P-Flow: Proxy-gradient Flows for Linear Inverse Problems

TL;DR

P-Flow introduces a proxy-gradient framework for linear inverse problems that stabilizes reconstruction, reduces computational costs, and maintains competitive performance under challenging conditions.

Contribution

The paper proposes P-Flow, a novel method using proxy gradients and Gaussian spherical projection to improve stability and efficiency in flow-based inverse problem solutions.

Findings

P-Flow achieves stable reconstructions without long-chain differentiation.

The method performs well under severe ill-posed conditions and high noise.

Theoretical analysis supports the stability and effectiveness of P-Flow.

Abstract

Generative models based on flow matching have emerged as a powerful paradigm for inverse problems, offering straighter trajectories and faster sampling compared to diffusion models. However, existing approaches often necessitate differentiating through unrolled paths, leading to numerical instability and prohibitive computational overhead. To address this, we propose P-Flow, a framework that stabilizes the reconstruction process by leveraging a proxy gradient to update the source point. This approach effectively circumvents the numerical instability and memory overhead of long-chain differentiation. To ensure consistency with the prior distribution, we employ a Gaussian spherical projection motivated by the concentration of measure phenomenon in high-dimensional spaces. We further provide a theoretical analysis for P-Flow based on Bayesian theory and Lipschitz continuity. Experiments across diverse restoration tasks demonstrate that P-Flow delivers competitive performance, especially under extreme degradations such as severely ill-posed conditions and high measurement noise.