Solving Inverse Problems with FLAIR
Authors: Julius Erbach
Dominik Narnhofer
Andreas Robert Dombos
Bernt Schiele
Jan Eric Lenssen
Konrad Schindler
Venue: NeurIPS 2025 poster ()
Keywords: inverse problems
diffusion models
flow models
stable diffusion
posterior sampling
variational inference
score distillation sampling
Primary area: deep_learning
TL;DR: We develop a variational approach to solve image inverse problems like super resolution, inpainting and deblurring for flow-based generative models.

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/.

Solving Inverse Problems with FLAIR
Julius Erbach 1.2
Dominik Narnhofer'
Andreas Dombos' Konrad Schindler'
Bernt Schiele?
Jan Eric Lenssen?
ETH Zürich
2 Max Planck Institute for Informatics, Saarland Informatics Campus
Abstract
Flow-based latent generative models such as Stable Diffusion 3 are able to gen- erate 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 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) 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 variational objective for flow matching that is agnostic to the type 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, tion scheme which the strength of the regularization is modulated according to off-line accuracy estimates. Results standard imaging benchmarks demonstrate that FLAIR consistently outperforms existing diffusion- and flow-based methods in terms of reconstruction quality and sample diversity. Source code available https://inversefllae www.a.......iiii..oooooo.o io/
introduce time-depentn dependent calibra-
1 Introduction
Flow-based generative models are at the core of modern image generators like Stable Diffusion or FLUX [14). Beyond image generation based on text prompts, these models have emerged as powerful data-driven priors for whole range of visual computing tasks. Their comprehensive representation of the visual world, learned from internet-scale training datasets, makes them an attractive alternative to traditional handerafted image priors. Often, they can be used without any task-specific retraining.
While it is evident that a model capable of generating photorealistic images should be suitable as prior (a.k.a. regularizer) for inverse imaging problems, a practical implementation faces several challenges. On the one hand, flw-based models normally operate in the diver--deessioall latent space of variational autoencoder (VAE), which means that the forward operator (the relationship between the observed, degraded image and the desired, clean target image) is no longer linear. On the other hand, the iterative nature of the generative process means that intermediate stages are corrupted with (time-dependent) random noise. Hence, one cannot explicitly evaluate their data likelihood, which renders the data term intractable. Moreover, learned generative models tend overly favor regions of the training distribution that have a high sample density. For test samples that fall in low-density regions, the prior will have a too strong tendency to pull towards outputs with higher a-priori likelihood, compromising fidelity to the input observations.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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-AR Vna no(2(,)) m(a,
Original
Po
Aly
ang mnin IV A(p=)l
Result
Figure 1: Starting from the adjoint based initialization, alternate between (i) regularizer updates via a flow-matching loss that aligns the velocity de of the variational distribution with the learned velocity field vo, and (ii) hard data consistency steps that project the current estimate onto the measurement manifold.
Here, we propose flow-based Jatent adaptive inference with deterministic re-noising (FLAIR), novel, training-free variational framework explicitly tailored to integrate flow-based latent diffusion models into inverse problem-solving. To the best of our knowledge, FLAIR is the first scheme that combines latent generative modeling, flow matching and variational inference into a unified formulation for inverse problems. Our main contributions are
A novel variational objective for inverse problems with flow-matching priors.
Deterministic trajectory adjustments guide the prior towards regions which are more consis- tent with the observed data.
Decoupled optimization of data and regularization terms, enabling hard data consistency,
A novel, time-dependent weighting scheme, calibrated via offline accuracy estimates, that adapts the regularization along the flow trajectory to match the changing reliability of the model's predictions, ensuring robust inference.
2 Related Work
Deep learning based priors. Deep learning-base methods typically follow one of two main
approaches: they either directly learn an inverse mapping [8, 30 [0] or aim to learn a suitable
prior, either through non-generative approaches like unrolled optimization networks [5] 29 36) or through generative models such as generative adversarial networks B3
7 46), or diffu-
sion performance in image generation tasks, sparking growing interest in leveraging them as priors for solving inverse problems, particularly through posterior sampling techniques.
40 52
respectively flow based models 31 33 The latter have demonstrated impressive
Posterior sampling. Although incorporating the implicit prior learned from denoising- based deep neural network seems straightforward, distinct challenges arise depending on the framework. While early stochastic approaches utilized blind universal denoisers to solve inverse problems (21). they rely heuristic scheduling and are limited to linear operators. Modern diffusion models formalize this process via fixed forward trajectories, yet this time-dependent structure renders the likelihood intractable [ A variety of approaches have been proposed for diffusion-basee posterior sampling 588
35 20]: enforcing the trajectory to stay on the diffusion in the spectral domain (5), utilizing range-null space decomposition during the reverse diffusion process (5 guidance by the pseudo-inverse of the forward operator (50)
19
respective noise manifold [ [2 61]. applying SVD
Many prior methods perform well in pixel space but are difficult to apply latent diffusion models due to VAE non- non-lincarity memory constraints. In order to cireumvent this issue, the authors of

