FLINT: Coupling Proximal Initialization and Bounded Stochasticity for Flow-Matching Inverse Problems

Flow matching has emerged as the state-of-the-art generative model and has been used for plug-and-play (PnP) priors to solve inverse problems in computational imaging. However, existing flow-based inverse solvers make simplifying approximations in posterior sampling and/or assume linear forward models. In contrast, principled diffusion-based methods exist but are computationally expensive. To circumvent these problems, we introduce FlowSGS, a flow-based posterior sampling method using Split Gibbs Sampling (SGS) with annealed coupling, and make flow-specific interventions to the prior sampling and annealing steps, theoretically grounded in the Stochastic Interpolants (SI) framework. Using a novel timestep correction and accelerated warm-up with one-step flow models, our method is faster and more accurate than the state of the art across a range of imaging problems. We demonstrate the first use of flow-based samplers for a nonlinear inverse problem (Fourier phase retrieval) and apply one-step flow models to principled Bayesian inverse imaging.

Flow models have recently emerged as powerful generative priors for solving inverse problems. Despite their success, most existing flow-based inverse problem solvers initialize the reconstruction process with random Gaussian noise. However, recent studies in image generation have shown that the choice of initial noise can significantly influence generation outcomes. The role of initialization as a distinct design choice in flow-based inverse problem solving remains comparatively underexplored. In this work, we systematically evaluate noise initialization strategies while holding the downstream flow-based inverse solver fixed. We propose Retrieval-Augmented Initialization with Dual-Guided Inversion (RA-DGI), a method that constructs improved initial noise by leveraging retrieved proxy clean images and dual-guided inversion. Extensive experiments across multiple inverse problems demonstrate that the quality of the initial noise substantially affects reconstruction fidelity and perceptual performance. RA-DGI outperforms standard random initialization and achieves superior results compared to inference-time scaling strategies, while requiring less computational overhead. These findings highlight initialization as a critical yet previously overlooked factor in flow-based inverse problem solvers.

A growing family of training-free solvers -- FlowDPS, FLOWER, PnP-Flow and their diffusion ancestors (DPS, DAPS) -- repurpose a pretrained flow-matching prior to solve imaging inverse problems by adding a measurement-guidance term to the deterministic probability-flow ODE. Despite strong empirical results, what these per-step corrections actually approximate -- and how far the resulting samples are from the true posterior $p(x\mid y)$ -- has not been characterized. We give a posterior-transport account of flow-based inverse problem solving. Our starting point is a simple but consequential fact: for a deterministic flow prior, Bayesian conditioning is realized entirely by a reweighting of the source distribution, not by a drift correction; pushing the reweighted source through the unmodified velocity field yields exact posterior samples. From this we show that trajectory-guidance solvers can be read as the minimum-kinetic-energy correction field needed to morph the unconditional source into the posterior, and that FlowDPS / FLOWER / PnP-Flow correspond to distinct zeroth-order / Gaussian / proximal approximations of this single object; we bound the resulting posterior bias in Wasserstein distance. A controlled $2$D study with a closed-form posterior confirms the theory decisively: source reweighting matches the true posterior to the Monte-Carlo floor on every metric, whereas trajectory guidance incurs $200$--$800\times$ larger error and collapses posterior modes, regardless of guidance strength. Guided by the analysis we propose a cheap, principled velocity-correction solver that is competitive across two in-domain priors (AFHQ, CelebA) and two out-of-distribution settings while, unlike point-estimate source-space optimizers, producing diverse posterior samples with uncertainty that correlates with reconstruction error.