The Prior-Force Family of Implicit Denoiser Priors and Its Decisions in Linear Inverse Problems
Abstract
Implicit priors given by Gaussian denoisers successfully regularize a wide range of inverse problems in approaches such as plug-and-play (PnP), regularization by denoising (RED) and score-based methods. Such a prior depends on the noise level, and how it selects among signals consistent with the data remains largely uncharacterized. To put implicit and explicit priors on the same footing, we define the prior-force family Wt=t−1(Dt−1−I) of any Gaussian denoiser Dt at noise variance t>0 (inverse in the graph sense), the normalized denoising residual as an operator on clean signals. At a common effective prior weight, the equilibrium equations differ only in their prior terms: PnP uses Wt, RED/score-based methods the score force (the normalized residual as an operator on noisy inputs), and maximum a posteriori (MAP) estimation the MAP force (the negative log-prior gradient). For the minimum mean-square error (MMSE) denoisers of a prior π, we compare the corresponding energies at three levels. Without observation, raising the noise level favors PnP minima that retain more posterior variance. On signals consistent with the data, the leading-order PnP energy for separated modes is the proximal hull of the MAP energy, which lets PnP select signals between modes; modes of equal data fit are ranked by peak height at small noise levels and by mass at large ones. For recovery, Wt at the truth bounds the local bias, and an oracle noise level balances this bias against denoiser error. These results show that the noise level is a parameter of the prior rather than a step size. Experiments with exact priors support these predictions, including the mixture components that PnP selects on unseen data.
Related
- Multi-split Multiple-sets Implicit Convex Feasibility Seeking As Split Inverse Problems
- Average Energy Dissipation Rates of Implicit-Explicit Runge-Kutta Methods for Gradient Flow Problems
- Outperformance Inverse Optimization: Learning Objective Functions that Outperform Agent Decisions
- Compact implicit high-resolution numerical scheme for multicomponent transport problems with Langmuir sorption
- Inertial Primal-Dual Dynamics Methods Featuring Implicit Hessian-Driven Damping for Convex Optimization Problems in Continuous and Discrete Time
BibTeX
@misc{liang2026priorforce,
title = {{The Prior-Force Family of Implicit Denoiser Priors and Its Decisions in Linear Inverse Problems}},
author = {Weijie Liang and Chuling Wen and Jian Lu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00183},
eprint = {2611.00183},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00183},
}
RIS
TY - UNPB
TI - The Prior-Force Family of Implicit Denoiser Priors and Its Decisions in Linear Inverse Problems
AU - Weijie Liang
AU - Chuling Wen
AU - Jian Lu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00183
UR - https://arxiv.org/abs/2611.00183
AB - Implicit priors given by Gaussian denoisers successfully regularize a wide range of inverse problems in approaches such as plug-and-play (PnP), regularization by denoising (RED) and score-based methods. Such a prior depends on the noise level, and how it selects among signals consistent with the data remains largely uncharacterized. To put implicit and explicit priors on the same footing, we define the prior-force family W_t=t^−1(D_t^−1−I) of any Gaussian denoiser D_t at noise variance t>0 (inverse in the graph sense), the normalized denoising residual as an operator on clean signals. At a common effective prior weight, the equilibrium equations differ only in their prior terms: PnP uses W_t, RED/score-based methods the score force (the normalized residual as an operator on noisy inputs), and maximum a posteriori (MAP) estimation the MAP force (the negative log-prior gradient). For the minimum mean-square error (MMSE) denoisers of a prior π, we compare the corresponding energies at three levels. Without observation, raising the noise level favors PnP minima that retain more posterior variance. On signals consistent with the data, the leading-order PnP energy for separated modes is the proximal hull of the MAP energy, which lets PnP select signals between modes; modes of equal data fit are ranked by peak height at small noise levels and by mass at large ones. For recovery, W_t at the truth bounds the local bias, and an oracle noise level balances this bias against denoiser error. These results show that the noise level is a parameter of the prior rather than a step size. Experiments with exact priors support these predictions, including the mixture components that PnP selects on unseen data.
ER -