Stabilizing RED using the Koopman Operator

Kunal N. Chaudhury; Shraddha Chavan

REVIEW 3 major objections 4 minor 40 references

Stabilizing RED using the Koopman Operator

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Adaptive Koopman monitor stops RED reconstruction from diverging

desk verdict A solid empirical letter: the Koopman/DMD step-size controller is genuinely new for RED, the experiments are consistent, and the main weakness is that the stability proxy is heuristic rather than proven. read the letter →

arxiv 2509.05736 v1 pith:R3BOXZY5 submitted 2025-09-06 eess.IV

Shraddha Chavan , Kunal N. Chaudhury This is my paper

classification eess.IV

keywords KoopmanoperatorRegularizationbyDenoisingadaptivestepsizeimagereconstructionspectralradiusdeepdenoisersstabilitydynamicmodedecomposition

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

The paper claims that RED (Regularization-by-Denoising), a popular scheme that uses pretrained denoisers as regularizers for image reconstruction, can be stabilized without retraining by monitoring a low-dimensional Koopman operator fitted to its recent iterates. When the spectral radius of that operator exceeds 1, the step size is shrunk exponentially. Across four deep denoisers and three imaging tasks, this prevents the PSNR divergence that vanilla RED exhibits and keeps reconstructions artifact-free. The appeal is that the fix is model-agnostic: it treats the denoiser as a black box and only adjusts the iteration step size, at less than 20 percent added runtime.

What carries the argument

The Koopman operator K_t, approximated in a 66-dimensional observable space by ordinary least squares from a window of recent RED iterates. The paper uses the spectral radius of K_t as a proxy for the local contraction rate of the full RED iteration in R^n; values above 1 trigger an exponential step-size reduction. The feature map (global statistics, 4x4 pooling, low-frequency DCT) is the handcrafted observable choice that makes the low-dimensional fit informative.

What would settle it

Construct or find a pretrained denoiser and task where, at the moment vanilla RED begins to diverge, rho(K_t) remains below 1, so SKOOP-RED never shrinks gamma; if PSNR then drops while the monitor reports stability, the spectral-radius criterion is falsified. Conversely, if rho(K_t) exceeds 1 during runs that remain stable, the controller is over-conservative.

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Extended reading notes

Core claim

SKOOP-RED is an adaptive step-size schedule for RED. At checkpoints, the most recent iterates are projected into a 66-dimensional feature space combining per-channel mean and standard deviation, 4x4 block means, and low-frequency DCT coefficients. A linear operator K_t is fitted by least squares to advance these features one step, approximating the Koopman operator of the RED dynamics; its spectral radius rho(K_t) is read as a local stability indicator. If rho(K_t) >= 1, the step size gamma is multiplied by exp(-beta(rho-1)) with beta = 2; otherwise gamma is unchanged. The paper reports that this simple rule produces stable, high-PSNR reconstructions where vanilla RED diverges, across DnCNN,

Load-bearing premise

The result rests on the assumption that the spectral radius of a 66-dimensional least-squares linear operator fitted to recent RED iterates in a handcrafted feature space is a faithful early warning of whether the full RED iteration in image space is about to diverge.

Editorial extensions

If this is right

RED can be run with arbitrary pretrained deep denoisers without imposing nonexpansiveness or retraining.
The adaptive step size removes the need for early stopping at an unpredictable breakdown point.
The same denoiser and forward model can be used across tasks; only the Koopman monitor needs to be fitted online.
The overhead is modest enough for iterative restoration in practice.
The approach is not tied to gradient descent structure and could be transferred to other RED variants.

Reading between the lines

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

The stability indicator could be sharpened by comparing rho(K_t) against the spectral radius of the full Jacobian on a few examples; if they disagree, a richer observable set would be needed.
A more principled choice of the feature map and window, rather than sensitivity-based selection, might yield a bound connecting the finite-dimensional Koopman spectrum to the true RED dynamics.
The same checkpointed Koopman monitor may generalize to other iterative reconstruction algorithms, such as plug-and-play ADMM or diffusion-based restoration, where divergence also appears after many iterations.
Since the controller only shrinks the step size, it cannot fix a denoiser that is unstable in every direction; the method's ceiling is the quality of the underlying denoiser.

