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FLOWER: A Flow-Matching Solver for Inverse Problems

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Inverse ProblemsImage ReconstructionGenerative ModelingFlow MatchingAncestral Sampling
Abstract:

We introduce Flower, a solver for linear inverse problems. It leverages a pre-trained flow model to produce reconstructions that are consistent with the observed measurements. Flower operates through an iterative procedure over three steps: (i) a flow-consistent destination estimation, where the velocity network predicts a denoised target; (ii) a refinement step that projects the estimated destination onto a feasible set defined by the forward operator; and (iii) a time-progression step that re-projects the refined destination along the flow trajectory. We provide a theoretical analysis that demonstrates how Flower approximates Bayesian posterior sampling, thereby unifying perspectives from plug-and-play methods and generative inverse solvers. On the practical side, Flower achieves state-of-the-art reconstruction quality while using nearly identical hyperparameters across various linear inverse problems.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper introduces Flower, a three-step iterative solver for linear inverse problems using pre-trained flow models. It resides in the 'Iterative Refinement and Trajectory Correction' leaf, which contains only three papers total (including Flower itself). This is a relatively sparse research direction within the broader taxonomy of 50 papers across 36 topics, suggesting the specific approach of iterative destination estimation, refinement, and time-progression is not yet heavily explored. The sibling papers focus on corrupted trajectory matching and iterative flow matching, indicating a small but coherent cluster of training-free refinement strategies.

The taxonomy reveals that Flower's leaf sits within the larger 'Training-Free Posterior Sampling Methods' branch, which also includes guidance-based approaches, plug-and-play frameworks, and Langevin dynamics methods. Neighboring leaves such as 'Guidance-Based Posterior Sampling' (four papers) and 'Plug-and-Play Flow Matching' (two papers) explore alternative training-free strategies that incorporate measurement likelihood through different mechanisms. The taxonomy's scope notes clarify that Flower's iterative refinement approach excludes single-pass guidance and proximal operator frameworks, positioning it as a distinct middle ground between pure guidance and optimization-based methods.

Among the 30 candidates examined, the first contribution (the three-step Flower solver) shows two refutable candidates out of ten examined, suggesting some overlap with prior iterative refinement work. The second contribution (Bayesian unification of plug-and-play and generative solvers) also has two refutable candidates among ten, indicating existing theoretical connections in the literature. The third contribution (theoretical framework for conditional posterior sampling) found zero refutable candidates across ten examined papers, appearing more novel within this limited search scope. These statistics reflect a focused semantic search, not an exhaustive survey of all related work.

Based on the limited search of 30 candidates, Flower appears to occupy a moderately explored niche within training-free flow-based inverse solvers. The iterative refinement approach has some precedent, but the specific three-step formulation and theoretical unification may offer incremental advances. The sparse population of its taxonomy leaf (three papers) and the modest refutation rates suggest room for contribution, though the analysis cannot rule out relevant work outside the top-30 semantic matches or in adjacent methodological areas.

Taxonomy

Core-task Taxonomy Papers
45
3
Claimed Contributions
30
Contribution Candidate Papers Compared
4
Refutable Paper

Research Landscape Overview

Core task: solving linear inverse problems using flow-matching models. The field has organized itself around several complementary directions. Training-Free Posterior Sampling Methods explore how to leverage pretrained flow models without additional optimization, often through iterative refinement or trajectory correction strategies such as those in Corrupted Trajectory Matching[5] and Iterative Flow Matching[33]. Training-Based and Hybrid Approaches instead fine-tune or augment models with task-specific supervision, while Theoretical Foundations and Methodological Extensions develop the mathematical underpinnings of flow matching and optimal transport. Domain-Specific Applications and Extensions tailor these techniques to areas like medical imaging or turbulence modeling, as seen in Turbulence Flow Matching[7]. Accelerated Sampling and Computational Efficiency focus on reducing the number of function evaluations, and Stochastic Control and Distribution Steering frame inverse problems through control-theoretic lenses. Classical and Alternative Inverse Problem Methods provide baseline comparisons from traditional optimization and variational approaches. A particularly active theme contrasts training-free guidance methods, which modify sampling trajectories on the fly, against methods that learn or refine the flow itself. Training-free approaches like FlowDPS[8] and Training-free Linear Inverses[10] offer flexibility and avoid retraining costs, but may require many iterative steps or careful tuning of correction schedules. FLOWER[0] sits within the Iterative Refinement and Trajectory Correction cluster, sharing conceptual ground with Corrupted Trajectory Matching[5] and Iterative Flow Matching[33]. While these neighbors emphasize correcting or matching corrupted trajectories to enforce measurement consistency, FLOWER[0] appears to refine the flow path iteratively to satisfy linear constraints. This positions it as a training-free method that balances the generality of pretrained priors with the precision needed for inverse problems, contrasting with heavier training-based schemes like Flow Matching Guidance[3] or domain-specialized fine-tuning efforts.

Claimed Contributions

Flower: A three-step flow-matching solver for linear inverse problems

The authors propose Flower, an iterative solver that operates through three steps: flow-consistent destination estimation using the velocity network, measurement-aware refinement via proximal projection, and time progression that re-projects the refined destination along the flow trajectory. This method achieves state-of-the-art reconstruction quality with nearly identical hyperparameters across various linear inverse problems.

10 retrieved papers
Can Refute
Bayesian analysis unifying plug-and-play methods and generative inverse solvers

The authors establish a Bayesian justification showing that Flower approximates posterior sampling from the conditional distribution. They demonstrate that the three steps collectively perform ancestral sampling along the conditional trajectory, linking plug-and-play approaches with approximate posterior sampling using generative models for linear inverse problems.

10 retrieved papers
Can Refute
Theoretical framework connecting flow-matching steps to conditional posterior sampling

The authors provide formal propositions demonstrating how each step of Flower relates to Bayesian inference. They show that the velocity network predicts conditional expectations, the refinement step samples from an approximate conditional posterior using the ΠGDM approximation, and the time-progression step performs ancestral sampling under independence assumptions between source and target distributions.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Flower: A three-step flow-matching solver for linear inverse problems

The authors propose Flower, an iterative solver that operates through three steps: flow-consistent destination estimation using the velocity network, measurement-aware refinement via proximal projection, and time progression that re-projects the refined destination along the flow trajectory. This method achieves state-of-the-art reconstruction quality with nearly identical hyperparameters across various linear inverse problems.

Contribution

Bayesian analysis unifying plug-and-play methods and generative inverse solvers

The authors establish a Bayesian justification showing that Flower approximates posterior sampling from the conditional distribution. They demonstrate that the three steps collectively perform ancestral sampling along the conditional trajectory, linking plug-and-play approaches with approximate posterior sampling using generative models for linear inverse problems.

Contribution

Theoretical framework connecting flow-matching steps to conditional posterior sampling

The authors provide formal propositions demonstrating how each step of Flower relates to Bayesian inference. They show that the velocity network predicts conditional expectations, the refinement step samples from an approximate conditional posterior using the ΠGDM approximation, and the time-progression step performs ancestral sampling under independence assumptions between source and target distributions.

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