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REVIEW 4 major objections 4 minor 33 references

Equivariant Denoisers for Image Restoration

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that a single π-equivariant plug-and-play algorithm, ERED, encompasses existing equivariant and stochastic schemes, and that its iterates converge almost surely on bounded trajectories, with bias bounded by the…

desk verdict A useful unifying framework for equivariant PnP, but the convergence proofs as written contain a sign error that makes them analyze a different update than Algorithm 1; likely fixable, but it must be fixed. read the letter →

arxiv 2412.05343 v2 pith:MZVDJOQA submitted 2024-12-06 eess.IV cs.CVstat.ML

Marien Renaud , Arthur Leclaire , Nicolas Papadakis This is my paper · ORCID
classification eess.IVcs.CVstat.ML MSC 68U1065K1090C1594A08
keywords imagerestorationplug-and-playequivarianceregularizationbydenoisingstochasticgradientdescentconvergenceanalysisTweedieformuladeepdenoiser
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes ERED, a stochastic plug-and-play algorithm that replaces a denoiser by its average over a group of transformations and treats that average as the score of a symmetric prior. The authors prove that when the exact MMSE denoiser is used and the iterates happen to stay inside a fixed compact set, the algorithm converges almost surely to critical points of the surrogate objective $F^{\pi}_{\sigma} = f + \lambda r^{\pi}_{\sigma}$. With an inexact, learned denoiser, the asymptotic gradient norm and the spread of objective values are bounded by a bias term that is controlled by the denoiser's error on the transformed images. They also show that as the denoising level goes to zero, critical points of the surrogate converge to critical points of the original variational problem whenever the prior is $\pi$-equivariant. Why it matters: this gives one theoretical template for stochastic and equivariant plug-and-play methods, so a user can choose transformations with some guarantee about where the algorithm is heading.

What carries the argument

The load-bearing object is the $\pi$-equivariant score identity built from Tweedie's formula. For the MMSE denoiser $D^*_{\sigma}$, $-\nabla \log p_{\sigma}(x) = (x-D^*_{\sigma}(x))/\sigma^2$; the paper defines $s^{\pi}_{\sigma}(x) = E_{G\sim\pi}[J_G^T(x)(G(x)-D_{\sigma}(G(x)))]/\sigma^2$ and the regularizer $r^{\pi}_{\sigma}$. The convergence argument rests on a theorem for biased stochastic gradient descent (recalled as Theorem 1 from [7]), which yields almost-sure convergence and bias bounds on the event $\Lambda_K$ where all iterates remain in a compact set. The critical-point transfer at $\sigma\to0$ rests on the identity $s = E_{G\sim\pi}[J_G^T(s\circ G)]$ for $\pi$-equivariant priors, which lets score convergence on the orbit ball become gradient convergence of the original objective.

What would settle it

Run Algorithm 1 with the exact MMSE denoiser on a $\pi$-equivariant prior with unbounded support and a data term with no coercivity, using a step sequence satisfying Assumption 1; if the iterates escape every compact set with positive probability, the event $\Lambda_K$ fails and the almost-sure claims do not apply to such runs, which would settle exactly how conditional the theorem is.

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Extended reading notes

Core claim

The central discovery is a unification: equivariant PnP and stochastic denoising regularization are not separate heuristics but instances of one construction, the $\pi$-equivariant denoiser $\tilde D_{\sigma}(x)=E_{G\sim\pi}[J_G^T(x)D_{\sigma}(G(x))]$, with the corresponding regularizer $r^{\pi}_{\sigma}$. Algorithm 1 is a biased stochastic gradient descent on $F^{\pi}_{\sigma}$. Under the paper's assumptions, Proposition 2 shows that with the exact MMSE denoiser the iterates converge almost surely on $\Lambda_K$ to the critical set of $F^{\pi}_{\sigma}$; Proposition 3 shows that with an inexact denoiser, $\limsup \|\nabla F^{\pi}_{\sigma}(x_k)\|$ and the objective spread are bounded by the asymptotic bias $\eta$, which in turn is bounded by $\lambda \sigma^{-2} E(\|J_G\|)\,\|D_{\sigma}-D^*_{\sigma}\|_{\infty,L}$ on the orbit ball $L$. Proposition 5 closes the loop: for a $\pi$-equivariant prior, every limit point of critical points of the surrogate as $\sigma\to0$ is a critical point of the original objective $F=f+\lambda r$.