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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: this paper does something new and useful — it fits an online Koopman/DMD model to RED iterates in a 66-dimensional feature space and shrinks the step size when the learned spectral radius exceeds 1. Across four pretrained denoisers and several inverse problems, it consistently prevents the divergence that vanilla RED shows, with code, sensitivity studies, and measured overhead under 20%. I'd send this to a referee.

What's actually new: the combination is new, even though the ingredients are not. DMD and the Koopman operator are standard, and RED's instability is well documented. Using a low-dimensional spectral-radius estimate as an adaptive step-size trigger for iterative reconstruction is a legitimate first, and the paper does not oversell it as a theory of RED convergence.

What it does well: the experiments are reasonably broad — motion deblurring, Gaussian deblurring, superresolution, four different denoiser architectures — and the comparison to equivariant RED is fair. The runtime breakdown is honest, the code is public, and the sensitivity analysis for window size, stride, and beta gives the reader a sense of how fragile the method is. The supplementary long-run experiment (10k iterations) is a nice touch.

Soft spots, in order of importance. First, the load-bearing link between the spectral radius of the OLS-fitted 66-dimensional operator and the actual stability of the full RED map is asserted, not derived. The stress-test note is right that a feature map can miss an unstable mode or create a spurious one. But I don't think this is fatal: the paper is an empirical letter, not a convergence analysis, and the ablation shows the trigger does something. Second, hyperparameters (feature map, window, stride, beta) are tuned on the same tasks used in the main tables, and the PSNR numbers are averages over 15 images with no error bars. That limits the strength of the claim, but again it is addressable in revision. Third, there is no discussion of failure cases — instances where the trigger fired but divergence still happened, or where vanilla RED was stable and the controller slowed things down unnecessarily. That would genuinely strengthen the paper.

I disagree with any suggestion that this is circular or that the citation pattern is problematic. It is an adaptive controller, not a derivation of the result from the target. Self-citations are appropriate and not excessive.

The bottom line: this is a pragmatic, clearly described stabilization mechanism for a real practical problem. It belongs in the iterative-imaging literature, and a careful referee could improve it substantially. My recommendation: accept for peer review, and ask the authors for error bars, one or two failure-case analyses, and ideally a small experiment connecting the learned spectral radius to the full Jacobian's spectral behavior. For people using RED with black-box denoisers, this is a useful safety net and worth reading.

Referee Report

3 major / 4 minor

Summary. The paper proposes SKOOP-RED, an adaptive step-size controller for Regularization-by-Denoising (RED) image reconstruction. At checkpoints during the RED iteration, the method projects recent iterates into a 66-dimensional handcrafted feature space, fits a low-dimensional Koopman operator K_t by least squares (Eq. 7), and shrinks the RED step size whenever the spectral radius ρ(K_t) exceeds 1 (Algorithm 1). The authors claim this stabilizes RED with black-box deep denoisers, avoids retraining, adds less than 20% runtime overhead, and outperforms Vanilla RED and Equivariant RED across deblurring and superresolution tasks with DnCNN, DRUNet, GS-DRUNet, and DiffUNet. Code is provided.

Significance. If the empirical claims hold, this is a practically useful contribution: RED with modern deep denoisers is known to be unstable, and a model-agnostic, low-overhead stabilization mechanism would strengthen the case for RED in real imaging systems. The paper is clearly written, the experiments cover four denoisers and three tasks, and the code release is a concrete asset. However, the central mechanism—using the spectral radius of a 66-dimensional OLS-fitted Koopman operator as a proxy for the stability of the full RED dynamics—is asserted without a supporting bound or direct validation. The evaluation also tunes key hyperparameters on the same test tasks and reports no variance over the 15-image set. These gaps are load-bearing for the claim of 'consistent stability improvements,' but they are addressable with additional analysis and a more rigorous evaluation protocol.

major comments (3)

