FlowLPS: Langevin-Proximal Sampling for Flow-based Inverse Problem Solvers

TL;DR

FlowLPS addresses inverse problems by leveraging pretrained latent-flow models through a training-free Langevin–proximal framework that first anchors estimates to the data manifold via Langevin dynamics and then aggressively optimizes toward the posterior mode. The method blends manifold-consistent sampling with proximal optimization, achieving a superior balance between reconstruction fidelity and perceptual quality. Extensive experiments on FFHQ and DIV2K across deblurring, inpainting, and super-resolution demonstrate state-of-the-art performance and favorable efficiency, especially with adaptive re-noising and dynamic budget strategies. This work provides a principled bridge between posterior sampling and optimization for latent-space priors, enabling robust, practical inference with pretrained flow models.

Abstract

Deep generative models have become powerful priors for solving inverse problems, and various training-free methods have been developed. However, when applied to latent flow models, existing methods often fail to converge to the posterior mode or suffer from manifold deviation within latent spaces. To mitigate this, here we introduce a novel training-free framework, FlowLPS, that solves inverse problems with pretrained flow models via a Langevin Proximal Sampling (LPS) strategy. Our method integrates Langevin dynamics for manifold-consistent exploration with proximal optimization for precise mode seeking, achieving a superior balance between reconstruction fidelity and perceptual quality across multiple inverse tasks on FFHQ and DIV2K, outperforming state of the art inverse solvers.

Figures (17)

• Figure 1: Qualitative results of FlowLPS. (a) Motion deblurring, (b) Gaussian deblurring, (c) Super-resolution ($\times12$) and (d) random inpainting results before and after FlowLPS. (e) and (f) are results for box inpainting.
• Figure 2: Comparison of representative inverse problem solvers: DPS, DDS, DAPS, and the proposed FlowLPS. (a) DPS performs a single gradient-based measurement correction at each timestep, but requires costly backpropagation through the denoising network. (b) DDS and PnP decompose this process into several gradient updates implicitly assuming a locally linear manifold. However, because the data manifold is highly non-linear, these gradient updates often drift off the manifold. (c) DAPS denoises ${\bm{x}}_t$ using PF-ODE steps and applies iterative Langevin dynamics to approximate the posterior distribution $p({\bm{x}}_0|{\bm{x}}_t,{\bm{y}})$. Such Langevin dynamics ensure posterior consistency but do not guarantee convergence toward the posterior mode. (d) In contrast, FlowLPS first performs a few Langevin dynamics steps to rectify the initial estimate and obtain a manifold-consistent posterior anchor, and then applies proximal optimization to seek the mode of the posterior near this anchor, achieving stable and high-fidelity reconstructions in latent space.
• Figure 3: Manifold-preserving update. The clean image is iteratively updated using Langevin optimization combined with hybrid noise sampling using pCN, ensuring that the resulting states remain on the data manifold.
• Figure 4: Qualitative comparison on FFHQ and DIV2K datasets.
• Figure 5: Effect of the number of Langevin steps ($N_L$). (a) Gaussian deblurring and (b) Super-resolution ($\times12$). Using no Langevin steps ($N_L=0$) leads to overly smooth and blurry reconstructions due to optimization getting stuck in poor local minima. Excessive Langevin updates ($N_L=15$) introduce instability and high-frequency noise. Moderate Langevin steps $N_L=5-7$ achieves an optimal balance, producing detailed restorations while maintaning measurement consistency.

Theorems & Definitions (6)

• Proposition 1
• Proposition 2
• Proposition 2
• proof
• Proposition 2
• proof