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ReSample [48] rely on enforcing hard data consistency through optimization and resampling during the reverse diffusion process. PSLD 42 introduces additional objectives terms to ensure that all
gradient updates point to the same optima in the latent space. FlowChef [39 incorporates guidance into the flow trajectory during inference, whereas FlowDPS (26) separates the update step into two
components: one for estimating the clean image and another for estimating the noise.
In contrast FLAIR follows another class of posterior sampling-basee methods, which integrate diffusion priors with inverse problems by directly optimizing variational objective that approximates the data posterior [4 This framework was recently extended by RSD [62]. which incorporates repulsion mechanism to promote sample diversity and applied to latent diffusion models. A known issue with this type of optimization is mode collapse [40], which leads blurry results for these
methods. Our method targets this problem by introducing a deterministic trajectory adjustment.
3 Background
3.1 Inverse problems
In many imaging tasks, such as inpainting [5]. super-resolution (38] or tomographic reconstruc-
tion 47 one aims to recover target signal R" from distorted observation y E R". The
observation is regarded as the result of applying a forward operator A IR" + Rmm to the target signal, corrupted by additive Gaussian noise y E R with standard deviation dy.
y Ax +
(1)
In most practical applications, the forward operator A is either non-invertible or severely ill- conditioned, making ①I generally ill-posed.
Variational methods solve ill-posed inverse problems by minimizing energy functional
E(x. v) D(x, v) R(x).
2)
recover the solution.
Interpreted probabilistically via Bayes' theorem, the posterior distribution p(zly) is proportional to the product p(y) p(x). In the negative log-domain, this yields the data term D(r, y) log p(y| x))
and the regularizer R(x)
log p(x). Handcrafted priors based on regularity assumptions like 13 9 have long been the standard, but have largely been replaced by deep learning-bai based methods in modern data-driven schemes.
sparsity 44 45
10
3.2 Flow based priors
Models based on flow matching 31 learn time-dependent vector field (z,,,, t) that continuously transforms samples from a simple initial distribution P1 (a)) to a complex target data distribution Pu(z). Formally, this transformation is described by solving the ordinary differential equation (ODE):
d (z) = (x), E [0,
dt
where (xx) represents the trajectory of sample, evolving smoothly from an initial value drawn t=1 toward target value at / 0.
Since the integrated ODE path maps the simple distribution P1 (z) to the complex target Po(x), the learned flow-based model captures the structure of the data and can therefore serve as a powerful prior for solving inverse problems. To make this approach tractable for high-resolution data, adopt the latent diffusion model (LDM) framework [41), which shifts the generative process to a lower-dimensional latent space using pretrained autoencoder with encoder E R" - IRd and decoder D : Rd -> IR", where d < n. However, applying such priors to inverse problems introduces challenges, as the non-linearity of the VAE disrupts the linear relationship between measurements and the target signal, resulting in a nonlinear forward operator.
3.3 Variational flow sampling
To solve inverse problems from Bayesian perspective, we aim to sample from the posterior
p(zolv) (o))zo))p(z0))
(4)
3

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where the likelihood is given by p(3]20)) N (AAzo, IIId), and p(zo) represents the prior modeled by the flow-based generative model.
Inspired by previous work [34 62] we introduce a variational distribution (( = (i og) to
approximate the true posterior ()) by minimizing their Kullback-Leibler divergence:
( w) E arg min (L(g(xolv)) (pppoo)))) (
(5)
Rewriting the KL. divergence by means of the variational lower bound leads to: [[((roow)))l((x)) + logp(v)
K [Llg(zoww)ll<<oluv) = (og +
(6)
D(r. 9)
(2))))
Since a single Gaussian cannot capture a multi-modal posterior, we simplify to a deterministic approximation, setting o2 Equivalently, this corresponds to single-particle approximation in the sense of Stein variational methods [32] As shown in [51], rewriting Equaatin 6 uunde this approximation and extending it to the time-dependent noisy posterior yields:
l f(
arg min E (aolv) ((to y)
w(t) (
| log ((a))))))) V. logp(x;)) de
20
D(a, v)
R(n)
The first term Equation describes the data term D(a, y) and the second the regularizer R(x)), where the integral ensures optimization over the entire diffusion trajectory. Notably, the latter constitutes a weighted score-matching objective, where V, log p(xt) represents the score function [52], which may be extracted from pretrained diffusion or flow model.
The score of the noisy variational distribution depends on the forward diffusion process and can be computed analytically.
Note that for w()) B(t) 22 the weighted score-matching loss recovers the gradient of the diffusion model's evidence lower bound, so that optimizing it yields the maximum likelihood estimate of the data distribution [51). However, optimizing [Equation 7is costly, it requires computing the gradient through the flow model. As shown in (54) this can be circumvented by reformulating the regularizer terms of the Wasserstein gradient flow:
WEX w(t(
V n4 R(x) E, (x)al))
V z log (xzlv))
V log p(
(8)
noisy variational distribution
sccree noisy prior distribution
Note that optimizing only the regularization term, without the data term, time is equivalent the objective of Score Distillation Sampling (SDS)
40
4 Method
Flow Formulation. The variational formulation in Equation is formulated for the score, but can be reformulated into a denoising or eo parameterization [52]34] However, we are interested in variational objective that depends on the velocity field (a t), which characterizes the probabilistic trajectory that connects the noise and data distributions.
We propose to replace the score-based regularizer in the standard variational objective with a flow matching formulation, resulting in the following objective function:
Proposition 1.
|yy - f(w))
arg min E Eq(zo b)) ((zal)))
+
XR(()) Eg(asls) [110 () ue((rle))2 dt (9)
203
D(x. w)
R(mr)
V Ma R(xx) Beg(ael) AA (t)) (x
(10)

