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

Weijie Liang; Chuling Wen; Jian Lu

stat.MLOct 30, 2026

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

Authors: Weijie Liang, Chuling Wen, Jian Lu

Organizations: Shenzhen Key Laboratory of Advanced Machine Learning and Applications, School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, China · National Center for Applied Mathematics Shenzhen (NCAMS), Shenzhen, 518055, China

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 $W_t=t^{-1}(D_t^{-1}-I)$ 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.

Figures & tables

method	prior term	evaluated at	energy (Bayes denoiser)	$t\to 0$
MAP	$\nabla g$	—	$g$	—
PnP family	$W_{t}$	denoised output	$R_{t}$ , curvature $\Sigma_{t}^{-1}-t^{-1}I$	$\nabla g$
RED / score	$\nabla E_{t}$	noisy input	$E_{t}$ , Moreau envelope of $R_{t}$	$\nabla g$

Table 1: Three forces for one prior. For a Bayes denoiser of a smooth prior, all three agree as

t\to 0

Figure 1: A Bayes denoiser whose prior force selects an improbable reconstruction ( Example 4.4 ). Prior

\pi=\tfrac{1}{2}\mathcal{N}(-(1,1),0.05\,I)+\tfrac{1}{2}\mathcal{N}((1,1),0.05\,I)

, measurement of the first coordinate, datum

0

. (a)–(c) Energies along the fiber

\{x_{1}=0\}

: the prior energy

g=-\log\pi

, the smoothed energy

E_{t}=-\log p_{t}

and the induced energy

R_{t}

with

\nabla R_{t}=W_{t}

, each minus its fiber minimum. (d) Global fiber minimizers against the noise variance

t

(logarithmic axis). Dots mark all local minimizers.

R_{t}

selects the midpoint, where the prior density is

9.1\cdot 10^{-5}

times its fiber maximum, for every

t\geq t_{R}=0.0452

. The prior energy

g

selects points near the modes. The smoothed energy

E_{t}

selects points away from the midpoint up to

t_{E}=0.95

result	statement	status	here
variance-ordering law	gap derivative $-(V_{t}(u_{1})-V_{t}(u_{2}))/(2t^{2})$ along branches, on fibers	new; pointwise [ 10 ]	Theorem 3.2
fiber stability	Schur-complement criterion $\Sigma_{K\mid N}\prec tI_{K}$	new	Theorem 4.3
proximal hull	$R_{t}=h_{t}(G_{\lambda})+o(\lambda^{2})$ ; core or bridge	new (statement)	Theorem 4.7
proximal hull, mixtures	$\lvert R_{t}-h_{t}(-\log\pi)\rvert\leq C$ , every separation	new	Theorem 4.7
height against mass	PnP score $=$ MAP score $+$ penalty on narrow modes	new	Theorems 4.10 and 4.11
selection bias	bounds by $P_{K}W_{t}(x^{*})$ , local and statistical	new	Theorems 5.1 and 5.2

Table 2: Results of this paper. “New (statement)”: the statement is new, but it can be assembled formally from known ingredients, which are cited next to it.

concept in the literature	object here	source	here
MMSE denoiser is proximal	$D_{t}=\operatorname{prox}_{tR_{t}}$	[ 9 , 13 ]	Proposition 2.2
denoiser as resolvent of a monotone operator	$D_{t}=(I+tW_{t})^{-1}$	[ 15 ]	Section 2.2.1
PnP and RED as monotone inclusions	( 4 ), ( 5 )	[ 16 ]	Proposition 2.1
fixed points of PnP-ADMM, DRS, consensus equilibrium	( 4 ), effective $\alpha$	[ 45 , 4 ]	Proposition 2.1
non-variational regularization	( 4 )	[ 20 ]	Proposition 2.1
upper Moreau envelope, weak convexity	$E_{t}$ Moreau envelope of $R_{t}$	[ 14 ]	Proposition 2.2

Table 3: Known structure used in this paper: concepts of the literature and the corresponding objects here.

statement	tested against	result
$\nabla D_{t}=\Sigma_{t}/t$ , ( 9 )	finite differences	$1.9\cdot 10^{-10}$
$\operatorname{Lip}D_{t}=R^{2}/t$ , attained at $0$ ( Remark 3.3 )	two-point prior, $t<R^{2}$	$\operatorname{Lip}D_{t}=2.0864=R^{2}/t$
rigidity ( Proposition 2.5 (ii))	drift of $W_{t}$ in $t$	$\leq 1.5\cdot 10^{-13}$ Gaussian; $0.3639$ otherwise
small-scale expansion ( 20 )	halving $\sigma=\sqrt{t}$	error ratios $16.2$ , $14.8$ (predicted $16$ )
$\operatorname{Fix}D_{t}=\operatorname{Crit}p_{t}$ ( Proposition 2.4 (i))	$24$ critical points	residual $1.7\cdot 10^{-13}$
congruence ( 19 ), equal inertia	finite-difference Hessians	$1.5\cdot 10^{-7}$ ; inertia equal at all $24$

Table 4: Structural identities against exact values. Relative errors unless stated.

