 | image translation quality | Helped |
| Add training across the entire simplex rather than only on the edge between digits 0 and 3 (ablation) | MNIST image translation model · MNIST digits 0 and 3 | image translation quality | Helped |
| Use different paths through the simplex to the same endpoint | Unconstrained multimarginal model · AFHQ, Oxford Flowers, CelebA | generated-sample variation | Mixed |
Surprising findings
- Learning only on simplex edges still leads to two-marginal transports influenced by all training data, due to neural network imperfection and implicit bias.
Limitations
- Learned path optimization restricted to a parametric Fourier-type class, and linear-in-velocity class not sufficient for optimal transport
- Evaluation is primarily qualitative with no quantitative metrics for translation quality
- Multi-way correspondence claims lack rigorous quantitative validation
Experimental setup
Datasets: MNIST AFHQ-2 Oxford Flowers CelebA
Benchmarks: MNIST AFHQ-Flowers-CelebA
Metrics: path length (Wasserstein-2 transport cost) qualitative sample inspection
Baselines: two-marginal stochastic interpolant edge-only multimarginal training
Builds on: stochastic interpolants flow matching rectified flow Benamou-Brenier dynamical transport
Closest papers by meaning
- Stochastic Interpolants with Data-Dependent Couplings (ICML 2024 · similarity 0.91)
- Multitask Learning with Stochastic Interpolants (NeurIPS 2025 · similarity 0.89)
- Stochastic Interpolants via Conditional Dependent Coupling (NeurIPS 2026 · similarity 0.88)
- Generative Modeling by Value-Driven Transport (NeurIPS 2026 · similarity 0.87)
- A solvable model of learning generative diffusion: theory and insights (NeurIPS 2025 · similarity 0.86)
- Generative Marginalization Models (ICML 2024 · similarity 0.86)
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