What Do Flow-Based Inverse Solvers Approximate? A Posterior-Transport View

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 define a canonical minimum-energy corrector that exactly transports the unconditional source to the posterior. Existing guidance methods can then be placed within the same correction framework as local surrogate correctors-differing in their endpoint estimate and mobility—and their discrepancy from the canonical corrector controls the terminal posterior bias in Wasserstein distance. A controlled $2$D study with a closed-form posterior confirms the theory decisively: source reweighting matches the two-sample Monte-Carlo reference on every metric, whereas trajectory guidance incurs substantially larger error-$6$–$14\times$ that reference on sliced-$W_2$ and $200$–$800\times$ on energy distance and MMD—and collapses posterior modes across all tested guidance strengths. 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 conditional samples whose sample-based uncertainty correlates with reconstruction error (moderately so once the inpainting mask is controlled for).