FlowADMM: Plug-and-play ADMM with Flow-based Renoise-Denoise Priors

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.

Image restoration faces a fundamental tradeoff: methods that minimize error produce blurry reconstructions, while those that maximize perceptual quality yield sharp but less faithful images. Existing approaches either commit to a single operating point on this distortion perception (DP) frontier or require paired-data supervision, auxiliary models, or hyperparameter tuning of the sampler to access different points. We show that flow map models, a recent extension of flow matching for few-step sampling that learns an average field, implicitly define a one-parameter family of denoisers that continuously spans the DP frontier. The lookahead parameter t acts as a control knob between the MMSE and perceptual regimes. For Gaussian targets, we prove that varying t exactly recovers the optimal DP frontier; for natural images, we observe similar behavior empirically. Within a Plug-and-Play solver, the same mechanism extends to general inverse problems, where it controls a tradeoff between perceptual alignment and data consistency. Despite the lack of exact optimality guarantees in this setting, a single trained flow map spans the DP tradeoff, matching or exceeding specialized baselines at both extremes. Extensive experiments on CelebA ($128\times 128$) and AFHQ ($256\times 256$) across several linear and nonlinear inverse tasks validate our findings.

Plug-and-Play (PnP) methods have become standard tools for solving imaging inverse problems by replacing the intractable maximum a posteriori (MAP) denoiser with the MMSE one. While this mismatch has been widely treated as unavoidable, recent works have sought to close this gap by targeting the MAP with diffusion-model scores. We show this is problematic in practice: learned scores do not match the true ones, so MAP-targeting iterations converge to cartoon-like images rather than realistic ones, and better results are obtained by stopping short of convergence. We turn this observation into a design principle and introduce ProxiMAP, an iterative MAP approximation whose noise schedule keeps the iterate's residual noise matched to the denoiser's training noise. This keeps the denoiser in-distribution where its score is reliable, and yields implicit early stopping that avoids the failure mode above. ProxiMAP is a modular drop-in replacement for MMSE denoisers in standard PnP algorithms and consistently sharpens reconstructions across deblurring, inpainting, super-resolution, and phase retrieval. Building on the same principle, we propose a hybrid variant that applies ProxiMAP only in the late iterations of PnP, where the denoiser is most reliable -- matching or exceeding the full-replacement variant at a fraction of the cost.