Title: Solving Inverse Problems with FLAIR
Venue: NeurIPS 2025
Accepted: yes
Keywords: inverse problems

diffusion models

flow models

stable diffusion

posterior sampling

variational inference

score distillation sampling
Abstract: Flow-based latent generative models such as Stable Diffusion 3 are able to generate images with remarkable quality, even enabling photorealistic text-to-image generation. Their impressive performance suggests that these models should also constitute powerful priors for inverse imaging problems, but that approach has not yet led to comparable fidelity.

There are several key obstacles: (i) the data likelihood term is usually intractable; (ii) learned generative models cannot be directly conditioned on the distorted observations, leading to conflicting objectives between data likelihood and prior; and (iii) the reconstructions can deviate from the observed data.
We present FLAIR, a novel, training-free variational framework that leverages flow-based generative models as prior for inverse problems. To that end, we introduce a variational objective for flow matching that is agnostic to the type of degradation, and combine it with deterministic trajectory adjustments to guide the prior towards regions which are more likely under the posterior. 
To enforce exact consistency with the observed data, we decouple the optimization of the data fidelity and regularization terms. Moreover, we introduce a time-dependent calibration scheme in which the strength of the regularization is modulated according to off-line accuracy estimates. 
Results on standard imaging benchmarks demonstrate that FLAIR consistently outperforms existing diffusion- and flow-based methods in terms of reconstruction quality and sample diversity. 
Source code is available at https://inverseflair.github.io/.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce FLAIR, a training-free variational approach to exploit flow-based generative models as priors for solving inverse problems. In particular, the authors approximate the true posterior with a Gaussian distribution and optimise the mean of this Gaussian.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\nThe authors build on the variational approach to solving inverse problems with diffusion models, extending it to latent flow models. They propose a novel reparameterised variational distribution in Proposition 2. The paper is generally well-written (with exception of some minor inconsistencies) and clearly organized. Given the development of large scale latent flow models, i.e., StableDiffusion 3, the development of scalable methods to use these models as priors for inverse problems is important. \n\nThey show remarkable experiments on both FFHQ and DIV2K for several linear inverse problems and even show results for image editing. Also, the authors include detailed comparisons and ablations. Further, even showing failure cases (Figure 9 in the Appendix) is a good standard for transparency. \n\n\n### Weaknesses\n\n- My main issue is the similarity of FLAIR with RED-diff [1]. Several equations are exactly the same, e.g., Equation (7) in the manuscript and Equation (7) in [1] or Equation (8) in the manuscript and Lemma 1 in [1]. In my opinion the differences to RED-diff should be discussed more.  \n- In Line 111 you write that you set $\\sigma_x^2 = 0$. However, for this choice the variational distribution $q(x_0 | y) = \\mathcal{N}(\\mu_x, \\sigma_x^2)$ is not defined and the mathematical framework breaks down, \n- I do not follow the argumentation for the posterior variance (starting line 223). Given that you set $\\sigma_x^2 = 0$ (or i guess in the numerical experiments $\\sigma_x^2 = $ some small constant?) you get an optimisation problem in Equation (5), where you approximate the true posterior with a Gaussian $\\mathcal{N}(\\mu_x, \\sigma_x^2)$. However, it is not evident why optimizing Equation (5) multiple times would yield samples from the posterior. I would expect that you obtain the same mean $\\mu_x$ everytime, given that you try to solve an optimisation task in Equation (5). Could you clarify the reasoning behind this claim? \n- Due to the non-linear nature of the VAE every linear inverse problem in pixel-space becomes a non-linear inverse problem in latent space. Given that you are therefore always dealing with non-linear inverse problems, it would be interesting to also include a non-linear forward operator $A$ in pixel-space (e.g., non-uniform deblurring or phase retrieval).  \n\nMinor points:\n- There are several inconsistencies regarding the notation. In Equation (2) you refer to the observations both with $z$ and with $y$. Same in Line 224. Further, you use both $A$ and $\\mathcal{A}$ in Section 3.1.