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arxiv: 2606.19802 · v1 · pith:NJRCVYLInew · submitted 2026-06-18 · 💻 cs.LG · cs.CV

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

Nicolas Zilberstein , Morteza Mardani , Santiago Segarra This is my paper

Pith reviewed 2026-06-26 18:15 UTC · model grok-4.3

classification 💻 cs.LG cs.CV
keywords flow map modelsdistortion-perception tradeoffimage restorationinverse problemsdenoisersplug-and-playflow matching
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The pith

Flow map models use a lookahead parameter to traverse the full distortion-perception tradeoff in image restoration.

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

The paper establishes that flow map models, which learn an average velocity field for few-step sampling, implicitly create a family of denoisers parameterized by a lookahead time t. This parameter allows continuous movement along the distortion-perception frontier, from minimum mean square error reconstructions to high perceptual quality ones. For Gaussian data, this is proven to achieve the optimal frontier, and for natural images it is shown empirically. The same mechanism is applied within Plug-and-Play solvers for inverse problems like deblurring and inpainting, controlling the balance between data fidelity and perceptual alignment.

Core claim

Flow map models implicitly define a one-parameter family of denoisers that continuously spans the distortion-perception frontier. The lookahead parameter t acts as a control knob between the MMSE and perceptual regimes. For Gaussian targets, varying t exactly recovers the optimal DP frontier; for natural images, similar behavior is observed empirically. Within a Plug-and-Play solver, the mechanism extends to general inverse problems.

What carries the argument

The flow map denoiser with its lookahead parameter t, which interpolates between different operating points on the distortion-perception plane by adjusting the prediction horizon in the learned average field.

If this is right

  • A single trained model can access multiple points on the DP frontier without retraining or auxiliary models.
  • In inverse problems, it allows trading off perceptual alignment and data consistency using the same mechanism.
  • Matches or exceeds specialized baselines at both extremes of the tradeoff.

Where Pith is reading between the lines

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

  • This approach might reduce the need for multiple specialized models in practical image restoration pipelines.
  • Extending the Gaussian proof to non-Gaussian cases could lead to theoretical guarantees for real-world images.
  • The method suggests that flow-based generative models have built-in flexibility for perception-distortion tradeoffs that other architectures might lack.

Load-bearing premise

The flow map model has learned a sufficiently accurate average field so that varying the lookahead parameter t produces the claimed continuous family of denoisers.

What would settle it

Training a flow map model on a dataset and then checking whether different values of t produce reconstructions that lie on or near the empirically measured optimal distortion-perception curve for that dataset.

Figures

Figures reproduced from arXiv: 2606.19802 by Morteza Mardani, Nicolas Zilberstein, Santiago Segarra.