Load-bearing premise

The whole convergence and bias theory is conditional on the event that the algorithm's own iterates never leave a fixed compact set, and the paper gives no condition that guarantees this event has positive probability.

Editorial extensions

If this is right

  • The same convergence theory now covers equivariant PnP, translation-based noising (SNORE), and other stochastic schemes as instances of ERED, so guarantees proven for one apply to the others.
  • For a fixed denoiser, the bias bound says that the asymptotic error of ERED is governed by how well the denoiser approximates the MMSE denoiser on the ball containing all transformed iterates, identifying where denoiser training effort should go.
  • When the prior is $\pi$-equivariant, the surrogate problem solved by ERED is not an unrelated objective: its critical points approach those of the original variational problem as $\sigma\to0$.
  • Choice of transformation set is not neutral: the experiments indicate flips and rotations improve deblurring and despeckling slightly, while subpixel rotation and combining all transformations do not, suggesting the image prior is only equivariant to certain groups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The boundedness event $\Lambda_K$ is the main gap; proving it from coercivity of $f$ or boundedness of the denoiser would turn the almost-sure-on-$\Lambda_K$ statements into unconditional almost-sure theorems.
  • One testable prediction of the theory is that the residual gradient norm of ERED should track the denoiser error on the orbit ball; measuring $\|D_{\sigma}-D^*_{\sigma}\|_{\infty,L}$ along a run and comparing with (21) would validate or falsify the bias model.
  • The framework suggests a practical diagnostic for symmetry: run ERED with different groups $\pi$ and compare restoration quality; a group that fails to improve indicates the image class is not $\pi$-equivariant, as the subpixel-rotation results suggest.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes Equivariant Regularization by Denoising (ERED), a plug-and-play image restoration framework in which the prior is encoded through an equivariant denoiser and a stochastic gradient update over a transformation group. The authors introduce a π-equivariant regularization term r^π_σ, relate it to previous equivariant PnP methods and to their own SNORE method, and analyze convergence of the resulting stochastic algorithm. They prove convergence to critical points of F^π_σ = f + λ r^π_σ when the exact MMSE denoiser is used (Proposition 2), give bias bounds for an inexact denoiser (Proposition 3), and study the behavior of critical points as the denoiser noise level σ tends to zero for π-equivariant priors (Propositions 4 and 5). Numerical experiments on deblurring, super-resolution, and despeckling show modest but consistent gains over RED.

Significance. The proposed framework is conceptually appealing: it unifies equivariant PnP, stochastic PnP, and SNORE under a single formalism, and it targets a real gap, namely the lack of convergence guarantees for equivariant stochastic PnP methods. The use of the Doucet-Tadic theorem is appropriate for biased stochastic gradient descent, and the paper explicitly states its assumptions and provides complete proofs in the appendix. The experimental section is reasonably broad, covering multiple restoration tasks and denoisers. However, the theoretical core as written contains sign inconsistencies that make Propositions 2 and 4 incorrect as stated relative to the definitions and algorithm used. These issues are local and repairable, but they must be fixed before the convergence claims can be accepted.