\n\n[1] Mardani et al. \"A variational perspective on solving inverse problems with diffusion models\" (ICLR 2024)\n\nQUESTIONS:\nI have a few questions and suggestions:\n- As you deal with latent flow models, it would be good to explicitly have the Encoder/Decoder of the VAE as part of the Algorithm 1. Currently the notation is a bit inconsistent, in Algorithm 1 you write $\\mu_x = A^T(y)$ and in line 146 you write $\\mu_x = E(A^T y)$.  \n- In line 178 you write that you use stochastic gradient descent to enforce hard data consistency. What is stochastic for the optimisation problem $\\arg \\min_{\\mu_x} \\| y - A(\\mu_x) \\|^2$? \n- You write in your abstract (line 13) \"enforce exact consistency with the observed data\". Given that you have noisy observation you almost never want to *exactly* fit the observations, because you do not want to overfit to the observation noise. \n- At several points you claim that FLAIR is able to recover atypical modes (e.g., line 254). I do not fully see that the experiments support this claim. Which of the results corresponds to atypical modes?\n- I think in Equation (8) the expectation over the time $t$ is missing (given the definition of $\\mathcal{R}(x)$ in Equation (7) as a time integral.) . \n \nExperiments:\n- In Table 2 you obtain one of the highest PSNRs (both on FFHQ and DIV2K) if you do not use HDC, DTA and CRW. Why could this be the case?\n- Ablation on Hard Data Consistency (Line 240). How do the observations $y$ influence the optimisation process (Algorithm 1) if you drop the HDC? Only through the initial input $\\mu_x = A^T(y)$?\n- What is $\\nu = 0.01$ in Table 1? In Line 204 you write that you use an observation noise of $0.5$%\n- I guess that the number of function evaluations of the forward operator $A$ and the Decoder of the VAE gets quite high, given that you solve $\\arg \\min_{\\mu_x} \\| y - A(\\mu_x) \\|^2$ in every iteration in Algorithm 1. Are there any ways to reduce the number of these function evaluations? If you would extend your method to other inverse problems (e.g., Computed Tomography) the evaluation of the forward operator $A$ can get quite expensive and this would be a bottleneck of the method.\n\nLIMITATIONS:\nThe authors address limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe author proposes FLAIR, a scheme that combines latent generative modeling, flow matching, and variational inference into a unified solution for inverse problems. Their designed DTA (Deterministic Trajectory Adjustment) and CRW (Calibrated Regularizer Weight) modules improves the performance of flow based image generation priors in solving image inverse problems. The proposed framework and modules are interesting. The author tested the proposed method on image restoration, image super-resolution (8x and 12x), and image motion deblurring tasks, and demonstrated that FLAIR visually outperforms other existing methods.\n\nSTRENGTHS AND WEAKNESSES:\nThe author proposed FLAIR, which provides a new solution for image inverse problem by combining flow-base image generative model priors, and achieve enhanced performance in visual. I am concerned about the innovativeness of this study and significance of experimental results (especially PSNR/SSIM metrics). See Questions and Limitations for detalis.\n\nQUESTIONS:\n1. Overall, this study appears to present a progressive innovation building on existing work. To enhance clarity, the authors could consider reorganizing or more prominently emphasizing the core insight or innovation of their approach. \n\n2. In abstract, the author proposes several challenges to existing methods, such as \"the encoding into a lower dimensional latent space makes the underlying (forward) mapping non-linear\" and \"learned generative models struggle to recover rare, atypical data modes during inference\". How does the FLAIR studies and addresses these challenges? What insights will it bring to this field?\n\n3. Regarding methodology, clarify how the weighting parameters within the Calibrated Regularizer Weight module are determined. Are these parameter settings, along with task-specific variables (inpainting box size, zoom-in factors, kernel size, SNR), treated as independent?  Suggest the author to provide more explanations or proofs for optimizing convergence using Calibrated Regularizer Weight and Deterministic Trajectory Adjustment.\n\n4. While FLAIR demonstrates strengths, its performance on PSNR/SSIM metrics does not currently show an advantage compared to other image generation methods; Suggest the author to clarify/explain this point. In addition, there is a lack of comparison with non-flow models (e.g., esrgan, deblurgan).