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The fow-matching, term that defines the regularizer arises by reparameterizing the variational dis- tribution to (z (g) MM((1 t)y t² I). This corresponds sampling via the deterministic map
(1ro <) = (1 1) mo + te, with ~ NN( T). By reformulating the score in terms of the target
velocity field uy we get:
V log ((rll)) = (1 - (xele) x
(11)
For the learned velocity Up (z 8) a similar approximation holds for a full derivation, see the supplementary material subsection A.3
u0 f) -tV, log p(z,) It
(12)
1-
We can therefore approximate the score of the noisy prior with our learned velocity field vi
V logp(x)) = (1 - t)vo (2,, 1) + + Iz
13)
Hard Data Consistency. Existing variational posterior sampling approaches [34 62 impose soft constraints on the data fidelity term ((z, y). In contrast, recent work 48) has demonstrated that, when sampling from latent diffusion models, enforcing hard data consistency generally leads to better reconstructions with improved visual fidelity. Our method shares this motivation, but differs in that we optimize over a variational distribution, ie.. we compute min Egtooly) |- log p(y [2)]] An additional advantage of this variational setup is that it allows to initialize the optimization variable with an adjoint based initialization ر E{A" A T y), with E being the encoder of the VAE and A the adjoint of the linear forward operator in pixel space. Other initialization strategies are also possible.
Accuracy Calibration. As our framework evaluates the trajectory at each time step, we aim to weight the regularizer's s contribution according to its reliability. The difficulty of the prediction task has been shown to depend on the network parameterization, as well as on the specific time step [22]. Since the regularization term R(x) in our approach is equivalent to the training objective of the pre-trained flow model, we can easily weight it by the expected model error, which we calibrate on a small set of images. Specifically,
sample N calibration images and compute the conditional flow matching objective for 100 linearly spaced time steps between and 1, then average the error over all images to obtain the expected model error each time step. Different functions of the model error can be chosen as weight for the regularizer. We choose:
Ave (t)
1 N
Σ
vo po( .t) u(( 9 [0)l²)
(14)
and set Are (t) = 0 for all t << 0.2, since the accuracy of SD3 is heavily degraded for low noise levels.
Deterministic Trajectory Adjustment. Score distillation sampling relies on the assumption that It = (1 - t) + le lies in a region of the learned prior that has reasonably high support/density. In practice, this is not always the case. When not tightly conditioned (usually with extensive text prompts), even the best available diffusion models assign low density to many plausible regions of the latent space, leading bad gradient steps. Therefore, we increase the probability of p(æ;) by additionally conditioning It on the estimated "end-point"
Proposition 2.
We introduce reparameterized variational distribution with mean that linearly interpolates between the posterior mean μlz and a model- guidel- ā1:
g(x+ v) N'' ((1 l)1zz + lori, (1)1))))
(15)
where क1 = It4he + (1-t (l))va(=+485 1 + +81) is a single-step velocity-based predictor, and [0, 1] controls the trade-off between deterministic guidance and random noise. This reparameter- ization induces the following reference velocity field:
ue () ad1 + v1 a²c It.
(16)
Intuitively, changing the formulation this manner ensures that the model relocates the sample to its expected position on the learned manifold rather than injecting arbitrary noise, which could drive it in a direction that has high prior likelihood but is not consistent with the observation. To further encourage exploration and avoid collapsing onto the trajectory of the adjoint measurement, we inject an additional stochastic component e during this process. A full derivation can be found in the supplementary material, subsecction A.3
5

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4.1 Algorithm
The following pseudo-code summarizes our method, integrating all the components discussed above.
We adapt the standard scheme [34 62 of linearly traversing time in a descending manner and stop at f = 0.2 as explained in (section 4 We choose a = 1-16 t. Gradient updates to enforce hard data consistency are performed using stochastic gradient descent. For further implementation details and
ablations, see (uussectionnn A.4
Algorithm 1: The FLAIR solver for inverse imaging problems
Input: P Pinntt AR. Ct y, A, Ug
Output: Mz
8 &~ NO0, I);
initial noise sample
for t to by At do
2t + (1 - t) uzz +tē;
b. sample noisy latent
(a () 4 8- E-xe. Ft. 1-
Va R 4- uo (Ire 1) uc (xx E);
Nr t uz - AR Vuo R;
b. update w.r.t. regularizer hard data consistency
Hs f- arg argmin 111 A(uz)) 21
é c~ N0, I);
&ι + It + (1 - t) 20 (22, t);
b predict deterministic noise b update noise estimate
8 - ab + v aze
Experiments
We evaluate the performance of FLAIR in a variety of inverse imaging tasks and compare it against several baselines, using the SD3 backbone without any fine-tumin tinettunigg We used several metries including SSIM (55), LPIPS [59] and patchwise FID [16] (pFID) to comprehensively assess the perceptual and quantitative quality of the reconstructions. FID is computed using Inception features on patches of 256x256 resolution. All experiments were performed on NVidia RTX 4090 GPU with 24GB of VRAM. For completeness we also show PSNR values, but point out that the metric is not well suited for our purposes: PSNR favors the posterior mean, while the goal of the variational approach is to sample from the posterior distribution. Accordingly. PSNR is known to prefer over- smoothed, blurry outputs over sharp ones (6) To demonstrate that our model can also produce accurate MMSE estimates, we performed ensemble predictions by running posterior sampling eight times and averaging the results. As shown in subsectioo A.ITI ensembling improves PSNR values while reducing LPIPS. This confirms that our samples are distributed around the posterior mean. Moreover, it shows that results closer to the posterior mean such those produced by baseline methods- - are perceptually farther from the ground truth (in LPIPS) compared to our samples.
5.1 Setup
Datasets. We utilize two high-resolution image datasets: FFHQ 23 and DIV2K (2) FFHQ consists of 70k diverse face images 1024x1024 resolution of which take the first 1000 samples. It is covering variations in age, pose, lighting, and ethnicity. DIV2K contains 800 high-quality images in 2K resolution that span a range of natural scenes with varied textures and structures.
Baselines. Our method is benchmarked against several recent inverse imaging solvers based on posterior sampling. Specifically, compare to ReSample [48), FlowDPS (26), FlowChef [39], and RSD [62]. The latter is used without repulsive term as it delivers better results. To ensure a fair and meaningful comparison, all methods are evaluated with the same number of function evaluations.
Problem Setting. We run and evaluate all methods at a fixed output resolution of 768 pixels. For single image super-resolution, we consider scaling factors of 8x and 12x. The corresponding low-resolution inputs are generated by bicubic downsampling. Motion blur is simulated with a blur kernel of size 61. For box inpainting, we mask large, continuous rectangles that cover approximately one third of the observation. All synthesized observations are corrupted with additive Gaussian noise, with standard deviation o2 of 0.5%.

