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arxiv: 2512.07150 · v2 · submitted 2025-12-08 · 💻 cs.LG · cs.AI· cs.CV

FlowLPS: Langevin-Proximal Sampling for Flow-based Inverse Problem Solvers

Jonghyun Park , Jong Chul Ye This is my paper

Pith reviewed 2026-05-17 00:38 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CV
keywords flow-based generative modelsinverse problemsLangevin dynamicsproximal optimizationlatent space samplingtraining-free solversimage reconstruction
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The pith

FlowLPS interleaves a few Langevin updates with proximal refinement to solve inverse problems using pre-trained flow models while preserving both fidelity and perceptual quality.

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

The paper tries to establish that a hybrid procedure can resolve the practical trade-off faced by training-free solvers for latent flow models in imaging inverse problems. Pure optimization quickly improves consistency with measurements but often loses perceptual realism when the nonlinear latent space makes results sensitive to the starting point. Pure stochastic sampling better explores the posterior but typically needs many iterations to reach sharp, consistent outputs. FlowLPS inserts a small number of Langevin steps at each reverse flow iteration to generate posterior-oriented initializations, then applies fast local proximal refinement and controlled re-noising to stabilize the trajectory. A reader would care because this lets existing generative priors be used directly on common tasks such as deblurring or inpainting without retraining and without sacrificing either measurement accuracy or visual realism.

Core claim

At each reverse step, FlowLPS uses a few Langevin updates to perturb the model-predicted clean estimate in posterior-oriented directions, supplying stochastic initializations for local refinement. It then applies MAP-style proximal refinement to improve measurement consistency from those initializations and employs controlled pCN-style re-noising to maintain trajectory coherence. Experiments on FFHQ and DIV2K across five linear inverse problems show the resulting reconstructions achieve a strong balance between measurement fidelity and perceptual quality, with further tests on pixel-space problems and phase retrieval.

What carries the argument

Langevin-Proximal Sampling, a per-step procedure that perturbs the current estimate with limited Langevin dynamics to create better starting points and then performs local proximal optimization for rapid consistency gains.

If this is right

  • Pre-trained flow models can be used for multiple linear inverse problems without any task-specific retraining.
  • The number of total reverse steps needed stays lower than pure sampling while avoiding the initialization sensitivity of pure optimization.
  • pCN-style re-noising keeps the reverse trajectory coherent enough for stable reconstruction across steps.
  • The same hybrid pattern extends at least to pixel-space inverse problems and phase retrieval.

Where Pith is reading between the lines

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

  • The same interleaving of limited stochastic perturbation and local refinement might transfer to other latent generative models that exhibit similar initialization sensitivity.
  • Varying the exact count of Langevin steps per reverse iteration could be tuned per problem type to further improve the fidelity-perception trade-off.
  • If the proximal step can be formulated for nonlinear measurements, the method could address a wider class of inverse problems beyond the linear cases tested.

Load-bearing premise

That a small number of Langevin updates in the latent space reliably produce posterior-oriented initializations that local proximal refinement can then improve without degrading perceptual realism.

What would settle it

Running FlowLPS on the same FFHQ or DIV2K inverse problems and finding that its reconstructions score worse on both perceptual metrics and data-fidelity measures than either pure proximal optimization or extended pure sampling would falsify the claimed balance.

Figures

Figures reproduced from arXiv: 2512.07150 by Jong Chul Ye, Jonghyun Park.