Figure 2: Decisions and preferences along the family; all horizontal axes are the noise variance

t

or functions of it. (a) Separation of the fiber decisions of

R_{t}

E_{t}

and

g

\{x_{1}=0.4\}

for the mixture with weights

0.6

0.4

, means

(0,0)

(1.5,1)

and covariances

\bigl(\begin{smallmatrix}1&0.3\\ 0.3&0.6\end{smallmatrix}\bigr)

\bigl(\begin{smallmatrix}0.5&-0.2\\ -0.2&0.8\end{smallmatrix}\bigr)

, against the first-order predictions ( 29 )–( 30 ) (lines, slope one); axis

t

. (b) Energy gap between two branches of minimizers of

R_{t}

against

t

, without observation (one-dimensional mixture with weights

0.8

0.2

and variances

1

0.01

) and on the fiber

\{x_{1}=0.2\}

of the mixture with weights

0.8

0.2

, means

(0,-2)

(0.3,2)

and covariances

\bigl(\begin{smallmatrix}0.5&0.3\\ 0.3&1\end{smallmatrix}\bigr)

\bigl(\begin{smallmatrix}0.5&0.1\\ 0.1&0.04\end{smallmatrix}\bigr)

; each switches once. (c) Measured

t

-derivative of the gap against the variance law ( 22 ),

40

values of

t

each; the line is the identity.

Figure 3: Height against mass. Columns: truth, data (blue rows unobserved), MAP of $\pi$ , and the score and PnP decisions at noise variance $t=0.09$ . Top: the decisions agree in the dominant posterior component. Below: the PnP decision has a different dominant component from the narrow MAP peak, while the score decision stays in the component of the peak. $H$ is the entropy of the posterior component probabilities.

condition	instances	scores	trivial	departures predicted	false alarms
new images	$40$	$95.0\,\%$	$73.3\,\%$	$32/32$	$4.5\,\%$
top rows observed	$30$	$97.8\,\%$	$81.1\,\%$	$17/17$	$2.7\,\%$
noise $0.05$	$30$	$96.7\,\%$	$80.0\,\%$	$18/18$	$2.8\,\%$
all	$100$	$96.3\,\%$	$77.7\,\%$	$67/67$	$3.4\,\%$

Table 5: Prediction of the PnP decision by the scores of Theorem 4.10 on

100

unseen instances (

300

cases), decisions recomputed from all component candidates. “Trivial”: the decision stays in the component of the MAP. A departure is a case in which the PnP decision lies in a different component from the MAP decision; “departures predicted” counts the departures that the scores flag, and “false alarms” is the fraction of cases without a departure in which the scores flag one.

Figure 4: Recovery and identification. (a) Error of Theorem 5.6 against

\delta

, with

t=\alpha=\delta

(so the axis is also the noise variance

t

), for a prior whose Hessian at the truth is

\operatorname{diag}(-1,1)

: linear in

\delta

, with quadratic expansion remainder. (b) One scale does not identify the prior stably: priors at total-variation distance bounded away from zero with exponentially close denoisers and prior forces ( Example 2.6 ); axis: oscillation frequency

k

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Pretrained denoisers provide a powerful way to incorporate image priors into restoration algorithms. Plug-and-Play and RED approaches exploit fixed-noise-level denoisers within first-order optimization schemes, with convergence guarantees, but often struggle to achieve high-quality reconstruction on severely ill-posed inverse problems. In contrast, recent state-of-the-art approaches leverage denoisers derived from flow- or diffusion-based generative models and evaluate them along a sequence of decreasing noise levels. While these methods achieve strong empirical performance, their convergence theory remains limited. In this paper, we bridge this gap by specifically designing an algorithm that combines denoisers at decreasing noise levels with a schedule tailored to ensure convergence. From a Bayesian perspective, we prove that our method converges to a

\textit{Maximum a Posteriori}

(MAP) estimate, under suitable assumptions. Subsequently, we apply our method to various ill-posed inverse problems and show that it surpasses convergent methods while competing with state-of-the-art empirical ones.

Abstract

Linear Inverse Problems

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