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For inference on the FFHQ dataset, we use predefined text prompt of the form "A high quality photo of a face", and for DIV2K "A high quality photo of concatenated with an image-specific description retrieved by applying DAPE 56 to the observation.
5.2 Experimental Results
Inverse Problems. Our experiments clearly demonstrate that FLAIR outperforms existing flow-based approaches in terms of all perceptual metrics, see seele
In the case of image inpainting, our method produces high-quality reconstructions that fully leverage the power of the generative model and blend naturally into the surrounding context, avoiding degradations and artifacts that we observe in the baselines. In particular, FlowDPS tends to produce implausible textures in the inpainted regions, while FlowChef regularly fails to generate semantically consistent content at all.
For single-image super-resolution, FLAIR consistently delivers the most perceptually convincing and realistic outputs. Notably, the FID scores remain low for both x8 and x12 magnification, indicating an effective usage of the generative prior to overcome the increasing ill-posedness. Again, FlowDPS suffers from blur and low texture quality, whereas FlowChef tends to lose semantic coherence.
In motion deblurring, FLAIR also restores sharper and semantically more credible content than competing approaches, which often suffer from residual blur inconsistent details. The boost in reconstruction quality is quantitatively reflected by all metrics, confirming that FLAIR reconstructs images with high fidelity. For further qualitative examples, see subsection A.13
Table 1: Quantitative results with 50 NFE and o, 0.5%.
SR x8
SR x 12
Mition Deblurring
Inpainting FIDI,
Meihod
LPIPS.
FIDJ
SSIM1
PSNR1 LPIPS)
PIDJ
SSIMI PSNR1| LPIPS}
FIDJ SSIM) PSNR1 LPIPS)
SSIM: PSNR
FFHQ 768x768
ReSample
0.400
55.6
0.155
26.37
0.474
0.786
25.47
0.457
82.9
0.788
25.45
70.8
0,827
21.83
FlowDPS
0,374
38.5
0.756
29.24
0.413
44.0
0.741
28.05
0.431
54.3
0.732
27.64
0.344
42.5
0.771
19.19
RSD
0.391
51.7
0.776
29.69
0.462
71.7
0.743
28.11
0.458
77.3
0.743
27.67
0.478
73.3
0.736
21.97
FlowChef
0.341
30.5
0.760
28.42
0.373
46.5
0.730
27.00
0.406
40.2
0.716
25.81
0.394
69.8
0.780
18.18
Ours
0.213
13.3
0.777
29.54
0.271
16.2
0.740
27.71
0.236
10.7
0.772
29.61
0.184
8.7
0.828
23.69
DIV2K
768
768
ReSample
0.625
0.643
0.562
20.85
0.556
79.7
21.79
FlowDSS
0.476
44.4
0.567
23.01
0.547
54.0
0.528
21.79
0.558
65.5
0.536
21.88
0.328
29.2
0.692
21.71
RSD
0.591
23.45
0.684
95.7
0.523
21.96
0.638
97.6
0.551
22.
0.464
63.9
0.678
23.23
FlowChef
0.490
16.5
0.539
21.84
0.525
43.8
0.492
20.52
0.561
49.6
0.486
19.90
0.489
58.3
0.659
20.87
Ours
26.5
0.607
23.30
0.421
32.1
0.525
21.39
0.315
21.1
24.44
0.163
0.815
Posterior Variance. To demonstrate that FLAIR does not suffer from mode collapse, we assess the posterior variance Var[zly] for the task of x12 Super Resolution, by drawing 32 samples for fixed observation y and computing their pixel-wise variance. We conduct that experiment for our FLAIR approach, for RSD with repulsive term, and for FlowDPS [26. The example in ooFiguure ilillustrates that FLAIR has the highest sample diversity, which is also reflected in the corresponding variance maps. Notably, the sample variance is concentrated in regions with high-frequency textures. This indicates that method reliably reconstructs the posterior, whose low-frequency part is, in the super-resolution setting, tightly constrained by the likelihood term.
Editing. Beyond image restoration, observe that method also performs remarkably well for text-based image editing, simply presenting suitable target prompts during inpainting. [Figure illustrates a variety of edited images generated from the same photograph with the help of the depicted masks and prompts.
Pixel Space Experiments We also implement FLAIR in pixel-space using the model from 33 trained on CelebA-HO resized to 256x256 px. We compare DDNM (53), DPS [T], Moment Matching 43 and ITGDM (50] We tuned the hyperparameters for all baselines, which we report subsection A.9 As shown in Table 5.2) our method also outperforms previous work in the pixel space, demonstrating its broader applicability.
7

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Box inpainting
observation
FlowDPS
FlowChef
FLAIR
ground truth
12 super- super-reesssseeeeeeeee resolution
observation
FlowDPS
FlowChef
FLAIR
ground truth
Motion deblurring
Figure 2: Qualitative comparison. FLAIR produces posterior samples of high perceptual quality while maintaining high data likelihood. Best viewed zoomed in.
Table 2: Quantitative results with 50 NFE and o. 0.5% In-painting and Super-resolution (x8).
Inpainting
SR
Method
LPIPS4
FID.
SSIM
PSNR1
LPIPS
FIDI
SSIM
PSNR
DDNM
0.158 + 0.042
26.9
0.732 + 0.037
18.31
2.94
0.199
0.052
31.9
0.635
0.079
23.59 + 1.64
DPS
0.195 + 0.064
30.2
0.689
0.077
20.49
2.811
0.172
0.658
0.088
MM
0.161 + 0.054
28.8
0.728 + 0.062
20.59
+ 3.37
0.172
0.051
29.1
0.669 + 0.083
24.65
1.97
HGGM
0.195 + 0.064
30.2
0.689 4 0.077
20.49 = 2.81
0.157
0.052
26.5
0.677 + 0.084
24.98 - 2.07
FLAIR
0.097 + 0.035
14.2
0.831
0.031
2.66 21.87
0.143
0.039
22.9
0.712
0.076
25.93
1.96
5.3
Ablation Studies
We systematically analyze the impact of key design choices in our method. Specifically, we ablate the deterministic trajectory adjustment, the use of hard data consistency, and the calibration of the regularizer weight for 12 super-resolution, using subset of 100 samples from the FFHQ and DIV2K datasets. Quantitative and qualitative results are shown in Table 3 and Figure 6 respectively.
Hard Data Consistency (HDC). Dropping the hard data consistency degrades both metrics, with PSNR being particularly affected due to poorer aligmment with the input observation, which is also evident in the visual example: the reconstruction is plausible but deviates from the observation.
Deterministic Trajectory Adjustment (DTA). The biggest performance drop compared to the full setup occurs when removing the deterministic trajectory adjustment, random noise sampling harms

