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Variational Schrödinger Diffusion Models
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Proposes VSDM, a simulation-free-forward Schrödinger bridge diffusion model that linearizes the forward score via variational inference, with stochastic-approximation convergence guarantees and CIFAR10/time-series experiments.
Problem
Schrödinger bridge diffusion models need intractable forward score functions and costly simulation-based implicit training losses, which hurts scalability and requires warm-up checkpoints for large-scale training.
Approach
Replace the nonlinear forward score of the Schrödinger bridge with a locally linear variational score A_t x, yielding a multivariate OU-type forward process with closed-form Gaussian transitions, and adaptively optimize A_t via stochastic approximation while training backward scores simulation-free.
Key results
- VSDM achieves FID 2.28 unconditional on CIFAR10 without warm-up, beating the SB-FBSDE (3.01) and other OT baselines such as DGflow (9.63) and DOT (15.78).
- On anisotropic spiral-8Y data, VSDM-10 reconstructs boundaries that SGM-10 fails to recover, and gives around 40% improvement in trajectory straightness.
- Theory: stochastic approximation gives E||A_t^(k) - A_t*||^2 <= 2 eta_k, and the adaptively generated samples have a total-variation bound of the form exp(-T)+(sqrt(dh)+eps_score+sqrt(eta_k))sqrt(T).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use VSDM instead of SB-FBSDE | CIFAR10 unconditional generation · CIFAR10 unconditional | FID: 3.01 -> 2.28 | Helped |
| Use VSDM instead of EDM | CIFAR10 unconditional generation · CIFAR10 unconditional | FID: 1.97 -> 2.28 | Hurt |
| Use VSDM instead of LSGM | CIFAR10 unconditional generation · CIFAR10 unconditional | FID: 2.10 -> 2.28 | Hurt |
| Use VSDM-10 instead of SGM-10 | generation on anisotropic spiral-8Y · spiral-8Y | trajectory straightness: around 40% improvement | Helped |
Surprising findings
- Increasing the SGM maximum beta to 20 or 30 hurts straightness and transport efficiency on the anisotropic data, so simply enlarging diffusion scale is not a fix.
Limitations
- The variational score gives sub-optimal transport in general nonlinear settings, so global OT optimality is lost.
- The prior is restricted to Gaussian distributions.
- Theory relies on assumptions such as local strong convexity and Lipschitz scores, and convergence holds only to a local equilibrium.
Experimental setup
Datasets: CIFAR10 spiral (synthetic) checkerboard (synthetic) Electricity (time series)
Benchmarks: CIFAR10
Metrics: FID straightness NFE-based sample quality
Baselines: SB-FBSDE DOT DGflow SDE (score SDE) ScoreFlow VDM LSGM EDM DDPM Gaussian Schrödinger bridge
Builds on: Schrödinger bridge FB-SDE EDM Diff-Instruct score-based generative models
Closest papers by meaning
- Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.90)
- Learning non-equilibrium diffusions with Schrödinger bridges: from exactly solvable to simulation-free (NeurIPS 2025 · similarity 0.90)
- Transport meets Variational Inference: Controlled Monte Carlo Diffusions (ICLR 2024 · similarity 0.90)
- Exponential Convergence Guarantees for Iterative Markovian Fitting (NeurIPS 2025 · similarity 0.90)
- Go With the Flow: Fast Diffusion for Gaussian Mixture Models (NeurIPS 2025 · similarity 0.90)
- Improved sampling via learned diffusions (ICLR 2024 · similarity 0.90)
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