Principled MAP estimation for inverse problems: bridging the gap between convergence and performance

Alexandre Lagier; Valentine Tosel; Anne Gagneux; Mathurin Massias; Ségolène Martin

cs.CVSep 29, 2026

Principled MAP estimation for inverse problems: bridging the gap between convergence and performance

Authors: Alexandre Lagier, Valentine Tosel, Anne Gagneux, Mathurin Massias, Ségolène Martin

Organizations: ENS de Lyon, CNRS, Université Claude Bernard Lyon 1, Inria, LIP UMR 5668, 69342 Lyon Cedex 07, France · Univ. Bordeaux, Inria, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France · Inria, ENS de Lyon, CNRS, Université Claude Bernard Lyon 1, LIP UMR 5668, 69342 Lyon Cedex 07, France

Abstract

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.

Figures & tables

Figure 1 : Evolution of the objective value across time of GAMMA, PnP-Flow and Approx-PGD

(1+\eta)

(with

\eta

controlling the number of sub-iterations). PnP-Flow diverges. GAMMA converges faster than Approx-PGD.

Method	Theory	Denoising			Deblurring			Super-res.			Rand. inpaint.			Box inpaint.
		$\sigma=0.2$			$\sigma=0.05$ , $\sigma_{\mathrm{b}}=3.0$			$\sigma=0.05$ , $\times 4$			$\sigma=0.01$ , $70\%$			$\sigma=0.05$ , $80\times 80$
		PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS
Degraded		20.00	0.683	0.148	27.78	0.736	0.129	10.25	0.174	0.819	11.95	0.188	1.032	22.26	0.740	0.212
PnP-Flow	No	32.74	0.915	0.056	34.85	0.940	0.047	32.05	0.913	0.056	34.86	0.962	0.018	32.02	0.946	0.040
Approx-PGD	Yes	26.98	0.718	0.083	23.28	0.691	0.170	15.86	0.369	0.370	16.21	0.391	0.355	6.460	0.501	0.498
GAMMA (noiseless)	Yes	30.19	0.834	0.052	26.80	0.638	0.283	18.78	0.524	0.205	19.95	0.657	0.297	24.02	0.747	0.205

Table 1 : Comparisons of methods on different inverse problems on the CelebA dataset. Results are averaged across 100 test images. Higher PSNR / SSIM is better, lower LPIPS is better.

Panel of Figure 2 — Figure 2 : Qualitative results on AFHQ-256. The full images table is in the Appendix, Figure 5 .

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

CelebA	PnP-Flow
		Denoising	Deblurring	Super-res.	Rand. inpaint.	Box inpaint.
CelebA	$N$ (number of steps)	100	100	100	100	100
	$\alpha$ (learning rate)	0.8	0.01	0.3	0.01	0.5
	Approx-PGD
	$\gamma$ (step size)	1.0	1.0	1.0	1.0	1.0
	$\kappa$ (noise level factor)	1.0	0.1	0.5	2.0	0.5

Appendix

Table 2 : Hyper-parameters used and GAMMA on CelebA and AFHQ-Cat datasets.

CelebA	GAMMA
		Denoising	Deblurring	Super-res.	Rand. inpaint.	Box inpaint.
	$N$ (number of steps)	100	100	500	500	100
	$\kappa$	1.0	1.0	2.0	1.5	1.2
	$\lambda_{0}$	$10^{-3}$	$10^{-3}$	$10^{-3}$	$10^{-3}$	$10^{-3}$
	$\beta$	1.0	1.0	1.0	1.0	1.0
AFHQ-Cat	GAMMA

Appendix

Table 3 : Hyperparameters used for GAMMA on the CelebA and AFHQ-Cat datasets.

Figure 3 : Ablation of the number

N_{\epsilon}

of noise samples

\epsilon

used to approximate

\mathbb{E}_{\epsilon}[\mathrm{MMSE}_{\sigma}(x+\sigma\epsilon)]

- AFHQ - Box inpainting

Figure 4 : The two schedules

\sigma_{k}=\frac{\sigma_{0}}{(k+1)^{\gamma/2}}

used in our experiments. Setting

N=2000

across all experiments ensures that

\sigma_{k}

is sufficiently close to

0

Method	Theory	Denoising			Deblurring			Super-res.			Rand. inpaint.			Box inpaint.
		$\sigma=0.2$			$\sigma=0.05$ , $\sigma_{\mathrm{b}}=3.0$			$\sigma=0.05$ , $\times 4$			$\sigma=0.01$ , $70\%$			$\sigma=0.05$ , $80\times 80$
		PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS
Degraded		20.00	0.292	0.550	24.43	0.531	0.452	11.77	0.220	0.877	13.44	0.221	1.090	21.68	0.726	0.219
PnP-Flow	No	32.27	0.876	0.164	29.48	0.795	0.317	29.02	0.811	0.171	34.97	0.939	0.037	28.63	0.911	0.107
Approx-PGD	Yes	28.21	0.679	0.236	18.42	0.464	0.526	7.08	0.059	0.690	18.36	0.404	0.510	17.86	0.717	0.354
GAMMA (noiseless)	Yes	30.03	0.751	0.102	14.23	0.311	0.613	26.60	0.703	0.361	22.09	0.593	0.326	21.91	0.603	0.277

Appendix

Table 4 : Comparisons of methods on different inverse problems on the AFHQ dataset. Results are averaged across 100 test images. Higher PSNR / SSIM is better, lower LPIPS is better.

Panel of Figure 5 — Figure 5 : Qualitative results on AFHQ-256.

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Flow Map Denoisers: Traversing the Distortion-Perception Plane for Inverse Problems

Nicolas Zilberstein, Morteza Mardani, Santiago Segarra

Rice University · NVIDIA Inc.

Iterative Denoising Methods Image Restoration Flow Map Inverse Problem Distortion Perceptual Quality

Open paper AskPDF arXiv

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.

Abstract

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