Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel

TL;DR

This paper introduces a novel gradient flow method based on the maximum mean discrepancy with a negative distance kernel for efficient posterior sampling and conditional generative modeling, demonstrating effectiveness in image generation and inverse problems.

Contribution

It develops a new particle flow approach using MMD with negative distance kernel, providing theoretical guarantees and practical algorithms for posterior sampling.

Findings

Effective in conditional image generation

Applicable to inverse problems like superresolution and inpainting

Theoretically proven to be a Wasserstein gradient flow

Abstract

We propose conditional flows of the maximum mean discrepancy (MMD) with the negative distance kernel for posterior sampling and conditional generative modeling. This MMD, which is also known as energy distance, has several advantageous properties like efficient computation via slicing and sorting. We approximate the joint distribution of the ground truth and the observations using discrete Wasserstein gradient flows and establish an error bound for the posterior distributions. Further, we prove that our particle flow is indeed a Wasserstein gradient flow of an appropriate functional. The power of our method is demonstrated by numerical examples including conditional image generation and inverse problems like superresolution, inpainting and computed tomography in low-dose and limited-angle settings.