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

2026-06-23

AI Summary

This work reveals the true objective approximated by current training-free streaming solvers for inverse problems—such as FlowDPS and FLOWER—and their substantial deviation from the true posterior distribution. Adopting a posterior transport perspective, it argues that under deterministic flow priors, Bayesian conditioning should be implemented via reweighting the source distribution, and for the first time establishes a theoretical connection between flow-based solvers and posterior optimal transport, showing that these methods essentially approximate the minimum kinetic energy correction field. Building on this insight, the paper proposes a more accurate source reweighting principle and an improved velocity field correction scheme. Experiments demonstrate that source reweighting faithfully recovers the true posterior, whereas existing trajectory-guidance approaches incur errors 200–800 times larger and fail to preserve multimodality; the new solver significantly enhances both reconstruction quality and diversity on benchmarks such as AFHQ and CelebA.

Abstract

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 \emph{deterministic} flow prior, Bayesian conditioning is realized entirely by a \emph{reweighting of the source distribution}, not by a drift correction; pushing the reweighted source through the \emph{unmodified} velocity field yields exact posterior samples. From this we show that trajectory-guidance solvers can be read as the minimum-kinetic-energy \emph{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, \emph{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.

Problem

• inverse problems
• flow-based solvers
• posterior approximation
• measurement guidance
• Wasserstein distance

Innovation

• posterior transport
• flow-based inverse solvers
• source reweighting
• velocity correction
• Wasserstein bias