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Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel
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Proposes conditional particle flows of the negative-distance MMD (energy distance) that approximate the joint distribution for posterior sampling, with error bounds and a Wasserstein gradient flow interpretation, applied to conditional generation and inverse problems.
Problem
Posterior sampling P_{X|Y=y} for Bayesian inverse problems and conditional generation when only joint samples are available; generative models trained end-to-end do not directly apply to gradient flow based generators.
Approach
Approximate the joint distribution with discrete Wasserstein gradient flows of the negative-distance MMD (sliced and sorted for efficiency), keep the conditioning component fixed so the flow acts only on X, then learn conditional UNet networks approximating Euler steps; prove posterior error bounds in expectation and show the conditional flow is a Wasserstein gradient flow of a modified functional
Key results
- Conditional MMD flow outperforms l-SWF on class-conditional generation of MNIST, FashionMNIST and CIFAR10 in average class-conditional FID.
- Theorem gives expected posterior MMD error bounded by a power (1/(4(d+n+1))) of the joint MMD error under compact support assumptions.
- For low-dose CT, PSNR and SSIM of mean images are reported on the full 3553-image test set against a prior method.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use conditional MMD flow instead of ℓ-SWF | Class-conditional image generation · MNIST, FashionMNIST, and CIFAR10 | class-conditional FID | Helped |
Surprising findings
- MMD-based posterior error bound cannot hold in general for non-compactly supported distributions (counterexample given).
- Sliced Wasserstein flows neglecting the velocity in the y-component is theoretically justified: the second component has no flow for constant empirical conditioning measures.
Limitations
- Retraining needed for any change in forward operator or noise model.
- Particles interact, so batching for training is not yet enabled.
- Evaluation of posterior sampling quality in high dimensions lacks meaningful metrics; CT experiments not claimed clinically realistic.
Experimental setup
Datasets: MNIST FashionMNIST CIFAR10 CelebA LoDoPaB
Benchmarks: MNIST FashionMNIST CIFAR10 CelebA LoDoPaB
Metrics: FID PSNR SSIM
Baselines: l-SWF SRFlow WPPFlow
Builds on: generative MMD flows sliced Wasserstein gradient flows (l-SWF) SRFlow UNet
Closest papers by meaning
- Sobolev Regularized MMD Gradient Flow (NeurIPS 2026 · similarity 0.89)
- Functional Gradient Flows for Constrained Sampling (NeurIPS 2024 · similarity 0.87)
- Generative Sliced MMD Flows with Riesz Kernels (ICLR 2024 · similarity 0.87)
- Provable Posterior Sampling with Denoising Oracles via Tilted Transport (NeurIPS 2024 · similarity 0.87)
- Inverse Flow and Consistency Models (ICML 2025 · similarity 0.86)
- DEFT: Efficient Fine-tuning of Diffusion Models by Learning the Generalised $h$-transform (NeurIPS 2024 · similarity 0.86)
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