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arxiv: 2605.16396 · v1 · pith:EKNED7MYnew · submitted 2026-05-12 · 💻 cs.CV · cs.LG

Beyond MMSE: Enhancing PnP Restoration with ProxiMAP

Kenta Vert , Giacomo Meanti , Scott Pesme , Michael Arbel , Julien Mairal This is my paper

Pith reviewed 2026-05-20 22:33 UTC · model grok-4.3

classification 💻 cs.CV cs.LG
keywords Plug-and-PlayImage restorationDenoiserMAP estimationDiffusion modelsPnP algorithmsNoise schedule
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The pith

ProxiMAP improves PnP restoration by scheduling noise so residual matches the denoiser's training level, avoiding cartoon artifacts from direct MAP targeting.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard PnP replaces the MAP denoiser with an MMSE one, but attempts to close the gap with diffusion scores produce cartoon-like images at convergence because learned scores deviate from true ones. The paper turns the observation that early stopping works better into a design rule: ProxiMAP runs an iterative MAP approximation whose noise schedule keeps the current iterate's residual noise at the exact level the denoiser saw during training. This keeps the denoiser in-distribution where its score is reliable and supplies implicit early stopping. The resulting modular replacement sharpens results on deblurring, inpainting, super-resolution and phase retrieval, and a hybrid that switches to ProxiMAP only late in the process matches full performance at lower cost.

Core claim

ProxiMAP is an iterative MAP approximation whose noise schedule is chosen so the residual noise in the current iterate exactly matches the noise level on which the denoiser was trained. Because the denoiser therefore stays in-distribution, its score remains reliable, and the iteration automatically stops short of the cartoon-like fixed point that direct MAP targeting reaches.

What carries the argument

ProxiMAP noise schedule that matches iterate residual noise to the denoiser's training distribution

If this is right

  • ProxiMAP serves as a drop-in replacement for MMSE denoisers inside existing PnP algorithms.
  • The method yields sharper reconstructions on deblurring, inpainting, super-resolution and phase retrieval.
  • A hybrid schedule that activates ProxiMAP only in late iterations matches or exceeds full replacement at reduced cost.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same noise-matching principle could be tested in other score-based inverse-problem solvers that currently rely on early stopping heuristics.
  • If the denoiser training distribution is known, the schedule can be derived without task-specific tuning, suggesting a parameter-light extension to new imaging modalities.
  • The implicit early stopping may reduce the need for separate convergence diagnostics in deployed PnP pipelines.

Load-bearing premise

Matching residual noise to the training distribution is enough to keep the denoiser reliable and prevent cartoon-like convergence.

What would settle it

If ProxiMAP iterations still converge to visibly cartoon-like images when run to full convergence, or if they produce no consistent improvement over standard MMSE PnP on the reported tasks, the central claim is falsified.

Figures

Figures reproduced from arXiv: 2605.16396 by Giacomo Meanti, Julien Mairal, Kenta Vert, Michael Arbel, Scott Pesme.

Figure 1
Figure 1. Figure 1: Conditional MMSE estimates (left) versus [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The MAP approximation procedure of Eq. (5) with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Iterates of Eq. (5) with the score from a diffusion model (left) or from a GMM (right). [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (left) PSNR and LPIPS (normalized so the MMSE estimator’s LPIPS is 1) across noise levels and step counts for ProxiMAP and three conditional samplers on ImageNet. ProxiMAP dominates the others by remaining closest to the (high-PSNR, low-LPIPS) corner. (right) Outputs of ProxiMAP for increasing K (the MMSE estimator is the K = 1 case). Results on FFHQ are in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of image restoration from ImageNet illustrating the detail enhancing effect of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Three PnP algorithms versus their ProxiMAP counterparts on four inverse problems. Lighter colors are baselines; darker colors are ProxiMAP variants. ProxiMAP shifts the Pareto fronts toward the optimal (high-PSNR, low-LPIPS) corner across nearly all settings. to only 7 additional calls to the denoiser. DAPS’s inner loop already runs an iterative sampling procedure, which we replace by 6 ProxiMAP iterations… view at source ↗
Figure 7
Figure 7. Figure 7: Comparing the conditional performance of a memorizing and a generalizing diffusion [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FFHQ Gaussian denoising results. LPIPS is normalized such that the MMSE denoiser has [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Baseline algorithms versus their full-replacement [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Trade-off between precision and computation cost for many hybrid variants of [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Same as Fig. 10, but on [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Qualitative results of DAPS on random inpainting. The [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example of gaussian deblur using DPIR on FFHQ. Some leftover residual noise is removed [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Qualitative visualization for different switch times on motion blur. Panel titles indicate [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Super resolution using DiffPIR. Here the full ProxiMAP variant seems to work better than the others (as predicted by [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Failure cases for ProxiMAP in DPIR on the super-resolution task. High-frequency artifacts appear along the edges (right column), but are removed when we use the Fast ProxiMAP variant. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes ProxiMAP as a modular drop-in replacement for MMSE denoisers in Plug-and-Play (PnP) algorithms for imaging inverse problems. It observes that direct MAP targeting with learned diffusion scores produces cartoon-like artifacts because the scores do not match the true posterior, and better results come from early stopping. ProxiMAP implements an iterative MAP approximation whose per-iteration noise schedule is chosen so that the residual noise in the current iterate matches the denoiser's training noise level; this is claimed to keep the denoiser in-distribution, yield reliable scores, and supply implicit early stopping. A hybrid variant applies the schedule only in late iterations. Improvements are reported on deblurring, inpainting, super-resolution, and phase retrieval.

