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

• URL: http://arxiv.org/abs/2606.24516v2
• Date: Sat, 27 Jun 2026 09:51:13 GMT
• Title: What Do Flow-Based Inverse Solvers Approximate? A Posterior-Transport View
• Abstract summary: We give a posterior-transport flow-based problem solving account.<n>We show that trajectory-guidance solvers can be read as the minimum-energy emphcorrection field needed to morph the unconditional source into the posterior.
• Score: 31.082191748525137
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Same Flow, Different Paths: Variance Reduction in Flow Matching [67.12978375116599]
  We study the choice of this path from an optimization perspective.<n>Our main finding is that the choice of path $g_t$ can fundamentally change the convergence rate of SGD.
  arXiv  Detail & Related papers  (2026-09-15T15:01:04Z)
• Fast and Faithful: Principled Conditional Flow Matching for Inverse Problems [8.966524060251965]
  Flow matching approaches to imaging inverse problems commonly incorporate measurements in two ways.<n>We propose a principled parametrization of the measurement-conditional velocity field to solve inverse problems.<n>Our method achieves state-of-the-art PSNR with $50times$ fewer evaluations function than the strongest flow baseline.
  arXiv  Detail & Related papers  (2026-09-11T15:12:11Z)
• Round-Trip Consistency: Bidirectional Diffusion Models Can Predict Their Own Rollout Errors [51.56484100374058]
  Autoregressive models accumulate error over long rollouts, yet at deployment there is no ground truth to measure it against.<n>We train a single conditional latent diffusion model that steps a dynamical system forward or backward in time via a direction flag.<n>We show that this bidirectionality supplies a measurement-free test-time error signal.
  arXiv  Detail & Related papers  (2026-08-01T13:49:46Z)
• The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems [28.97171921966014]
  Generative models are increasingly used to solve partial differential equation (PDE) inverse problems.<n>We show that this widely adopted recipe samples the wrong distribution.<n>We introduce textbfCoCoS, a measure-aware constrained sampler that targets the correct co-area posterior.
  arXiv  Detail & Related papers  (2026-06-03T12:30:03Z)
• Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability [13.640937605089865]
  Diffusion-based posterior samplers use pretrained diffusion priors to sample from measurement- or reward-conditioned posteriors.<n>This paper introduces a tractable surrogate path connecting the true posterior to a standard Gaussian and comparing it to the sampler's path.<n>A Feynman-Kac representation then expresses the Radon-Nikodym correction as an explicit path expectation, identifying which posterior regions are over- or under-sampled.
  arXiv  Detail & Related papers  (2026-05-07T16:37:29Z)
• Spherical Flows for Sampling Categorical Data [6.88204255655161]
  We study the problem of learning generative models for discrete sequences in a continuous embedding space.<n>Experiments compare the vMF path against geodesic and Euclidean alternatives.<n>The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.
  arXiv  Detail & Related papers  (2026-05-07T03:34:00Z)
• Posterior Augmented Flow Matching [64.1559809786948]
  Posterior-Augmented Flow Matching (PAFM) is a generalization of flow matching (FM)<n>We show that PAFM improves over FM by up to 3.4 FID50K across different model scales.
  arXiv  Detail & Related papers  (2026-05-01T17:59:59Z)
• Efficient Approximate Posterior Sampling with Annealed Langevin Monte Carlo [36.938523047101555]
  We study the problem of posterior sampling in the context of score based generative models.<n>We show that one can tractably sample from a distribution that is simultaneously close to the posterior of a noised prior in KL divergence and the true posterior in Fisher divergence.
  arXiv  Detail & Related papers  (2025-08-11T05:25:24Z)
• FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems [51.99765487172328]
  Posterior sampling for inverse problem solving can be effectively achieved using flows.<n>Flow-Driven Posterior Sampling (FlowDPS) outperforms state-of-the-art alternatives.
  arXiv  Detail & Related papers  (2025-03-11T07:56:14Z)
• On the Wasserstein Convergence and Straightness of Rectified Flow [54.580605276017096]
  Rectified Flow (RF) is a generative model that aims to learn straight flow trajectories from noise to data.<n>We provide a theoretical analysis of the Wasserstein distance between the sampling distribution of RF and the target distribution.<n>We present general conditions guaranteeing uniqueness and straightness of 1-RF, which is in line with previous empirical findings.
  arXiv  Detail & Related papers  (2024-10-19T02:36:11Z)
• Amortizing intractable inference in diffusion models for vision, language, and control [89.65631572949702]
  This paper studies amortized sampling of the posterior over data, $mathbfxsim prm post(mathbfx)propto p(mathbfx)r(mathbfx)$, in a model that consists of a diffusion generative model prior $p(mathbfx)$ and a black-box constraint or function $r(mathbfx)$.<n>We prove the correctness of a data-free learning objective, relative trajectory balance, for training a diffusion model that samples from
  arXiv  Detail & Related papers  (2024-05-31T16:18:46Z)
• Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariance [52.093434664236014]
  Recent diffusion models provide a promising zero-shot solution to noisy linear inverse problems without retraining for specific inverse problems. Inspired by this finding, we propose to improve recent methods by using more principled covariance determined by maximum likelihood estimation.
  arXiv  Detail & Related papers  (2024-02-03T13:35:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.