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~ p(x1y)
E [zr [y]
Std y
Ours
FlowDPS
RSD
Zoomed-in reconstructions for x12 Super Resolution. We show posterior samples (col. 1 1-4) of FLAIR, FlowDPS, and RSD, posterior mean and standard deviation (over 32 samples, col. 0.16
Figure 3:
5,6).
"clown with red nose. makeup Figure 4: Edited images shown alongside original, with prompts: "A high resolution portrait of a..."
Original image
"man wearing aviator glasses."
"man weating
"man with a Mike Tyson facial
the gradient updates in low-density regions of the prior. The reconstruction appears overly smooth and lacks texture details.
Calibrated Regularizer Weight (CRW). Replacing our calibrated regularizer weight with A (t) = also has a strong impact on perceptual quality: the result is visibly blurred if one ignores the changing accuracy of the regularizer along the flow trajectory.
Table 3: Ablation study for x 12 super-resolution on DIV2K and FFHQ. Model components are individually switched on off.
HDC
DTA
CRW
FFHQ
DIV2K LPIPS
LPIPS 1
PSNR
PSNR
/
0.259
27.45
0.427
21.05
X
/
0.297
27.17
0.467
20.82
X
V
0,432
27.20
0,622
21.69
0.363
28.58
0.583
21.98
X
0.392
28.33
0.605
21.99
Legend. HDC: Hard Data Consistency; DIA:
Deterministic Trajectory Adjustment: CRW: Calibrated Regularizer Weight, / = included, X = ablated.

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Box inpainting
observation
IIGDM
MM
FLAIR
ground truth
X super-resolution
observation
IIGDM
MM
FLAIR
ground truth
Figure 5: Qualitative comparison. FLAIR in pixel space produces posterior samples.
Ground south
HIXV, DTA,, CRW/
HЕCK DTA, CKW
HDCV, DTAX, CRWV
HIVV. DTAY, CRWX
НВСX DTAX CRWX
Figure 6: Qualitative samples from the ablation study on 12 Super Resolution.
Conclusion and Limitations
We have presented FLAIR, a training-free variational framework for inverse problems that uses a flow-based generative model as its image prior. By combining the power of (latent) flow-based models with a principled reconstruction the posterior distribution, FLAIR addresses key limitations of existing methods. First, it is able to target the generation towards images, which match the observation, by aiding the cegradationnagnosss Second, it enables hard data consistency without sacrificing sample diversity, by decoupling the data consistency constraint from the regularization, while adaptively reweighting the latter according to its expected accuracy, calibrated offline. Experiments with different image datasets and tasks confirm that FLAII consistently achieves higher reconstruction quality than existing baselines based on either flow matching or denoising diffusion. Notably, our proposed method achieves, the same time, excellent perceptual quality, close adherence to the input observations, and high sample diversity.
flow matching loss with deterministic noise vectors.
Evidently, FLAIR inherits the limitations of the underlying generative model. These include biases caused by the selection training data, constraints w.r.t. the output resolution, and a limited ability to recover nut-of-distribute needed to control the deterministic trajectory adjustment. We note that high fidelity image restoration methods potentially be misused for unethical image manipulations.
modes. Furthermore, our approach introduces additional parameers
Acknowledgments This work was funded, in part, by the Max Plank ETH Center for Learning Systems and Huawei Technologies Oy (Finland) Co. Ltd.
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Answer: [NA]
Justification: We did not use LLMs any important, original non-standard component of the core methods.
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21

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Supplementary Material
In the following, we provide detailed line-by-line derivations of the mathematical formulations used in the paper, as well as additional implementation details and experimental results.
A
Derivations
A.1 Derivation of flow-based variational formulation
The linear conditional flow and it's corresponding velocity defined as:
the (To c) = (1 -1-1) t) xo + tc, (0, U (2, di3
(17)
(18) The score of the noisy variational distribution can be analytically computed with:
dt (1) (sra)))l))
((<xlu) = -V1(1 1) 12 I):
(19)
c
Va, log (xxx (y) = t
(20)
We compute deh (arole -0o + and v. (asele le) and insert it into Equation 18
Up (xzle) 1-6
(21)
Solving Equation 21 for e and inserting in Equation 20 gives:
Va. log ((rl))) = (1 (z,le)
(22)
t
For the learned velocity uo (2x, t) similar approximation holds:
vo (1rr, t) .. log p(x+) 2t.
(23)
t
Hence, we can approximate the score of the noisy prior with our learned velocity field ng
(1 - t)va (2e, 1) + 25.
V 2. logp(x.)
(24)
and we see that for (tt) we obtain the conditional flow matching objective for R(x). We
therefore set (1) = T-7
+ and end up at our final objective:
II [(
Eycreled [Ilvo(2e,)) t) - (x((aa)))]]] dt
arg min (g(o)))) g( v))
(25)
2
D(z,y)
R(x))
Again, the gradient step for the regularizer becomes:
n. R(x)= Er,a(ss))) [vo( us (xalcc)
(26)
A.2 Derivation of trajectory adjusted flow-based variational formulation
To achieve the proposed trajectory adjustment, we modify the forward process to:
i Zt+dt + (1 t (d))vo(#r4al) dt),
(27)
Ft = - (1- + (as: + v1 a²c)
(28)
where é is the noise vector prediction from the last optimization iteration. This induces variational distribution:
q(E, v) = - ((1 t)p + taž,, fa - a a²)4)
(29)
22