Figure 1
Figure 1. Figure 1: PnP with average denoisers on a 2D mixture-of-Gaussians inverse problem (y = Hx + σϵ, with H a rotation+scaling operator and σ = 0.3). Ds,t denotes the average denoiser (defined formally in Section 3.1). The lookahead t controls the DP trade-off: at t = s the denoiser is MMSE and PnP reconstructions concentrate at the posterior mean, which for this symmetric bimodal posterior lies between the two modes, yi… view at source ↗
Figure 2
Figure 2. Figure 2: Analysis of the perception-distortion DP tradeoff and variance restoration on CelebA 128 × 128, quantified as RMSE/Var ratio vs FID. (a) DF tradeoff at different noise levels. Each curve is swept by varying the lookahead parameter t using a single trained model. (b) The near-collapse of curves across noise levels indicates a universal relationship between variance restoration and perceptual quality (FID) i… view at source ↗
Figure 3
Figure 3. Figure 3: Across five linear inverse problems on AFHQ (Gaussian/motion deblurring, box/random inpainting, SR×4), we sweep the lookahead t within a PnP solver and report RMSE (distortion) versus FID and KID (perception). Each marker corresponds to a value of t, with small lookaheads favoring low distortion and large ones favoring low FID. A single trained model spans the full frontier across all tasks, without retrai… view at source ↗
Figure 4
Figure 4. Figure 4: DP behavior on JPEG compression. Results are in in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results on CelebA and AFHQ. PnP-flow with t = 1 generates sharper results than with t = s, and closer to posterior sampling methods. Non-linear inverse problems. Finally, we evaluate the DP tradeoff on JPEG compression, a non￾linear degradation, with quantization factor 10. We study the effect of the number of PnP iterations; the results are shown in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ablation of PnP-flow as a function of the number of steps with Poisson noise and Gaussian deblurring C.3 Distortion-perception of CelebA We include here the same plot as in Section 2.3; it is shown in [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distortion-perception tradeoff traced by a single flow map on CelebA. Same setup as [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Distortion-perception tradeoff traced by a single flow map on CelebA. Same setup as [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Gaussian deblurring with kernel size 61 × 61 and σb = 3, and σ = 0.05 22 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Motion deblurring with kernel size 61 × 61 and σb = 1, and σ = 0.1 23 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Super resolution ×4, and σ = 0.05 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Random inpainting with 90% drop rate, and σ = 0.01 25 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Box inpainting with mask 80 × 80, and σ = 0.01 26 [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: JPEG compresion 27 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: CelebA - Super resolution ×4, and σ = 0.05 28 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: CelebA - Motion deblurring 29 [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Gaussian deblurring with Poisson noise 30 [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Colorization 31 [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Larger version of the images in Fig. 3 [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Progression of random inpainting with lookahead [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Progression of random inpainting with lookahead [PITH_FULL_IMAGE:figures/full_fig_p033_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Progression of SR ×4 with lookahead t = s meas t=20 t=40 t=60 t=80 t=100 GT [PITH_FULL_IMAGE:figures/full_fig_p034_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Progression of SR ×4 with lookahead t = 1 C.7 Performance as a function of the number of PnP iterations We study how the two endpoint lookaheads, t = s and t = 1, depend on the number of PnP iterations N. Results for AFHQ are reported on super-resolution ×4 ( [PITH_FULL_IMAGE:figures/full_fig_p034_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: AFHQ - Super-resolution ×4. Ablation of PnP-flow as a function of the number of steps. 0 100 200 300 400 500 600 700 Number of steps 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 FID Gaussian deblurring FID t = s t = 1.0 0 100 200 300 400 500 600 700 Number of steps 16 18 20 22 24 R M S E Gaussian deblurring RMSE t = s t = 1.0 [PITH_FULL_IMAGE:figures/full_fig_p035_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: AFHQ - Gaussian deblurring. Ablation of PnP-flow as a function of the number of steps. [PITH_FULL_IMAGE:figures/full_fig_p035_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: AFHQ - Box inpainting. Ablation of PnP-flow as a function of the number of steps. [PITH_FULL_IMAGE:figures/full_fig_p035_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: CelebA - Motion deblurring. Ablation of PnP-flow as a function of the number of steps. [PITH_FULL_IMAGE:figures/full_fig_p036_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Performance as a function of the number of steps [PITH_FULL_IMAGE:figures/full_fig_p036_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Reconstruction for phase retrieval 3. Semigroup structure. Ds,t(x) = γ Ds,u(x) + (1 − γ) Du,t(Xs,u(x)), (21) with γ = (1 − t)(u − s) (1 − u)(t − s) ∈ [0, 1]. (22) Proof. (1) Flow-map relation. Using the parameterization, Ds,t(x) − x = (1 − s) v(x, s, t), Xs,t(x) − x = (t − s) v(x, s, t). Eliminating v gives Xs,t(x) = x + t − s 1 − s [PITH_FULL_IMAGE:figures/full_fig_p037_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Visual comparison for α = 0.3. From left to right: clean image, noisy input, and reconstructions at increasing lookahead. The estimator smoothly transitions from a blurry conditional mean (MMSE) to a sharp sample, progressively restoring texture, contrast, and color saturation. D.3 Experiment on Mixture-of-Gaussians D.3.1 MMSE denoiser for a mixture of Gaussians Setup. Let p0 = N (0, I) and p1 = 1 2N (µa … view at source ↗
Figure 31
Figure 31. Figure 31: Distortion–perception comparison on a 2D mixture of Gaussians at three noise levels [PITH_FULL_IMAGE:figures/full_fig_p041_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: The composite bound s LD(t)LF exceeds 1 across all (s, t), so Tt is never a global contraction — empirical convergence relies on the local analysis of Theorem 2. LD(t) is estimated via power iteration on ∂Ds,t/∂x at noisy samples (worst case over 20). The bound increases with t, with steepest growth at low noise (large s). On the validation of assumption (38). Empirical validation of (38) and the t-depend… view at source ↗
Figure 33
Figure 33. Figure 33: Two forward models are tested: (a) isotropic rotation H1 = 0.8 R(π/6) with λ = 0.1, and (b) anisotropic operator H2 = R(π/6) diag(1.0, 0.3) R(π/4) with λ = 0.5. Black: measurements y. Gold crosses: ground truth x1,i. Yellow: Wiener/MAP estimator. Red dots: Tikhonov fixed point x ⋆ (t) predicted by Eq. (40). Red ×: PnP iterates after convergence (matching theory to machine precision, ∥xˆ − x ⋆ (t)∥mean < 5… view at source ↗
read the original abstract

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.

