Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms

Audrey Repetti; Beno\^it Mal\'ezieux; Matthieu Kowalski; Thomas Moreau

REVIEW 2 major objections 4 minor 62 references

Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A single inner denoising step provably converges to the full PnP solution

desk verdict Useful idea, but the central analysis theorem has a step-size condition that is too weak; a valid counterexample exists and the paper's own appendix supplies the fix. read the letter →

arxiv 2411.13276 v3 pith:YIBXSRRH submitted 2024-11-20 math.OC cs.CVeess.IVeess.SP

Matthieu Kowalski , Benoît Malézieux , Thomas Moreau , Audrey Repetti This is my paper · ORCID

classification math.OCcs.CVeess.IVeess.SP MSC 90C5965K1068T0768U1094A08

keywords Plug-and-Playforward-backwardalgorithmunrollinganalysisdenoisersynthesisprimal-dualwarmrestartdictionarylearning

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 asks whether a Plug-and-Play forward-backward (FB-PnP) algorithm still converges to the right solution when the denoiser inside it is computed with only a single unrolled sub-iteration instead of being solved to high accuracy. For two dictionary-based denoisers—an analysis denoiser built from dual forward-backward steps and a synthesis denoiser built from forward-backward steps—the authors show that with a warm-restart strategy the one-sub-iteration algorithm is algebraically identical to known primal-dual or forward-backward schemes. Consequently, one step and infinitely many steps of the inner denoiser lead to the same asymptotic solution of the outer inverse problem. The paper also proves that the synthesis denoising problem is itself a proximity operator, and gives convergence results for a smoothed Moreau-Yosida version with any finite number of inner steps. If correct, this removes the need to run many inner denoising iterations to guarantee Plug-and-Play convergence.

What carries the argument

The central object is the unrolled denoiser. The analysis denoiser is $G^A_{L,\lambda,v}(u_0)=v-\Gamma^*\widetilde{G}^A_{L,\lambda,v}(u_0)$, where $\widetilde{G}^A$ is $L$ compositions of the dual-FB layer $T^A_{\lambda,v}(u)=\mathrm{prox}_{\sigma g^*_\lambda}(u-\sigma\Gamma^*(\Gamma u-v))$. The synthesis denoiser is $G^S_{L,\lambda,v}(z_0)=D\widetilde{G}^S_{L,\lambda,v}(z_0)$, where $\widetilde{G}^S$ composes $L$ FB layers $T^S_{\lambda,v}(z)=\mathrm{prox}_{\zeta g_\lambda}(z-\zeta D^*(Dz-v))$. The proof identifies the case $L=1$ with a warm-started inner variable and shows, by direct algebra, that the outer iteration reduces to either the scaled Loris-Verhoeven primal-dual algorithm or to forward-backward iterations in the sparse-code variable. The surrounding arguments use Fenchel-Rockafellar duality and the infimal post-composition identity $D\triangleright g_\lambda=(g^*_\lambda\circ D^*)^*$.

What would settle it

On the compressive-sensing toy example of Section 5.1, run the analysis PnP algorithm (2.7) with $L=1$ and the same fixed dictionary, once with warm restart and once with $u_k$ reset to zero at every $k$, then compare both limits to the reference solution obtained with $L=10^4$. If the zero-restart version converges to the same reference solution, the claimed dependence on warm restart is not necessary; if the warm-restart version does not converge to that reference, the theorem's conclusion is false.

Watch

Extended reading notes

Core claim

For a fixed dictionary, running FB-PnP with exactly one sub-iteration of an unrolled analysis denoiser, with the dual variable carried across outer iterations, is equivalent to the Loris-Verhoeven primal-dual algorithm, and converges to a minimizer of $\frac{1}{2}\|Ax-y\|^2 + g_\lambda(\Gamma x)$. With one sub-iteration of an unrolled synthesis denoiser, the iteration is equivalent to forward-backward on the sparse-code problem $\frac{1}{2}\|ADz-y\|^2 + g_\lambda(z)$, and its image under $D$ converges to a minimizer of the analysis-form problem with $D\triangleright g_\lambda$. Hence, for these dictionary-based denoisers, one-step PnP and fully iterated PnP have the same asymptotic solution.