---

leading to score of
1
Zx log (( ( = [2 (21- a²)
tt1 a²== w1
(30)
The velocity field is again computed by Equation 18 We start by defining the flow:
(1 ()) (1 t):0 + (a.) + v - a²E
(31)
The resulting derivative reads
d c0== - Io + V a²c,
2/ ( = a.1
(32)
dt
and the inverse becomes
30 = v/ (zz c)= It. taxt
tv 2
(33)
t
Plugging these results into Equation 18
Ue (e ()) aK81 v1 - a Zt.
(34)
1 t
A.3 Derivation of Score from Flow
The score matching objective reads as:
1
Va, (nx) arg min 100 w(() 10 (0(())))) a(t)² (x+ m( ())
0
(35)
where,
o1)2 (Ir p(z0, f))
V., log pr zo),
(36)
with pr (zt ..) = ((H I).. Note that as usual we assume 44 (zo) being linear in Zo-
Equation 35 is solved by:
Zx, (x)))())) Bp((zolzs) Vz, loglo(r)
(37)
and can be written as:
log (((e)) = (xx (((Err x)).t))
(38)
olt)²
In the case of OT matchingg we obtain
It = (1 - t) + tr
(39)
Ii ~ (0, Id) and p(æ)(=1)) = =((1 - 1) t²). The optimal velocity under the flow matching loss given by:
o vx(r,, 1) Rlz xz)-
(40)
Expressing 21 2t (1-11) we can insert into Equation 40 and obtain:
E f
xt = Ie [1[21 Do rz].
(41)
Inserting in Equation 38
leads to:
= Et (1 () - [T(21 20 rz))
V log (x()))
(42)
t2
which for o² (xx, E[xr 2n
xxa] reads
(1 t) (x + 24
log ((gp(1))
(43)
23

---

A.4 Implementation details
Flow Model and Regularizer Settings. As flow matching model, we us Stable Diffusion 3.5- Medium, which has been released under the Stability Community License. The classifier-free guidance scale is set 2 for all experiments. To minimize the regularization term, we use stochastic gradient descent with learning rate of 1.
Data Likelihood Term. We use stochastic gradient descent for the minimization of the data term towards hard data consistency. For numerical stability, the squared error is summed over all measure- ments instead of computing the mean. The learning rate has be adjusted accordingly, compensate for the varying number of measurements Moreover, the minimization is terminated with early stopping once the likelihood term reaches 10 4 len(y), not overfit the noise in the image observation.
Super-resolution. We employ bicubic downsampling the forward operator, implemented in [53. The learning rate set to 12 for x 12 reper-resolution and to for x8 super-resolution.
Motion Deblurring, A different motion blur kernel is created for each sample using the MotionBlur package (8) available via github, with kernel size 61 and intensity 0.5. The learning rate for our data term optimizer is set to 10-1
Inpainting. For inpainting on FFHQ we always use the same rectangular mask at fixed position, chosen such that it roughly masks out the right side of the face (Figure 3 For DIV2K we also use fixed mask for all samples, consisting of six randomly generated rectangles (Figure 6)
Data. We use the publicly available Flickr Faces High Quality dataset 244, which is realeased under the Creative Commons BY 2.0 License and the DIV2K dataset [3]. which is released under research only license. For FFHQ we use the first 1000 samples of the evaluation dataset and for DIV2K we use the 800 training samples. We downscale both datasets 768 768 px by applying bicubic sampling so that the shorter edge of the frame has 768 px and apply central cropping afterwards.
A.5 Baselines
For comparability, all baselines use Stable Diffusion 3.5-Medium and the same task definitions as in A.4
FlowDPS [26] The standard DPS implementation [27] is applied with 50 NFE, classifier-free guidance scale of 2, and step sizes of 15 for inpainting and 10 for all other tasks.
FlowChef 39
Additionally, 27 is employed for FlowChef as well, using 200 NFE for inpainting and 50 NFE for all other tasks, classifier-free guidance scale of 2, and a step size of for all tasks.
62 We implement RSD for flow-mater wwwchingnn models by applying Proposition 1 with w(() L, resulting in weighting term consistent with the original RSD approach. However, we omit the pixel-space augmentation as it negatively affected performance when combined with the SD3 VAF. Consistent with the original findings from RSD, we observed that incorporating the repulsive term improves sample diversity but reduces fidelity. Therefore, we set the repulsive term to for all results presented in the table, employing it exclusively for comparing posterior variances.
Repulsive Score Distillation (RSD)
ReSample 48 We re implement ReSample for flow-matching by setting a; = -+(1-1)² Further-
(1-0)²
more, we compute the hard-data consistency at every iteration as larger skip steps seem to harm performance. We set the learning rate of the data term optimizer to 15 for all inverse problems.
PSLD (42 Our attempt to adapt PSLD following (27 using 500 NFE, a classifier- free guidance scale of 2, and step sizes of I (x 12 super-resolution), 0.5 (x8 super-resolution and motion deblurring) and 0.1 (inpainting)- not yield meaningful results.
A.6 Regularizer weighting
Figure disspaas the mean and standard deviation of the conditional flow matching loss LCFM as function of I, estimated over 100 samples. The loss function starts with high values at = 1, decreases over time, but then starts to rise again, and when reaching / x 0.2 even exceeds its initial value. The rising loss when approaching = 0 is due, in part, to the increasing difficulty of distinguishing high-frequency image content from residual noise. Another factor is that near t = 0 the model
24

