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FLOWER: A Flow-Matching Solver for Inverse Problems

Mehrsa Pourya, Bassam El Rawas, Michael Unser

TL;DR

Flower is a flow-matching solver for linear inverse problems that leverages a pre-trained velocity network to produce measurements-consistent reconstructions and to sample from the posterior ${p({\bf X}_1|{\bf Y}={\bf y})}$ under a linear forward model ${\bf y}={\bf H}{\bf x}+{\bf n}$. It implements a three-step loop: flow-consistent destination estimation, measurement-aware refinement via a proximal step, and time progression along the flow, enabling ancestral sampling along the conditional trajectory. A Bayesian analysis shows how the method yields valid posterior samples under reasonable assumptions, connecting plug-and-play concepts with posterior sampling. Empirically, Flower achieves state-of-the-art performance on standard flow-matching inverse problems (e.g., CelebA and AFHQ-Cat) with nearly identical hyperparameters across tasks and offers robust reconstruction quality with controlled computational cost.

Abstract

We introduce Flower, a solver for 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 inverse problems.

FLOWER: A Flow-Matching Solver for Inverse Problems

TL;DR

Flower is a flow-matching solver for linear inverse problems that leverages a pre-trained velocity network to produce measurements-consistent reconstructions and to sample from the posterior under a linear forward model . It implements a three-step loop: flow-consistent destination estimation, measurement-aware refinement via a proximal step, and time progression along the flow, enabling ancestral sampling along the conditional trajectory. A Bayesian analysis shows how the method yields valid posterior samples under reasonable assumptions, connecting plug-and-play concepts with posterior sampling. Empirically, Flower achieves state-of-the-art performance on standard flow-matching inverse problems (e.g., CelebA and AFHQ-Cat) with nearly identical hyperparameters across tasks and offers robust reconstruction quality with controlled computational cost.

Abstract

We introduce Flower, a solver for 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 inverse problems.

Paper Structure

This paper contains 33 sections, 4 theorems, 40 equations, 9 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Flow Matching
  4. Proximal Operator
  5. Method
  6. Application to other source–target couplings.
  7. Relation to plug-and-play.
  8. The role of $\gamma$.
  9. Related Works
  10. Numerical Results
  11. Toy Experiment
  12. Benchmark Experiments
  13. Conclusion
  14. Ethics Statement.
  15. Acknowledgements
  16. ...and 18 more sections

Key Result

Theorem 1

Let ${\bf{x}}_0$ be a sample from $p_{{\bf{X}}_0 \mid {\bf{Y}} = {\bf{y}}}$. If ${\bf{x}}_t$ is a sample from $p_{{\bf{X}}_t \mid {\bf{Y}} = {\bf{y}}}$, then the sample ${\bf{x}}_{t+ \Delta t}$ from $p_{{\bf{X}}_{t+\Delta t} \mid {\bf{X}}_t = {\bf{x}}_t, {\bf{Y}} = {\bf{y}}}$ follows $p_{{\bf{X}}_{t

Figures (9)

  • Figure 1: Overview of the three steps in Flower. Starting from an initial sample $\mathbf{x}_0 \sim p_{\mathbf{X}_0}$ at time $t$, the method: Step 1 predicts a flow-consistent destination $\hat{\mathbf{x}}_1(\mathbf{x}_t)$; Step 2 refines this destination using the measurements via a proximal step and associated uncertainty sampling to obtain $\tilde{\mathbf{x}}_1(\mathbf{x}_t, \mathbf{y})$; and Step 3 updates the trajectory along time by interpolating $\tilde{\mathbf{x}}_1(\mathbf{x}_t, \mathbf{y})$ with new noise ${\boldsymbol{\epsilon}} \sim p_{\mathbf{X}_0}$. The $N$-time repetition of these steps yields the final reconstruction $\mathbf{x}_1$.
  • Figure 2: Temporal evolution of 2D Flower and comparison with true posterior for noise variance $\sigma_n = 0.25$.
  • Figure 3: Visual comparison for flow-matching inverse solvers.
  • Figure 4: Solution path of Flower for box inpainting.
  • Figure 5: Temporal evolution of 2D Flower and comparison with true posterior for noise variance $\sigma_n = 0.75$.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Theorem 1
  • proof : Proof of Theorem \ref{['theorem:sampling']}
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • proof
  • proof