The Prior-Force Family of Implicit Denoiser Priors and Its Decisions in Linear Inverse Problems

Weijie Liang, Chuling Wen, Jian Lu

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…