Load-bearing premise

The central equivalences hold only when the inner variable, $u_k$ for analysis or $z_k$ for synthesis, is carried from one outer iteration to the next as the warm start for the single inner step; if the denoiser were re-initialized from zero at each outer step, the algebraic identities that make one-step PnP match the full solver would break.

Editorial extensions

If this is right

One inner sub-iteration suffices for asymptotic correctness, so PnP implementations can be much cheaper per outer iteration without changing the target problem.
The analysis PnP with one step is exactly a scaled primal-dual algorithm, so convergence-rate and step-size results for Loris-Verhoeven apply verbatim.
The synthesis PnP with one step is exactly forward-backward on the sparse-code variable $z$, so the outer fixed point inherits the FB convergence theory and sparsity structure.
For the smoothed Moreau-Yosida objective, any finite number $L$ of inner steps converges, provided the approximate proximity operator contracts the previous error by a factor $\alpha_L < 1/\sqrt{2}$.
In the deep-dictionary experiments, training the analysis denoiser with only one sub-iteration gave reconstruction quality comparable to training with twenty, suggesting single-layer training is sufficient in practice.

Reading between the lines

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

The equivalence suggests that a deep-unrolled PnP network with one layer per outer step can be viewed as a convex optimization algorithm, so its fixed point, not just its trained output, is interpretable; this reading goes beyond the paper's explicit statements.
The same warm-restart algebra may extend to other proximal algorithms, such as Douglas-Rachford or ADMM, when the inner solver is unrolled, which would be a natural testable extension.
The result suggests a practical protocol: instead of tuning the number of inner iterations for accuracy, tune it for speed and rely on warm restart to preserve the solution; this could be checked on larger inverse problems with learned dictionaries.
If the equivalence holds for learned dictionaries as well as fixed ones, then stability of the PnP iteration may be assessed through the convex problem it implicitly solves, which would connect these unrolled networks to fixed-point analysis of deep equilibrium models.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper shows that a single sub-iteration of an unrolled analysis or synthesis denoiser, with warm restart, lands on a known primal-dual or FB iteration. That is a genuinely useful observation, and the algebraic derivations are correct. But Theorem 2.5, one of the two pillars of the one-step-equals-infinite-steps claim, states a step-size condition that is insufficient. I checked the stress-test counterexample: A=0.1, y=0, Gamma=2, g=abs, lambda=1, tau=1, sigma=0.4. Both stated inequalities hold (tau<2/||A||^2 and sigma<2/||Gamma||^2), yet the iterates oscillate and do not converge to the unique solution x=0. The proof maps the algorithm to Loris-Verhoeven but then applies the wrong convergence condition. The paper's own Appendix A gives the correct condition, sigma<1/||Gamma Gamma^*||, which excludes the counterexample. Since Theorem 2.5 is load-bearing, this is a real flaw. It is fixable by restating the condition to match Appendix A and the LV theorem. What is genuinely new: the L=1 equivalences for both analysis and synthesis (Theorems 2.5 and 3.8) and the Moreau-envelope result (Theorem 4.3) are new statements, even though some intermediate propositions are known. The synthesis equivalence (Theorem 3.8) checks out; the derivation is clean and the condition tau zeta < 2/||AD||^2 is correct. The paper is honest about what is known, and the numerical experiments are reproducible (code on GitHub). Soft spots, in proportion: Theorem 2.5 is the big one. Theorem 4.3's proof asserts coercivity of h by citing [4, Cor. 11.16 & 11.17], but the assumptions do not guarantee it; for example, if g~=0, the Moreau envelope is not coercive. That is a smaller gap, fixable by adding a coercivity assumption on f~ or on the smoothed objective. The numerical section uses convergence loosely for intermediate L, a minor wording issue. Citation pattern is fine; self-citations are contextual and the relevant unrolling/PnP literature is covered. Who it is for: researchers working on PnP convergence, unrolled optimization, or efficient dictionary-based denoisers. The paper deserves a serious referee, but not as is. I would send it to review with a clear request to correct the step-size condition in Theorem 2.5 and patch the coercivity step in Theorem 4.3. The core idea is sound and likely right after those fixes.