major comments (4)
  1. [Section 3.2, Eqs. (7)-(8); Section 4.3, Eq. (25) and Proposition 4] The definition of s^π_σ is inconsistent with the proof of Proposition 4. Equation (7) defines s^π_σ(x) = -E[J_G^T(x)∇log p_σ(G(x))], which is the gradient of r^π_σ, not the score s = ∇log p. For a π-equivariant prior, s = +E[J_G^T (s∘G)], so as σ→0 one gets s^π_σ → -s. Proposition 4 claims ∥s - s^π_σ∥_{∞,K} → 0, which is generally false under definition (7). The proof's Eq. (25) instead uses s^π_σ = +E[J_G^T ∇log p_σ(G)], i.e. the opposite sign. This affects Proposition 5 as well, since its proof relies on Proposition 4 to conclude ∥F - F^π_σ∥_{∞,K} → 0. The authors should adopt one consistent sign convention throughout and adjust Eqs. (7), (8), (25), and Proposition 4 accordingly.
  2. [Section 4.1, Eq. (15) and proof of Proposition 2] There is a sign mismatch between Algorithm 1 and the update analyzed in Proposition 2. With the exact MMSE denoiser, Tweedie's formula gives (G(x)-D*_σ(G(x)))/σ^2 = -∇log p_σ(G(x)); substituting into Algorithm 1 line 6 yields x_{k+1} = x_k - δ∇f(x_k) + δλ J_G^T(x_k)∇log p_σ(G(x_k)). Equation (15), however, states x_{k+1} = x_k - δ_k∇f(x_k) - λδ_k J_G^T(x_k)∇log p_σ(G(x_k)), with the opposite sign. The subsequent definition of ξ_k in the proof of Proposition 2 is consistent with Eq. (15) but not with Algorithm 1. As written, Proposition 2 proves convergence for an update that maximizes the prior log-likelihood rather than for the proposed ERED algorithm. This is fixable by flipping the sign in Eq. (15) and in the definition of ξ_k, but the current proof does not establish the stated result.
  3. [Section 4.1, definition of Λ_K and Propositions 2-3] The convergence results are stated 'almost surely on Λ_K', where Λ_K is the set of random seeds for which all iterates stay in a fixed compact K. The paper calls this the boundedness assumption but gives no conditions on f, G, or the denoiser that ensure Λ_K has positive probability, nor any argument that trajectories remain bounded under Assumptions 1-4. Thus the central guarantees are conditional on an unverified property of the algorithm's own dynamics. Please either provide sufficient conditions for almost-sure boundedness, or explicitly state and discuss this as a limitation that must be assumed separately.
  4. [Section 3.2, Proposition 1] Proposition 1 states that r^π_σ is π-equivariant, but 'π-equivariant' is defined only for densities (Definition 2), and r^π_σ is not a density. The proof also takes logarithms of r^π_σ, which is not guaranteed to be positive. Please reformulate the property that is actually being proved for the regularizer and justify the proof under the stated assumptions on G and π.
minor comments (4)
  1. [Section 1] The sentence 'As shown in [16], natural images densities tend to be invariant to some set of transformations such as rotations or flips have been studied in.' is incomplete and should be rewritten.
  2. [Section 5] The text refers to 'Table 5' when presenting results that appear in Table 1; also, 'PNSR' is used instead of 'PSNR' in several table headers.
  3. [Section 3.3, Eq. (12)] The notation for the noising-denoising example uses σ both for the translation magnitude and for the denoiser noise level; this is potentially confusing and should be clarified.
  4. [Appendix B.2-B.3] The constants appearing in the proofs of Lemmas 1 and 2 are introduced with inconsistent notations (C vs C2, and subscripted constants); a pass for notational consistency would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence and critical-point results are derived from stated assumptions and an external theorem, not from their own conclusions.

full rationale

The paper's theoretical chain is not circular. The π-equivariant regularizer r^π_σ and score s^π_σ are defined from first principles (Eqs. 6-8) using the Tweedie identity, and Proposition 1 is proved from Haar-measure invariance. Propositions 2 and 3 are obtained by checking the hypotheses of an external stochastic-approximation theorem (Doucet and Tadic 2017, Theorem 2.1), with the noise fluctuation estimates (Lemmas 1-2) proved from Assumptions 1-4; the bias bound (23) follows from the definition of the estimator bias η, not from a fitted constant. Proposition 4 uses the π-equivariance identity of Remark 1 and an external regularity result ([15]), and Proposition 5 is a standard Kuratowski limit argument. The boundedness event Λ_K is a standard conditional event in stochastic approximation, not a self-referential definition of convergence. The only self-citation (SNORE, [23]) is used as a special case and as an experimental baseline; none of the new convergence or critical-point claims rests on that citation. A reviewer-visible sign inconsistency between Algorithm 1 and Eq. (15) concerning the placement of the Tweedie regularizer term is a correctness concern (the proofs may analyze a different update), but it is not circular: it does not make the conclusion equivalent to the hypotheses by construction. Accordingly, no circular step is flagged.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The theoretical core rests on standard tools (Haar measure, Tweedie, Doucet-Tadic) and two domain assumptions: the image prior is pi-equivariant and the algorithm's iterates stay bounded. The main unverified load-bearing premise is the boundedness assumption. No new entities are introduced.