---

operates in highly sensitive regime where small prediction errors can cause disproportionatel large deviations from the target, making accurate flow estimation particularly challenging in the final stages of the trajectory. We therefore modulate the regularization term according to the model error. Different weighting functions for (fCFM) could be chosen that fulfill the condition AR(t=))
We simply take the reciprocal of the model error AR()) - LCFM, as the regularization weight while t 2 0.2, then it to 0 for t < 0.2. An alternative would be shift the reciprocal of LCFM, by C-1 M, such that (1) = LCFM, LCFM, In Table we compare our default weighting with this variant, denoted as Ashift.
1.00
Corat
0.75
Coss standard deviation
loss
0.50
0.25
0.8
0.6
0.4
0.2
0.0
Figure I: The Flow-Matching loss over time
Table 1: Quantitative results with 50 NFE and o, 0.01. We compare different weighting functions
(x())) based on the model eπrоr
SR
SR x12
Motion Dehlurring
Inpainting
Method
LPIPS)
FIDJ
SSIM? PSNR1
LPIPS.
HDDL
SSIMP PSNR
LPIPS] RDJ
SSIM
PSNR1
LPIPS)
HIJ
SSIM?
PSNR
FFHQ 768x 768
Avhife
0.246
27.4
0.793
29.91
0.286
24.4
0.766
28.19
0.237
14.5
0.790
29.84
0.180
8.2
0.828
23.58
Ours
0.777
29.54
0.236
10.7
0.184
0.828
23.69
DIV2K 768x768
Awhefe
0.379
30.4
0.625
23.58
0.434
37.5
0.522
21.40
0.337
25.8
0.664
24.54
0.151
0.819
23.79
Ours
0.353
26.5
0.607
23.30
0,421
32,1
0.525
21.39
0.315
21.1
0.653
24.44
0.163
11.0
0.815
23.75
A.7
Effect of captioning
Given the diversity of DIV2K, use DAPE (5 to generate captions for and include them in the prompt A high quality photo of (DAPE caption). For FFHQ we always prompt with A high quality photo of a face.. The effect of the text prompt to increase the likelihood of our sample under the prior of the (pre-trained, frozen) image generator. For comparison, we also ran experiments without data specific captions, where we always used the generie prompt high quality photo. Results are shown in Table 2
Table 2: Quantitative results with 50 NFE and a. 0.01. We compare our version with data-specific captions and a version without captions.
FFHQ
DIV2K
Method
LPIPS
FID,
SSIM1
PSNR
LPIPS
FIDJ
SSIM
PSNR
wo captions
0.278
17.0
0.734
27.66
0.488
51.5
0.546
21.82
Ours
0.271
16.2
0.740
27.71
0.421
32.1
0.525
21.39
A.8 Additional Experimental Results
We present the experimental results from the main paper in Table now augmented with sample-wise standard deviations for all metrics except FID.
25

---

Table 3: Quantitative results with 50 NFE and o, 0.01 Super-resolution (x8 and x12).
SR
SR x12
Method
LPIPS]
FIDI
SSIM
PSNR
LPIPS.
FID!
SSIMI
PSNR1
FFHQ
768x768
ReSample
0,400 = 0.069
55.6
0.815
0,051
26.37 + 1.00
0.474
0.078
80.3
0.786 = 0.056
25.47 = 1.16
FlowDPS
0.374 + 0.107
38.5
0.756
0.075
29.24
2.04
0,413
0.107
44.0
0.741
:: 0.074
28.05 + 2.06
0.776
71.7
FlowChef
0.341 = 0.083
30,5
0.760
0.064
28.42 + 2.22
0.373
0.084
46.5
0.730
= 0.068
27.00 = 2.07
Ours
0.213 = 0.056
13.3
0.777
0.051
29.54
2.02
0.271
0.071
16.2
0.740
= 0.058
27.71
= 2.00
DIV2K 768 768
ReSumple
0.533 0.130
55.0
0.625
0.132
22.34
2.27
0.643
0.152
88.1
0.562 : 0.151
20.85 + 3.02
FlowDPS
0.476
0.567
0.139
23.01
3.01
0.547
0.139
0.146
21.79
RSD
0.539 +: 0.121
60.9
0.591
0.124
23.45 + 2.96
0.684
0.137
95.7
0.523 + 0.132
21.96 + 2.86
FlowChef
0.490 + 0.116
36.5
0.339
0.137
21.84 2.96
0.525
0.118
43.8
0.492
4 0.145
20.52 + 2.85
Ours
0.353
26.5
0.607
0.127
23.30
2.90
0.421
0.131
32.1
0.525
0.136
21.39 +
Table 4: Quantitative results with 50 NFE and 0.01 Mootin deblurring and in-painting.
Motion Deblurring SSIM
In-painting
Method
LPIPS
FID
PSNR1
LPIPS,
FID|
SSIMI
PSNR
FFHQ
168x700
ReSample
0.457 + 0.087
82,9
0.788
0.058
25.45 +: 1.46
0.366
0.053
70.8
0.827 + 0.033
21.83 + 1.68
FlowDPS
0.431 + 0.117
54.3
0.732
0.078
27.64
2.20
0,344
0.060
42.5
0.771
0.048
19.19 + 3.19
RSD
0.458 0.098
77.3
0.743
0.059
27.67
+ 2.47
0.478
0.082
73.3
0.736
0.048
21.97
2.58
FlowChef
0.406 + 0.093
40.2
0.716
0.072
25.81
2.61
0.394
0.069
69.8
0.780
+ 0.051
18.18 + 2.84
Ours
0.236 = 0.070
10.7
0.772
0.055
29.61
2.24
0.184
0.038
8.7
0.828 : 0.029
23.69 + 2.77
DIV2K 768 768
ReSample
0,556 =: 0.146
79.7
0.617 = 0.134
21.79
2.52
0.285
0.073
51.9
0.796 = 0.067
22.68 + 1.84
FPo
0.558 + 0.153
65.5
0,536 + 0.148
21.88
3.02
0.328
0.103
29.2
0.692 1 0.112
21.71 + 2.67
FlowChef
0.561 = 0.123
49.6
0.486
0.148
19.90
3.06
0.489
0.148
58.3
0.659 = 0.131
20.87
+ 2.65
Ours
0.315 = 0.107
21.1
0.121 0.653
24.44 1: 3.05
0.163
0.053
11.0
0.815 1 0.054
23.75
+ 2.74
A.9 FLAIR Pixel Space
Table 5: Quantitative results with 50 NFE and a, 0.5% 11-%--n--ainnigg In-painninng and Super-resolution (x8).
Inpuinting
SR x8
Method
LPIPS|
FID.
SSIM1
PSNR1
LPIPS]
FID.
SSIM
PSNR
DDNM
0.158 + 0.042
26.9
0.037
18.31 2.94
0.199 : 0.052
31.9
0.635 0.079
23.59
1.64
DPS
0,195 1 0.064
30.2
0.689 1 0.077
20.49
2.81
0.172 1: 0.058
27.8
0.658 :: 0.088
24.59 : 2.04
MM
0.161 + 0.054
28.8
0.728
+ 0.062
20.59
+.39
0.172
- 0.051
29.1
0.669
+ 0.083
24.65
+ 1.97
HIGM
1 0.195
30.2
1 0.077
20.49
0.157
0.052
26.5
0.677
0.084
1 24.98
FLAIR
0.097 1: 0.035
14.2
0.831 0.031
21.87
2.66
0.143 0.039
22.9
0.712 0.076
25.93
1.96
We additionally implement method including DTA and AR() = (0 for t 0.2) in pixel space using the flow model from [33], trained on CelebA-HO resized (256x256). For comparison, we rephrase score based baselines to flow following (26] and evaluate all on 1000 samples from the dataset on super- repertresolution and inpainting. The methods hyperparameter tuneed to DDNM (53 (likelihood weight 4 for inpainting 1 for SR8), DPS [11 (64 512), Moment Matching 4 (4 8), ITGDM [50] (64 8) and pixel space FLAIR (0.5 32 and regularizer weight 0.4). As shown in Table 5) our method outperforms previous works also in pixel space, demonstrating its broader applicability.
A.10 Runtime Analysis
We compare the runtime and memory consumption of our method to the baselines. As our hard data consistency can strongly influence the runtime, we also provide measurements with the number of data term steps < 5 and additionally fast version using VAi"" of SD3, To validate that the
26