Referee Report

2 major / 4 minor

Summary. The paper studies forward-backward plug-and-play (FB-PnP) algorithms in which the proximity operator is replaced by a sub-iterative, unrolled Gaussian denoiser built from a fixed analysis or synthesis dictionary. For the analysis denoiser, the paper claims that with one sub-iteration and a warm restart the FB-PnP algorithm is equivalent to the Loris-Verhoeven primal-dual algorithm and converges to a solution of (2.6); for the synthesis denoiser, it claims equivalence with forward-backward iterations on the synthesis sparse-coding problem. The paper also analyzes a Moreau-envelope-smoothed version of the problem under a sufficient-decrease condition, and presents numerical experiments on compressive sensing and deep dictionary learning image restoration.

Significance. If correct, the paper would provide a clean and practically valuable statement: a single inner sub-iteration, warm-started from the previous outer iterate, can yield the same asymptotic solution as fully converged denoisers in FB-PnP. This would substantially simplify training and analysis of shallow unrolled PnP networks. The synthesis part of the paper is solid: Proposition 3.1 and Theorem 3.2 give a correct equivalence between the synthesis denoising problem and a proximity operator, and Theorem 3.8 correctly identifies the L=1 synthesis update as an FB iteration in the sparse-code domain. The algebraic proofs are direct and do not rely on fitted parameters, which is a strength. However, the central analysis-denoiser result, Theorem 2.5, has an insufficient step-size condition and is false as stated, and Theorem 4.3 has an unproven coercivity assertion. These issues affect load-bearing claims, so the paper cannot be accepted in its current form.

major comments (2)

[Section 2.2, Theorem 2.5 and Table 2] The stated convergence condition 0 < σ < 2||Γ||_S^{-2} is insufficient for the claimed result. Take A = 0.1, y = 0, Γ = 2, g_λ = λ|·| with λ = 1, τ = 1, and σ = 0.4. These values satisfy both conditions in Theorem 2.5 (τ = 1 < 200 and σ = 0.4 < 0.5), but algorithm (2.7) with L = 1 generates the iterates x_{k+1} = 0.99 x_k − 2 clip(0.792 x_k − 0.6 u_k, −1, 1), u_{k+1} = clip(0.792 x_k − 0.6 u_k, −1, 1). Starting from (x_0, u_0) = (1, 0) this yields x_1 = −0.594, x_2 = 1.303, x_3 = −0.710, x_4 = 1.297, ..., which does not converge to the unique minimizer x = 0 of 0.005 x^2 + 2|x|. The proof invokes the Loris-Verhoeven theorem, but the condition for that theorem, as correctly stated in the paper's own Appendix A, is σ < 1/||ΓΓ*|| (which equals 0.25 in this example), and it excludes the counterexample. Therefore Theorem 2.5 is false as stated; the step-size condition must be corrected to match the Loris-Verhoeven condition, and Table 2 must be updated accordingly. This is load-bearing because Theorem 2.5 is the basis for the one-step/infinite-step equivalence for the analysis denoiser.
[Section 4, Theorem 4.3] In the proof of Theorem 4.3, after deriving ||∇h(x_k)|| → 0, the paper states 'Since h is convex, coercive (see [4, Cor. 11.16 & 11.17])' and concludes that (x_k) is bounded and converges to a minimizer. However, no coercivity assumption on f or g_λ appears in the theorem statement, and the Moreau envelope of a non-coercive function need not be coercive (for instance, if f = 0 and g_λ is an affine function on a subspace). Without coercivity, ∇h(x_k) → 0 does not imply boundedness of (x_k), so the cluster-point argument is unsupported. The authors should either add an explicit coercivity assumption on the objective in (4.4) or prove that the Lyapunov descent inequality implies boundedness of the iterates. This gap affects the convergence result for the smoothed problem, which is one of the main contributions.