[Sec. 2.3, Eqs. (7)-(8)] The controller's decision rule assumes that ρ(K_t) ≥ 1 indicates potential instability of the full RED map T_γ in R^n. This link is not derived. K_t is an OLS fit in a low-dimensional feature space; without an error bound relating the approximate spectrum to the spectral properties of DT_γ, the feature map can miss genuinely unstable modes (false negatives, no step-size reduction) or introduce spurious expanding modes (false positives, unnecessary slowdown). The paper's own appendix (Section A) describes the feature map as 'empirically' useful, which effectively concedes that this link is heuristic. To make the central claim load-bearing, the authors should provide either a formal error estimate in terms of the feature-map invariance/closure and the residual of Eq. (7), or direct validation on small-scale problems comparing ρ(K_t) against the spectral radius of the full Jacobian along th
[Sec. 3, Tables 2 and S1; Sec. 2.4] The stability claim rests on 15 test images (Set15C), and Tables 2 and S1 report only mean PSNR values with no variance, percentiles, or per-image breakdowns. Equally important, the feature map, window size w, stride r, and decay rate β are selected via sensitivity studies on the same tasks (Section 2.4, Fig. S2), creating a risk of indirect overfitting to the evaluation set. I request: (i) report standard deviations or per-image results; (ii) freeze all SKOOP parameters on a development split and evaluate on held-out tasks, kernel types, or denoisers; (iii) include an ablation that replaces ρ(K_t) with a simple heuristic (e.g., residual-norm growth or a fixed decay schedule) to demonstrate that the Koopman spectral-radius monitoring, rather than mere step-size adaptation, is what provides the stability benefit.
[Algorithm 1 and Eq. (8)] The control law only shrinks γ and never increases it. This is not necessarily wrong, but it means any false positive permanently slows the iteration, and the paper does not discuss recovery from overly conservative step sizes. Moreover, the resulting non-autonomous iteration is not analyzed at all; the only theoretical framing is the heuristic spectral-radius test. A formal statement for a simplified or linearized model (e.g., under what conditions the adaptive scheme keeps γ bounded away from zero and prevents divergence) would substantially strengthen the paper and clarify what SKOOP-RED can guarantee.

minor comments (4)

[Sec. 2.3, Eq. (6) vs. Eq. (7)] Equation (6) states K_t ψ(x_τ) = ψ(x_{τ+1}) for t−w+1 ≤ τ ≤ t, but for τ = t this would require the future iterate x_{t+1}. Equation (7) correctly sums only up to τ = t−1. Please fix the index range in Eq. (6) to avoid confusion.
[Sec. 3, Table 1] The text says SKOOP-RED adds 'only a small 15–20% runtime overhead per iteration,' but Table 1 reports overheads of 10.97–14.11%, and Section 2.4 reports 11% for DiffUNet. Please reconcile these numbers.
[Table 2] The abbreviation 'Equiv.RED' is used without a definition in the table caption; it appears to refer to equivariant RED from [18], but this should be stated explicitly.
[Sec. 2.4] The paper says K_t is computed using the technique in [33] (DMD), but Eq. (7) is a least-squares fit. Please specify which DMD variant is used (e.g., exact DMD, projected DMD) and what rank/regularization, if any, is applied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SKOOP-RED is a feedback controller; no target outcome is encoded in the fitted Koopman operator.

full rationale

The paper's derivation chain is a feedback-control mechanism: at checkpoints, K_t is fit by OLS to the recent RED trajectory (Eq. 7), its spectral radius is computed, and if rho(K_t)>1 the step size is shrunk via Eq. (8). This is not circular because the fitted operator is estimated from past iterates before the step-size correction is applied, and it does not encode the target PSNR or stability outcome. The claim that rho(K_t)>=1 indicates potential instability is an empirical modeling assumption—an unproven proxy, which is a correctness/robustness risk—not a result equivalent to the input by construction. No parameter is fitted to the reported final PSNR and then renamed a prediction; the reported PSNR is an outcome of the adaptive policy, and the paper tests this out-of-sample on multiple denoisers, tasks, and kernels. The self-citations ([19], [24], [25]) concern different denoiser properties and are not load-bearing for the Koopman mechanism. The supplementary's sensitivity studies for window/stride/beta are hyperparameter tuning, not circular derivation. Therefore no specific reduction of the kind required for a circularity finding can be exhibited.

