Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving

Recent advances in inverse problem solving have increasingly adopted flow priors over diffusion models due to their ability to construct straight probability paths from noise to data, thereby enhancing efficiency in both training and inference. However, current flow-based inverse solvers face two primary limitations: (i) they operate directly in pixel space, which demands heavy computational resources for training and restricts scalability to high-resolution images, and (ii) they employ guidance strategies with prior-agnostic posterior covariances, which can weaken alignment with the generative trajectory and degrade posterior coverage. In this paper, we propose LFlow (Latent Refinement via Flows), a training-free framework for solving linear inverse problems via pretrained latent flow priors. LFlow leverages the efficiency of flow matching to perform ODE sampling in latent space along an optimal path. This latent formulation further allows us to introduce a theoretically grounded posterior covariance, derived from the optimal vector field, enabling effective flow guidance. Experimental results demonstrate that our proposed method outperforms state-of-the-art latent diffusion solvers in reconstruction quality across most tasks. The code will be publicly available at GitHub.

Recently, flow matching [41,42] has gained prominence as a compelling alternative for generative modeling. By parameterizing transformation dynamics with ordinary differential equations (ODEs), these models can generate arbitrary probability paths, including those grounded in optimal transport (OT) principles [41]. This flexibility enables the design of straight-line generative trajectories, leading to more efficient training and sampling compared to diffusion-based approaches [43,44]. Motivated by these capabilities, several recent works have explored the use of flow-based priors for inverse problems, achieving faster and higher-quality solutions across diverse tasks [45][46][47][48][49][50]. Nevertheless, existing methods still suffer from two key drawbacks: (1) they operate in pixel space, which restricts scalability to high-dimensional data and limits generalizability across different types of inverse problems; and (2) they adopt guidance techniques originally developed for diffusion models, which estimate posterior covariances independently of the learned prior. This disconnect may steer the sampling trajectory away from high-probability regions, leading to degraded sample quality and reduced fidelity, with slower convergence often observed when adaptive ODE solvers are employed.

To address these limitations, we propose LFlow (Latent Refinement via Flows), a framework that utilizes latent flow matching to solve linear inverse problems without additional training. By applying flow matching in latent space, LFlow achieves enhanced computational efficiency and enables more scalable and effective inverse solutions in reduced-dimensional domains. Additionally, we introduce a well-founded, time-dependent variance for the latent identity posterior covariance, formulated using Tweedie's covariance formula and the optimal vector field under the assumption of a Gaussian latent representation. This posterior covariance is explicitly informed by the pretrained optimal vector field, ensuring that guidance remains consistent with the generative dynamics. Our empirical evaluations demonstrate that images inferred via latent ODE sampling along conditional OT paths exhibit superior perceptual quality compared to those generated through latent diffusion-based probability paths.

Our primary contributions are as follows:

• Methodological: We propose a training-free framework based on latent flow matching and posterior-guided ODE sampling for solving linear inverse problems, significantly outperforming latent diffusion-based approaches in both efficiency and reconstruction quality.

• Analytical: We derive a principled correction to the pretrained latent flow using the measurement likelihood gradient and introduce an analytically justified, time-dependent posterior covariance to improve sampling accuracy and convergence speed.

• Empirical: We validate the performance of LFlow through extensive experiments on image reconstruction tasks, including deblurring, super-resolution, and inpainting, achieving state-of-the-art results without requiring substantial problem-specific hyperparameter tuning.