minor comments (4)

[Section 3.3, proof of Theorem 3.8] The statement after (3.26) says convergence holds 'if 0 < τ ζ < 2||AD||_S^2' but the exponent should be negative, i.e., 0 < τ ζ < 2||AD||_S^{-2}, matching the theorem statement and Table 2.
[Section 5.1, Figures 1 and 2] The text claims that 'the trajectory per iteration is the same independently of the value of L' for the analysis denoiser and 'fairly the same' for the synthesis denoiser. The theorems only guarantee identical limit sets, not identical iterates for different L; this phrasing should be clarified to avoid overstating the experimental evidence.
[Section 3.1, Example 3.3] In the displayed equation (3.15), the set F^S_λ(D) is defined with a superscript S and the text after the equation refers to 'F S λ (D)'; the subscripts and superscripts should be made consistent for readability.
[Appendix A] The Loris-Verhoeven algorithm in Appendix A states the condition σ < 1/||Γ*Γ||, but the main text Theorem 2.5 uses the different bound 0 < σ < 2||Γ||_S^{-2}. These discrepancies should be reconciled in the revised manuscript, since the attached proof of Theorem 2.5 relies on the appendix condition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the one-step PnP equivalences are direct algebraic rewritings of independently established primal-dual and forward-backward algorithms, not consequences of their own conclusions.

full rationale

The paper's central claims are structural equivalences proven by direct algebra. Theorem 2.5 rewrites algorithm (2.7)-(2.8) with L=1 into the scaled Loris-Verhoeven primal-dual update (2.10)-(2.12) and then invokes the independent convergence theorem [38, Thm. 1]. Theorem 3.8 rewrites (3.20)-(3.21) into the standard FB iteration (3.26) with step-size τζ and invokes [20, Thm. 3.4]. In both cases the target minimization problem is not assumed in the proof; the algorithm is exhibited as an instance of a known scheme with quoted external convergence guarantees. Proposition 3.1 and Theorem 3.2 use standard Fenchel-Rockafellar duality and Moreau identities from [4], not the paper's conclusions. The self-citations that appear (e.g., [35], [47] in Remark 2.3) are contextual attributions of unrolled denoiser designs and are not load-bearing for any theorem. I flag two non-circular correctness concerns that do not affect the circularity score: (a) Theorem 2.5 states σ<2||Γ||^{-2}, while Appendix A's Loris-Verhoeven condition is σ<1||Γ*Γ||^{-1}; a scalar instance such as A=0.1, y=0, Γ=2, τ=1, σ=0.4 satisfies the stated hypothesis but violates the stricter condition, so the cited theorem may not apply exactly as stated. (b) The proof of Theorem 3.8 silently uses x_k=Dz_k, which requires an initialization x0=Dz0 not stated in algorithm (3.20). Neither issue makes a claim equivalent to its own input, so the derivation chain is not circular.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim is a mathematical equivalence; it pulls from standard convex analysis and known convergence theorems. The main non-standard assumption is the sufficient decrease condition (4.10) in Theorem 4.3, which is not verified in practice. Also, the learned dictionaries in the experiments are fitted to data but are not part of the theoretical result.

free parameters (2)

Sufficient decrease constant α_L in (4.10) = unknown, required 0<α_L<1/√2
Theorem 4.3 assumes the approximate prox operator contracts the error by α_L. The paper notes in Remark 4.4(i) that determining L to ensure this is difficult in practice, so α_L is effectively a non-verifiable parameter of the theorem.
Learned dictionaries Γ and D in DDL experiments = 50 convolutional filters of size 5×5, trained on BSDS500
Fitted by MSE loss in Section 5.2 to produce the numerical demonstrations; not needed for the theoretical equivalence results, which assume fixed dictionaries.

