FLOWER: A Flow-Matching Solver for Inverse Problems

2025-09-30

AI Summary

Generating data-consistent reconstructions for inverse problems using pretrained flow models remains challenging. Method: This paper proposes FLOWER, a general Bayesian posterior sampling framework based on flow matching. It unifies flow matching with variational inference via flow consistency estimation, feasibility projection constrained by the forward operator, and iterative time-step optimization. Contribution/Results: FLOWER establishes the first theoretical connection between flow matching and Bayesian posterior sampling and integrates the plug-and-play (PnP) paradigm into a unified generative solver framework. It achieves state-of-the-art reconstruction quality across diverse inverse problems—including computed tomography (CT), magnetic resonance imaging (MRI), and phase retrieval—while exhibiting strong hyperparameter robustness. Crucially, FLOWER enables cross-task transfer using only a pretrained flow model, eliminating the need for task-specific fine-tuning.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Solving inverse problems using flow-matching methods
• Producing consistent reconstructions from observed measurements
• Unifying Bayesian sampling with generative inverse solvers

Innovation

• Flow-matching solver for inverse problems
• Iterative three-step reconstruction procedure
• Unifies plug-and-play and generative methods