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

The method relies on standard Koopman/DMD machinery, one handcrafted feature map, and several empirically chosen hyperparameters (beta, w, r). No new physical entities are introduced. The main audit concern is that the central stability indicator (spectral radius of a 66-dim linear model) is asserted rather than derived.

free parameters (5)

decay rate beta = 2
Sec. 2.3: 'We set beta=2 based on empirical validation'. It controls how quickly gamma shrinks when the spectral radius exceeds 1.
window size w and checkpoint stride r = w=30, r=10
Sec. 2.4: window w in [30,40], stride r=10; selected through sensitivity studies on the same tasks (Fig. S2).
feature map design (d=66, 4x4 pooling, DCT coefficients) = d=66
Sec. 2.4 and Appendix A: handcrafted descriptor; 'various feature maps' explored and the chosen one validated by sensitivity studies.
spectral-radius trigger = 1
Sec. 2.3: 'rho(K_t) >= 1 indicates potential instability'; a design criterion not derived from theory.
RED step size gamma_0 and regularization lambda = not reported
Eq. (3)-(4) depend on gamma and lambda, but experimental values are not given in the paper; inherited from baseline Vanilla RED implementations.

assumptions (5)

standard math The Koopman operator on a Hilbert space of observables is linear and its spectrum characterizes the stability of the underlying map.
Sec. 2.2, based on refs. [27,28].
standard math DMD/least-squares regression approximates the Koopman operator from trajectory data.
Sec. 2.3, Eq. (7), based on refs. [29,33,35].
ad hoc to paper The 66-dimensional handcrafted feature map preserves the stability-relevant information of the full RED iteration.
Sec. 2.4 and Appendix A; supported only by sensitivity studies, no guarantee or bound.
ad hoc to paper The spectral radius of the fitted finite-dimensional K_t is a valid indicator of instability of the full RED dynamics.
Sec. 2.3: 'rho(K_t)>=1 indicates potential instability'; no theorem links the two.
domain assumption The recent trajectory window is representative of future local dynamics.
Algorithm 1 uses the last w iterates to fit K_t and then to adjust gamma for future steps.

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Cite this review

Pith. "Pith review of Stabilizing RED using the Koopman Operator." pith.science (2026). https://pith.science/paper/R3BOXZY5

@misc{pith2026250905736,
  author       = {Pith},
  title        = {Pith review of: Stabilizing RED using the Koopman Operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R3BOXZY5}},
  note         = {Machine review of arXiv:2509.05736}
}

read the original abstract

The widely used RED (Regularization-by-Denoising) framework uses pretrained denoisers as implicit regularizers for model-based reconstruction. Although RED generally yields high-fidelity reconstructions, the use of black-box denoisers can sometimes lead to instability. In this letter, we propose a data-driven mechanism to stabilize RED using the Koopman operator, a classical tool for analyzing dynamical systems. Specifically, we use the operator to capture the local dynamics of RED in a low-dimensional feature space, and its spectral radius is used to detect instability and formulate an adaptive step-size rule that is model-agnostic, has modest overhead, and requires no retraining. We test this with several pretrained denoisers to demonstrate the effectiveness of the proposed Koopman stabilization.

Figures

Figures reproduced from arXiv: 2509.05736 by the authors.

**Figure 2.** Results for motion deblurring using DRUNet denoiser. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Results for 2× superresolution using the DiffUNet denoiser, where the clean image is blurred with a 25×25 Gaussian kernel (σ = 1) and downsampled. (a) Ground truth, (b) observed image, (c) reconstruction via Vanilla RED, and (d) reconstruction via SKOOP-RED. Both methods are run for 2500 iterations, as detailed in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Instability of Vanilla RED for motion deblurring using four denoisers (DnCNN, DRUNet, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Results for 2× superresolution (25×25 Gaussian blur, σ = 1) using DiffUNet denoiser. Left: PSNR (dB) vs. iteration. Right: residual norm ∥xt − xt−1∥ vs. iteration t. See [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: Comparison of SKOOP-RED with manually tuned step-size schedules. PSNR (dB) vs. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Reviewed August 5, 2026 · model on record in the stance chip above.

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