Figure 1
Figure 1. Figure 1: Qualitative results of FlowLPS. (a) Motion deblurring, (b) Gaussian deblurring, (c) Super-resolution (×12) and (d) random inpainting results before and after FlowLPS. (e) and (f) are results for box inpainting. Abstract Deep generative models have become powerful priors for solving inverse problems, and various training-free methods have been developed. However, when applied to latent flow models, existing… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of representative inverse problem solvers: DPS, DDS, DAPS, and the proposed FlowLPS. (a) DPS performs a single gradient-based measurement correction at each timestep, but requires costly backpropagation through the denoising network. (b) DDS and PnP decompose this process into several gradient updates implicitly assuming a locally linear manifold. However, because the data manifold is highly non… view at source ↗
Figure 3
Figure 3. Figure 3: Manifold-preserving update. The clean image is itera￾tively updated using Langevin optimization combined with hybrid noise sampling using pCN, ensuring that the resulting states remain on the data manifold. where ρt ∈ [0, 1). The acceptance probability is always 1 since the target distribution p(x1) is standard normal. In our practical implementation, we initialize the chain with the model’s noise estimate… view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative comparison on FFHQ and DIV2K datasets [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effect of the number of Langevin steps (NL). (a) Gaussian deblurring and (b) Super-resolution (×12). Using no Langevin steps (NL = 0) leads to overly smooth and blurry reconstructions due to optimization getting stuck in poor local minima. Excessive Langevin updates (NL = 15) introduce instability and high-frequency noise. Moderate Langevin steps NL = 5 − 7 achieves an optimal balance, producing detailed r… view at source ↗
Figure 6
Figure 6. Figure 6: Quantitative impact of Langevin dynamics steps (NL). While pixel-level metrics decline with increasing NL, perceptual metrics improve until NL ≈ 5 − 7. This sweet spot balances fidelity and perceptual quality. Excessive stochasticity (NL > 10) degrades all metrics. We experimented with varying the number of Langevin steps NL on 200 FFHQ images. We first set the total Langevin-proximal optimization step NP … view at source ↗
Figure 8
Figure 8. Figure 8: Reconstruction MSE error vs. Timestep t. (a) Deblur￾ring, (b) Super-resolution. The plot shows the relative MSE of the estimated clean image D(z ∗ 0|t ) for deblurring (averaged over Gaus￾sian/Motion) and super-resolution. The error minimizes around t ≈ 0.2 for deblurring and t ≈ 0.3 for super-resolution, justifying the use of task-specific truncated time schedules. A detailed summary of hyperparameters fo… view at source ↗
Figure 9
Figure 9. Figure 9: Visual artifacts from excessive Langevin steps. (a), (b) Gaussian deblurring, (c), (d) Motion deblurring. NL = 0 yields over-smoothed results. NL = 15 introduces speckled, high-frequency noise. NL = 6 produces clean, high-fidelity images. for pCN steps (4 → 1). This strategy aims to balance early￾stage exploration with late-stage exploitation. As shown in [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Effect of multiple pCN steps. Increasing pCN steps leads to progressive over-smoothing [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance vs. Proximal Steps (NP ). Performance gains show diminishing returns beyond NP ≈ 10, suggesting that NP = 15 serves as a robust operating point for maximum quality, while lower NP remains viable for efficiency. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Visual evidence of generative bias in FLAIR. Top: Gaussian Deblurring, Bottom: Super-Resolution. While FLAIR generates sharp images, it ignores the measurement’s underlying structure in favor of its generative prior. Note how FLAIR al￾ters facial features (e.g., eye shape), prioritizing the generation of "plausible" faces over the recovery of the true identity. In contrast, FlowLPS faithfully recovers the… view at source ↗
Figure 13
Figure 13. Figure 13: Box Inpainting results on FFHQ and DIV2K. [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Super-Resolution(×12 results on FFHQ and DIV2K. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Gaussian deblurring results on FFHQ and DIV2K. [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Motion deblurring results on FFHQ and DIV2K. [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Random Inpainting results on FFHQ and DIV2K. [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
read the original abstract

Deep generative models are powerful priors for imaging inverse problems, but training-free solvers for latent flow models face a practical finite-step trade-off. Optimization-heavy methods quickly improve measurement consistency, but in highly nonlinear latent spaces, their results can depend strongly on where local refinement is initialized, often degrading perceptual realism. In contrast, stochastic sampling methods better preserve posterior exploration, but often require many iterations to obtain sharp, measurement-consistent reconstructions. To address this trade-off, we propose FlowLPS, a training-free latent flow inverse solver based on Langevin-Proximal Sampling. At each reverse step, FlowLPS uses a few Langevin updates to perturb the model-predicted clean estimate in posterior-oriented directions, providing stochastic initializations for local refinement. It then applies local MAP-style proximal refinement to rapidly improve measurement consistency from the Langevin-updated estimate. We additionally use controlled pCN-style re-noising to stabilize the reverse trajectory while retaining trajectory coherence. Experiments on FFHQ and DIV2K across five linear inverse problems show that FlowLPS achieves a strong balance between measurement fidelity and perceptual quality, with additional experiments on pixel-space inverse problems and phase retrieval.

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 introduces FlowLPS, a training-free solver for linear inverse problems with latent flow models. At each reverse diffusion step, it performs a small number of Langevin updates on the model-predicted clean latent to generate posterior-oriented perturbations, followed by local proximal (MAP-style) refinement to enforce measurement consistency; controlled pCN-style re-noising is used to stabilize the trajectory. Experiments on FFHQ and DIV2K across five linear inverse problems (plus pixel-space and phase-retrieval cases) report that the method achieves a favorable balance between measurement fidelity and perceptual quality.