\n\n5. While noise is modeled as additive Gaussian (Line 78), real-world image inverse problems often involve non-Gaussian noise.\n\n6. Can the author provide the training time of the prior models and the inference time on each task.\n\n7. Given that prior work (e.g., Singanallur V. Venkatakrishnan et al., Plug-and-Play Priors for Model Based Reconstruction) has explored decoupling data fidelity and regularization via optimization algorithms, could the authors elaborate on how their hard data consistency module specifically differs from this established concept of decoupled optimization?\n\nLIMITATIONS:\nIn the Discussion section, the authors should also provide future optimization directions regarding the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents FLAIR (Flow-based Latent Adaptive Inference with deterministic Re-noising), a training-free variational framework that uses a flow-matching latent diffusion prior (Stable Diffusion 3) to solve inverse imaging tasks—single-image super-resolution, inpainting, and motion deblurring. Hard data-consistency (HDC) alternates regularizer updates with projections onto the measurement manifold, ensuring exact fidelity to the observation and Time-dependent calibrated weighting (CRW) scales the regularizer along the diffusion trajectory using offline error statistics\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n1. First integration of flow-matching priors and calibrated weighting into a unified inverse-problem solver.\n2. This paper shows that a single latent flow prior can tackle multiple inverse problems without retraining.\n3. Experiments on FFHQ and DIV2K show lower LPIPS/FID and sharper visuals than ReSample, FlowDPS, RSD and FlowChef across ×8/×12 SR, inpainting and deblurring.\n4. Ablations confirm that DTA and CRW are crucial\n\n**Weaknesses:**\n1. DTA resembles classifier-free guidance; CRW is a heuristic weighting fitted offline hence incremental novelty.\n2. Per-image solver needs 50 function evaluations on an RTX 4090—runtime not quantified.\n3. Applicability limited to VAE-based latents of SD3; extension to pixel-space flows not shown.\n\nQUESTIONS:\n1. Provide mean ± std over ≥ 3 random seeds and paired t-tests versus FlowDPS for LPIPS/FID.\n\n2. Have you run a MOS or 2AFC study? If not, include at least NIQE or aesthetic-LPIPS to rule out metric over-fitting.\n\n3. Does FLAIR handle non-Gaussian noise or compressive sensing (non-linear) operators? A small experiment would strengthen claims of “degradation-agnostic.”\n\n4. How does CRW compare to a linear t weighting or a constant λ on all tasks?\n\nLIMITATIONS:\nAuthors acknowledge dependence on SD3 biases and new hyper-parameters but should also discuss: (i) high energy footprint of 50-step optimization on a GPU, and (ii) failure cases when the forward model is strongly non-linear.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper proposes FLAIR, which is a training-free variational framework for solving inverse problems using flow-based latent generative models. It replaces traditional score-based regularization with a flow-matching loss between the learned velocity field and a deterministic reparameterized trajectory.

STRENGTHS AND WEAKNESSES:
Strengths: 
- The paper is well written, with a clear structure and smooth presentation.
- It introduces flow matching into a variational inference framework.
- The reconstructed images exhibit fine details with strong perceptual quality. 
- The experimental section is thorough with comprehensive baseline comparisons and well-designed ablation studies.

Weaknesses: 
- The core components, such as flow matching, variational inference, and hard data consistency, are all drawn from prior works. The paper mainly assembles these ideas into a cohesive system without introducing fundamentally new algorithms or theoretical insights. 
- The method introduces several non-trivial hyperparameters, such as $\lambda_{\mathcal{R}}(t)$, $\alpha$, and gradient descent learning rate. A sufficient sensitivity analysis for different tasks would be helpful. 
- Although the method is training-free, it involves repeated evaluation of learned velocity fields, gradient steps, and projection steps, which could be computationally expensive. A runtime or memory comparison with baseline methods would be helpful. 
- L179 should be subsection A.4, not A.3. L222 should be subsection A.9, not A.5.

QUESTIONS:
See Weakness.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-03