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 paper claims that flow map models, which learn an average field for few-step sampling, implicitly define a one-parameter family of denoisers controlled by the lookahead parameter t. This family continuously spans the distortion-perception (DP) frontier: for Gaussian targets the paper proves that varying t exactly recovers the optimal DP frontier, while for natural images similar behavior is observed empirically. The same t-mechanism is then used inside Plug-and-Play solvers for general inverse problems, where it controls the tradeoff between perceptual alignment and data consistency despite the acknowledged lack of exact optimality guarantees. Experiments on CelebA (128×128) and AFHQ (256×256) across linear and nonlinear tasks are presented to support the claims.

Significance. If the empirical results hold, the work supplies a simple, single-model mechanism for traversing the DP plane without paired supervision, auxiliary networks, or sampler hyperparameter search. The explicit Gaussian proof is a clear strength; the empirical extension to images and PnP solvers, while caveated, would still be useful if the observed curves reliably reach competitive extremes.

major comments (2)
  1. [Abstract / Gaussian proof section] Abstract and the section presenting the Gaussian result: the proof establishes exact recovery of the optimal frontier only for Gaussian targets; the extension to natural images and PnP rests on the unverified premise that the trained flow map has learned an average field whose error is small enough for t to trace the actual frontier rather than an arbitrary curve. The manuscript should quantify this approximation error (e.g., via residual norms or comparison against known optimal points where available) to make the load-bearing assumption explicit.
  2. [PnP experiments section] The PnP inverse-problem experiments: while the paper states that the mechanism 'matches or exceeds specialized baselines at both extremes' despite missing optimality guarantees, the central claim that a single model 'continuously spans the DP tradeoff' requires showing that the family produced by t is not merely a monotonic curve but lies near the frontier; additional plots comparing the obtained (distortion, perception) pairs against multiple strong baselines at intermediate t values would be needed to substantiate this.
minor comments (2)
  1. [Methods] Notation for the lookahead parameter t should be introduced with a clear equation reference in the methods section rather than only in the abstract.
  2. [Figures] Figure captions for the DP-plane plots should explicitly state the exact distortion and perception metrics used and whether error bars reflect multiple random seeds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract / Gaussian proof section] Abstract and the section presenting the Gaussian result: the proof establishes exact recovery of the optimal frontier only for Gaussian targets; the extension to natural images and PnP rests on the unverified premise that the trained flow map has learned an average field whose error is small enough for t to trace the actual frontier rather than an arbitrary curve. The manuscript should quantify this approximation error (e.g., via residual norms or comparison against known optimal points where available) to make the load-bearing assumption explicit.

    Authors: We agree that exact optimality holds only for the Gaussian case, with the image and PnP results being empirical. To address the concern, we will add a new paragraph (and associated figure) in the Gaussian section that quantifies the residual norm of the learned flow map on held-out Gaussian samples and compares the empirical DP curve on images against the known MMSE and perceptual extremes. revision: yes

  2. Referee: [PnP experiments section] The PnP inverse-problem experiments: while the paper states that the mechanism 'matches or exceeds specialized baselines at both extremes' despite missing optimality guarantees, the central claim that a single model 'continuously spans the DP tradeoff' requires showing that the family produced by t is not merely a monotonic curve but lies near the frontier; additional plots comparing the obtained (distortion, perception) pairs against multiple strong baselines at intermediate t values would be needed to substantiate this.

    Authors: We accept that intermediate-t points require explicit comparison to establish proximity to the frontier. We will add a new figure (and corresponding text) in the PnP experiments section that plots the full DP trajectories for several inverse problems together with the same strong baselines evaluated at matched distortion or perception levels. revision: yes

Circularity Check

0 steps flagged

No significant circularity; Gaussian proof is external and image results are empirical observations

full rationale

The paper's core derivation for Gaussian targets is a mathematical proof that varying the lookahead t recovers the optimal DP frontier, stated against an external benchmark rather than reducing to the model's own fitted parameters. For natural images the text reports empirical similarity without claiming exact recovery, and the PnP extension is explicitly qualified by 'despite the lack of exact optimality guarantees.' No self-definitional equations, fitted-input predictions, or load-bearing self-citations appear in the abstract or described claims. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that flow map models learn an average vector field whose lookahead variation produces the claimed family; no free parameters or invented entities are introduced beyond the standard flow-matching setup.

axioms (1)
  • domain assumption Flow map models learn an average field that can be queried at different lookahead horizons
    Invoked when the abstract states that the models 'implicitly define a one-parameter family of denoisers'

pith-pipeline@v0.9.1-grok · 5752 in / 1370 out tokens · 31474 ms · 2026-06-26T18:15:33.725438+00:00 · methodology

discussion (0)

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