FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems

TL;DR

FlowDPS proposes a principled flow-based posterior sampling framework for linear inverse problems by decomposing the flow ODE into denoised and noisy components via a flow-version Tweedie formula. It embeds likelihood data-consistency into the denoised path while injecting stochastic noise into the noisy path, producing DDIM-like posterior updates that align with traditional diffusion posterior sampling but within a flow setting. The method supports latent-flow variants for high-resolution, text-conditioned restoration and achieves state-of-the-art results without extra training across four inverse problems. The work clarifies the relationship between flow models and posterior sampling, provides theoretical tools (Tweedie-based decomposition, DDIM form) and practical algorithms, and demonstrates broad applicability to flow-based backbones such as Stable Diffusion 3.0 and FLUX. The practical impact is rapid, high-quality reconstruction for degraded measurements in high-resolution imaging, with a general framework that can extend to other flow-based generative models.

Abstract

Flow matching is a recent state-of-the-art framework for generative modeling based on ordinary differential equations (ODEs). While closely related to diffusion models, it provides a more general perspective on generative modeling. Although inverse problem solving has been extensively explored using diffusion models, it has not been rigorously examined within the broader context of flow models. Therefore, here we extend the diffusion inverse solvers (DIS) - which perform posterior sampling by combining a denoising diffusion prior with an likelihood gradient - into the flow framework. Specifically, by driving the flow-version of Tweedie's formula, we decompose the flow ODE into two components: one for clean image estimation and the other for noise estimation. By integrating the likelihood gradient and stochastic noise into each component, respectively, we demonstrate that posterior sampling for inverse problem solving can be effectively achieved using flows. Our proposed solver, Flow-Driven Posterior Sampling (FlowDPS), can also be seamlessly integrated into a latent flow model with a transformer architecture. Across four linear inverse problems, we confirm that FlowDPS outperforms state-of-the-art alternatives, all without requiring additional training.

Figures (17)

• Figure 1: Geometry of FlowDPS. (a) Unconditional sampling of flow models where denoising and renoising are performed alternatively. (b) Posterior sampling of flow models where the data consistency offset is added to the denoised estimate. Orange arrow denotes the likelihood gradient in Eq. (\ref{['eqn:likelihood_grad']}).
• Figure 2: Qualitative comparison for linear inverse problems with DIV2K validation set. Insets show an enlarged view of the highlighted yellow boxes
• Figure 3: Sampling trajectories for various choices of interpolation scale $\gamma$. Insets show an enlarged view of the highlighted yellow boxes.
• Figure 4: Qualitative results regarding ablation of stochastic noise.
• Figure 5: Evolution of $-\beta_t$ during sampling.

Theorems & Definitions (6)

• Proposition 1: Tweedie Formula
• Proposition 2
• Proposition 2: Tweedie Formula
• proof
• Proposition 2
• proof