WHAT DO FLOW-BASED INVERSE SOLVERS APPROXIMATE? A POSTERIOR-TRANSPORT VIEW

This paper investigates the theoretical foundations of training-free, flow-based inverse problem solvers (e.g., FlowDPS, FLOWER, PnP-Flow). While these methods empirically succeed at image restoration, their underlying mechanism—adding a measurement-guidance term to a deterministic probability-flow ODE—lacks a formal characterization. The authors seek to understand what these per-step corrections actually approximate and how they relate to the true Bayesian posterior.

Using a posterior-transport perspective, the authors demonstrate that for deterministic flow priors, Bayesian conditioning is fundamentally a source-space operation. They prove that the true posterior can be obtained by reweighting the source distribution and pushing it through the unmodified velocity field. By framing the trajectory-guidance solvers as approximations of a canonical minimum-kinetic-energy correction field, the authors provide a unifying view of existing methods and derive a Wasserstein distance bound for the resulting posterior bias. They validate these findings using a 2D study with a closed-form posterior and propose a new, principled velocity-correction solver.

• Source Reweighting: Conditioning is realized entirely by reweighting the source distribution; no drift correction is theoretically required for exact posterior sampling.
• Solver Classification: Existing solvers (FlowDPS, FLOWER, PnP-Flow) are identified as distinct, local approximations of the optimal, minimum-kinetic-energy correction field.
• Mode Collapse: Trajectory-guidance solvers inherently struggle with multimodal posteriors, often collapsing mass onto a single ridge regardless of guidance strength, whereas source reweighting preserves the true posterior structure.