Significance. If the empirical balance is robustly verified, the work would offer a practical compromise between optimization-heavy and full-sampling approaches for flow-based inverse solvers, reducing sensitivity to initialization while limiting iteration count. The controlled re-noising mechanism is a concrete algorithmic contribution that could generalize to other latent diffusion or flow pipelines.

major comments (2)
  1. [§3.2] §3.2 (Langevin-Proximal Sampling procedure): The central claim that a few Langevin updates reliably produce posterior-oriented initializations from which proximal refinement improves fidelity without eroding realism rests on an unverified assumption in highly nonlinear latent spaces. No mixing-time analysis, effective sample size diagnostics, or trajectory diagnostics are supplied to bound the required number of steps or to show that the short stochastic trajectory points toward high-posterior regions rather than remaining local noise.
  2. [§4] §4 (Experiments): The reported balance on FFHQ and DIV2K lacks error bars, statistical significance tests, and systematic ablations on the number of Langevin updates per reverse step (a free parameter listed in the method). Without these, it is difficult to confirm that the observed improvement is attributable to the proposed initialization rather than to the proximal step alone or to dataset-specific tuning.
minor comments (2)
  1. [§3] The notation for the proximal operator and the re-noising schedule could be made more explicit with a single equation block that defines all symbols used in the algorithm pseudocode.
  2. [Figures 2-4] Figure captions should explicitly state the number of Langevin steps and proximal iterations used for each method variant shown.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the recommendation for major revision. We address each major comment point by point below, indicating planned changes to the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Langevin-Proximal Sampling procedure): The central claim that a few Langevin updates reliably produce posterior-oriented initializations from which proximal refinement improves fidelity without eroding realism rests on an unverified assumption in highly nonlinear latent spaces. No mixing-time analysis, effective sample size diagnostics, or trajectory diagnostics are supplied to bound the required number of steps or to show that the short stochastic trajectory points toward high-posterior regions rather than remaining local noise.

    Authors: We agree that stronger empirical support for the short Langevin trajectories would improve the manuscript. While a full mixing-time analysis lies outside the scope of this practical, training-free method, we will add trajectory diagnostics (e.g., plots of measurement consistency and perceptual metrics over the Langevin steps) and an ablation on the number of updates to demonstrate that the perturbations consistently improve the starting point for proximal refinement. These additions will be included in the revised §3.2 and supplementary material. revision: partial

  2. Referee: [§4] §4 (Experiments): The reported balance on FFHQ and DIV2K lacks error bars, statistical significance tests, and systematic ablations on the number of Langevin updates per reverse step (a free parameter listed in the method). Without these, it is difficult to confirm that the observed improvement is attributable to the proposed initialization rather than to the proximal step alone or to dataset-specific tuning.

    Authors: We acknowledge that the experimental section would be strengthened by greater statistical rigor. In the revised manuscript we will report error bars over multiple random seeds for all quantitative tables, add paired statistical significance tests for the main comparisons, and include a systematic ablation study varying the number of Langevin updates per reverse step on both FFHQ and DIV2K. These changes will clarify the contribution of the Langevin initialization. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic procedure validated on external benchmarks

full rationale

The paper describes FlowLPS as a training-free algorithmic combination of a few Langevin updates in latent space followed by proximal MAP refinement, plus controlled pCN re-noising. This is presented as a practical procedure whose performance is measured on public datasets (FFHQ, DIV2K) across five linear inverse problems. No equations, fitted parameters, or self-citations are shown to reduce the claimed balance of fidelity and perceptual quality to a quantity defined by the same experiments. The derivation chain consists of standard Langevin dynamics and proximal operators applied to flow models; it does not collapse by construction to its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard assumption that pretrained flow models encode useful priors and on a small number of algorithmic hyperparameters whose values are not derived from first principles.

free parameters (2)
  • number of Langevin updates per reverse step
    Chosen to provide stochastic initializations without excessive compute; value is not derived and must be set by the user.
  • proximal refinement strength
    Controls how aggressively measurement consistency is enforced after Langevin perturbation.
axioms (1)
  • domain assumption Pretrained latent flow models provide a useful approximation to the data prior for posterior sampling in inverse problems.
    Invoked when the method uses the flow model to generate clean estimates that are then refined.

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