free parameters (5)
  • step size delta = delta=1.5 for GS-DRUNet deblurring, delta=2.0 for super-resolution and DnCNN, delta=0.01 for despeckling
    Chosen per method and task by grid search maximizing test PSNR; the reported gains depend on these tuned values.
  • regularization weight lambda = lambda=0.15 for RED and 0.17 for ERED in GS deblurring; 0.12 vs 0.15 or 0.14 for DRUNet; 100 for despeckling
    Grid-searched per method on the test set, so comparisons are tuned upper bounds.
  • denoiser noise level sigma = sigma=7/255 for RED and 8/255 for ERED in GS deblurring; 8/255 for despeckling; 13/255 for ERED and 11/255 for RED in…
    Grid-searched separately for each method; the bias bound (23) depends inversely on sigma squared.
  • Monte-Carlo samples for infinite transforms = 10
    Equation (9) is approximated with 10 random transformations for subpixel rotation and translation; finite-sample noise is not analyzed.
  • number of iterations N = 400 for deblurring, 100 for despeckling, 200 for super-resolution
    Set by hand per experiment; no convergence criterion based on the theory is used.
assumptions (6)
  • standard math G is a compact Hausdorff topological group and pi its right-invariant Haar measure, making the objective equivariant.
    Used for Proposition 1 in Section 3.2; finite groups with counting measure satisfy it, but subpixel rotations implemented by interpolation may not be true group actions.
  • standard math Tweedie's formula expresses the noisy prior score through the MMSE denoiser, -nabla log p_sigma(x) = (x - D*_sigma(x))/sigma^2.
    Equation (3) in Section 2 is the bridge that turns any denoiser into a score estimate.
  • standard math The Doucet-Tadic Theorem 2.1 for biased stochastic gradient descent (stated as Theorem 1 in Appendix B.1) applies to ERED.
    Propositions 2 and 3 verify its Assumption 7 and import its conclusion; the paper does not re-derive the theorem.
  • domain assumption The true clean-image prior p is pi-equivariant in the sense log p = E_G~pi[log(p composed with G)].
    Required for the score identity in Remark 1 and for the critical-point convergence in Propositions 4 and 5; natural images are only approximately invariant.
  • domain assumption The iterates remain in a compact set K (the boundedness assumption), meaning the random seed lies in Lambda_K.
    Stated before Proposition 2 in Section 4.1; no conditions are given that make Lambda_K non-negligible.
  • domain assumption The clean prior has compact support (manifold hypothesis), yielding sub-polynomial score growth in Assumption 1(c).
    Invoked under Assumption 1(c) with reference [6]; used in Lemma 1 and Lemma 2 to control noise fluctuations.

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Cite this review

Pith. "Pith review of Equivariant Denoisers for Image Restoration." pith.science (2026). https://pith.science/paper/MZVDJOQA

@misc{pith2026241205343,
  author       = {Pith},
  title        = {Pith review of: Equivariant Denoisers for Image Restoration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZVDJOQA}},
  note         = {Machine review of arXiv:2412.05343}
}
read the original abstract

One key ingredient of image restoration is to define a realistic prior on clean images to complete the missing information in the observation. State-of-the-art restoration methods rely on a neural network to encode this prior. Moreover, typical image distributions are invariant to some set of transformations, such as rotations or flips. However, most deep architectures are not designed to represent an invariant image distribution. Recent works have proposed to overcome this difficulty by including equivariance properties within a Plug-and-Play paradigm. In this work, we propose a unified framework named Equivariant Regularization by Denoising (ERED) based on equivariant denoisers and stochastic optimization. We analyze the convergence of this algorithm and discuss its practical benefit.

Figures

Figures reproduced from arXiv: 2412.05343 by the authors.

Figure 1
Figure 1. Deblurring (a motion blur kernel with input noise level σy = 5/255) and de￾speckling (number of looks 50) with RED and ERED with a GS-denoiser trained on natural images or SAR images (respectively). The set of transformations for ERED is random flip. ERED produces a better qualitative result than RED. Method PSNR↑ SSIM↑ N↓ RED [24] 32.25 0.84 400 ERED rotation [30] 32.53 0.85 400 ERED translation 32.44 0.85 400 ERED… view at source ↗
Figure 2
Figure 2. Super-resolution with RED and ERED with a super-resolution factor of 2 with a GS-denoiser trained on natural images. The set of transformation for ERED is random flip. Qualitative results of ERED and RED are very similar. Despeckling On [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Despeckling with RED and ERED with a number of look of L = 50 with a GS-denoiser trained on SAR images. The set of transformation for ERED is random flip. ERED produces a better qualitative result than RED. We note a slight improvement of +0.1 dB for equivariant methods compared to RED. These deblurring experiments with various denosiers suggest that the slight improvement brought by equivariance does not appear to … view at source ↗

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