Significance. If the central construction holds, the work supplies a practical design principle for noise scheduling that narrows the MMSE-MAP gap without requiring new training or heavy computation. The hybrid variant further improves efficiency while preserving gains. The explicit link between residual-noise matching and avoidance of the cartoon attractor is a concrete, testable contribution to the PnP and diffusion-based restoration literature.

major comments (2)
  1. The central claim (abstract and §3) that matching residual noise variance to the training distribution is sufficient to keep the denoiser reliably in-distribution rests on an assumption whose load-bearing status is not fully isolated. The skeptic analysis correctly notes that structured artifacts or correlations in the current estimate may still shift the effective input distribution even at matched noise level; the manuscript should supply an ablation that varies iterate structure while holding noise level fixed, or quantitative diagnostics (e.g., distribution-distance metrics between iterates and training patches) to rule out this alternative explanation.
  2. §4 (experimental section): the reported gains are described as consistent across four tasks, yet the strength of evidence for the noise-matching principle versus simple early-stopping is not quantified. A direct comparison of ProxiMAP against an MMSE baseline stopped at the same effective iteration count (or same residual norm) would clarify whether the schedule itself, rather than the implicit stopping, drives the improvement.
minor comments (2)
  1. Notation for the noise schedule parameter (e.g., how σ_t is chosen from the training schedule) should be made fully explicit in the algorithm box or pseudocode so that the method is immediately reproducible.
  2. Figure captions for the qualitative results could include the exact iteration at which each method is shown, to make the early-stopping effect visually verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments help clarify how to strengthen the isolation of the noise-matching principle and the comparison to early stopping. We address each major comment below and commit to revisions where appropriate.

read point-by-point responses

  1. Referee: The central claim (abstract and §3) that matching residual noise variance to the training distribution is sufficient to keep the denoiser reliably in-distribution rests on an assumption whose load-bearing status is not fully isolated. The skeptic analysis correctly notes that structured artifacts or correlations in the current estimate may still shift the effective input distribution even at matched noise level; the manuscript should supply an ablation that varies iterate structure while holding noise level fixed, or quantitative diagnostics (e.g., distribution-distance metrics between iterates and training patches) to rule out this alternative explanation.

    Authors: We agree that the current experiments do not fully isolate noise-level matching from possible distribution shifts induced by structured artifacts in the iterate. To address this, the revised manuscript will include a new ablation that holds the residual noise variance fixed while varying iterate structure (e.g., by injecting iterates from alternative PnP runs or different initializations). We will also report quantitative diagnostics such as patch-wise Wasserstein distance and FID between the effective input distribution at each step and the denoiser training distribution. These additions will directly test whether noise matching alone suffices to keep the denoiser in-distribution. revision: yes

  2. Referee: §4 (experimental section): the reported gains are described as consistent across four tasks, yet the strength of evidence for the noise-matching principle versus simple early-stopping is not quantified. A direct comparison of ProxiMAP against an MMSE baseline stopped at the same effective iteration count (or same residual norm) would clarify whether the schedule itself, rather than the implicit stopping, drives the improvement.

    Authors: We acknowledge that the present results do not yet quantify the incremental benefit of the ProxiMAP schedule over an MMSE baseline that is simply halted at a comparable iteration count or residual norm. In the revised version we will add this controlled comparison for all four tasks. The MMSE baseline will be terminated either after the same number of iterations used by ProxiMAP or when its residual norm matches the final residual norm observed under ProxiMAP; the resulting PSNR/SSIM and perceptual metrics will be reported side-by-side. This will make explicit whether the noise schedule contributes beyond implicit early stopping. revision: yes

Circularity Check

0 steps flagged

ProxiMAP derivation introduces an independent scheduling principle with no reduction to fitted inputs or self-citations

full rationale

The paper's central construction starts from the external empirical observation that direct MAP targeting with learned diffusion scores produces cartoon-like artifacts and that early stopping improves results. It then proposes a noise schedule that matches residual noise variance to the denoiser's training distribution as a design principle for keeping the denoiser in-distribution and providing implicit early stopping. This step does not reduce to any self-definitional equation, does not rename a fitted parameter as a prediction, and does not rely on load-bearing self-citations or uniqueness theorems from prior author work. The derivation remains self-contained against the stated observations and is validated empirically across restoration tasks rather than being forced by its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5771 in / 1130 out tokens · 45416 ms · 2026-05-20T22:33:26.273249+00:00 · methodology

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Reference graph

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