assumptions (5)

domain assumption g is a proper, lower-semicontinuous, convex function (g ∈ Γ0(RN)).
Assumed throughout the paper (Models 2.2, 3.5, and Section 4) to ensure the existence of proximity operators and the validity of Fenchel-Rockafellar duality.
standard math Standard convex analysis results from Bauschke & Combettes [4]: Moreau identity, Fenchel-Rockafellar duality, prox calculus, infimal post-composition properties.
Used in Propositions 3.1, 3.2, and the equivalence proofs of Theorems 2.5 and 3.8.
standard math Convergence of the FB algorithm (Combetes-Wajs [20, Thm. 3.4]) and the scaled Loris-Verhoeven algorithm [38, Thm. 1].
These external theorems are the convergence engines for the L=1 equivalences. Note the step-size condition in Theorem 2.5 is inconsistent with the Loris-Verhoeven condition stated in Appendix A.
ad hoc to paper Sufficient decrease condition (4.10): ||eG^{L,λμ−1}(x_{k+1},u_k) - u‡_{x_{k+1}}|| ≤ α_L ||u_k - u‡_{x_{k+1}}|| with α_L<1/√2.
Introduced in Theorem 4.3 to handle an arbitrary number of sub-iterations in the smoothed problem. The paper acknowledges it is hard to verify in practice.
ad hoc to paper Coercivity of h(x)=f(x)+μ g_λ(x) envelope, asserted in the proof of Theorem 4.3.
The theorem's stated assumptions (f convex Lipschitz-differentiable, g convex lsc proper) do not guarantee h coercive in general; the proof cites [4, Cor. 11.16 & 11.17] without establishing the precondition. This is an unflagged assumption.

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Pith. "Pith review of Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms." pith.science (2026). https://pith.science/paper/YIBXSRRH

@misc{pith2026241113276,
  author       = {Pith},
  title        = {Pith review of: Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YIBXSRRH}},
  note         = {Machine review of arXiv:2411.13276}
}

read the original abstract

In this work we study the behavior of the forward-backward (FB) algorithm when the proximity operator is replaced by a sub-iterative procedure to approximate a Gaussian denoiser, in a Plug-and-Play (PnP) fashion. In particular, we consider both analysis and synthesis Gaussian denoisers within a dictionary framework, obtained by unrolling dual-FB iterations or FB iterations, respectively. We analyze the associated minimization problems as well as the asymptotic behavior of the resulting FB-PnP iterations. In particular, we show that the synthesis Gaussian denoising problem can be viewed as a proximity operator. For each case, analysis and synthesis, we show that the FB-PnP algorithms solve the same problem whether we use only one or an infinite number of sub-iteration to solve the denoising problem at each iteration. To this aim, we show that each "one sub-iteration" strategy within the FB-PnP can be interpreted as a primal-dual algorithm when a warm-restart strategy is used. We further present similar results when using a Moreau-Yosida smoothing of the global problem, for an arbitrary number of sub-iterations. Finally, we provide numerical simulations to illustrate our theoretical results. In particular we first consider a toy compressive sensing example, as well as an image restoration problem in a deep dictionary framework.

Figures

Figures reproduced from arXiv: 2411.13276 by the authors.

**Figure 2.** Convergence profile of algorithm (3.20) with L ∈ {1, 20, 50, 100}. Solutions (x ∗ , u∗ ) are pre-computed considering algorithm (3.20) with L → ∞. L = 1) with the output of algorithm (2.7) for L = 104 . Furthermore, although the results from Theorem 2.5 only hold for L = 1, we show empirically that they remain true for 1 < L < +∞ for our problem by considering L ∈ {20, 50, 100}. In [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 3.** Example of outputs obtained with the unfolded SD and AD for different configurations. [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Blur kernel used in the experiments to model operator [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]

**Figure 5.** Figure 5: Convergence profiles with respect to iterations for the different FB-PnP algorithms. AD [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]

**Figure 6.** Figure 6: PSNR profiles with respect to iterations for the different FB-PnP algorithms. AD and SD [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]

**Figure 7.** Figure 7: Examples of FB-PnP outputs obtained with the different denoisers tested, for two im [PITH_FULL_IMAGE:figures/full_fig_p032_7.png]

**Figure 8.** Figure 8: Example of FB-PnP outputs obtained with SD for [PITH_FULL_IMAGE:figures/full_fig_p033_8.png]

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