---

Box inpainting
observation
IIGDM
MM
FLAIR
ground truth
X super-resolution
observation
IIGDM
MM
FLAIR
ground truth
Figure 2: Qualitative comparison. FLAIR in pixel space produces posterior samples.
usage of the tiny VAE or less steps does not degrades the performance noticeably we also provide a metrics for x12 Super Resolution 100 samples of FFHQ:
Table Comparison of different methods in terms of runtime and memory usage. We validate the use of less data steps and a "tiny VAE" on x x12 super resolution on 100 samples of the FFHQ dataset.
Method
Runtime (s)
Memory (MB)
LPIPS |
PSNR f
Resample
88.02
19009.2
0.461
25.31
FlowDPS
34.15
12228.6
0.404
27.74
RSD (no repulsion)
21.19
12400.0
0.462
28.11
FlowChef
15.23
12227.72
0.361
26.56
FLAIR (HDC, large VAE)
172.34
12389.4
0.259
27.42
FLAIR (HDC, tiny VAE)
40.77
5960.2
0.256
27.59
FLAIR data term steps, tiny VAE)
22.46
5960.2
0.264
27.61
A.11 Ensembling Experiment
To highlight that our model can also used to obtain good MMSE estimates, we also conducted ensemble predictions by running posterior sampling 8 times and averaging the result. The results show that ensembling increases PSNR values, but reduces L.IPS and confirms that our samples are indeed distributed around the posterior the baseline methods are perceptually further away (LPIPS) from the ground truth compared to our samples.
and that results very close to the posterior mean like
A.12 Statistical Relevance
Our method is training-free -fren, and the variance in reconstructed images is intentional, reflecting the stochasticity of our sampling process rather than instability. All methods are evaluated with identical random seeds to ensure fair comparison. We compute metrics over 1000 samples FFHQ and 800 samples for DIV2K. Perceptual FID (pFID) is evaluated on 256 x 256 patches, resulting in 9000 and 7200 samples, respectively. Table in Appendix A.8 reports means and standard deviations over multiple samples.
27

---

Table 7: Quantitative results Super-resolution (x8 and x We report PSNR and LPIPSI. For ensembling we averaged independent predictions of the corresponding methods. It can seen that PSNR improves for all methods when ensembling. However, FLAIR shows the biggest gain, which means that our samples are indeed distributed around the mean and feature a higher variance compared to the baselines.
SR X
SR x 12
Method
PSNRI
LPIPS|
PSNR1
LPIPS
DIV2K
FlowDPS
22.53
0.4837
21.47
0.5524
FlowDPS (8x ensemble)
23,28
0.5157
22.06
0.5995
FlowChef
21.44
0.4898
20.20
0.5228
FlowChef (8x ensemble)
22.60
0,5502
21.60
0.5931
FLAIR
22.83
0.3627
21.05
0.4270
FLAIR (8x ensemble)
23.79
0.4244
22.27
0.4930
FFHQ
FlowDPS
29.02
0.3659
27.74
0.4036
FlowDPS (8x ensemble)
30.12
0.3267
28.65
0.3749
FlowChef
28.05
0.3303
26.56
0.3609
FlowChef (8x ensemble)
29.54
0.3267
28.15
0.3602
FLAIR
29.36
0.2028
27.42
0.2594
FLAIR (8x ensemble)
30.94
0.2457
29.00
0.2999
We also evaluated 100 FFHQ samples and 80 DIV2K samples, sampling three reconstructions per input for each method. We report the mean of each metric across all samples and the standard deviation of the means.
Table 8: Statistical evaluation on FFHQ for x8 Super Resolution. We report mean + standard
deviation over reconstructions per input.
Method
LPIPS 4
FID
SSIM t
PSNR t
FlowDPS
0.370 0.0012
70.7 1.45
0.755 + 0.0010
28.98 + 0.008
RSD
0.4678 + 0.0001
102.9 + 0.05
0.7362 + 0.0001
28.45 + 0.001
FlowChef
0.3316 0.0055
63.5 1.45
0.7593
+ 0.0027
28.12 + 0.072
Ours
0.2039 + 0.0048
40.5 + 0.94
0.7970 + 0.0228
29.74 + 0.668
Table 9: Statistical evaluation on FFHQ for x12 Super Resolution. We report mean + standard
deviation over reconstructions per input.
Method
LPIPS 1
FID
SSIM t
PSNR 1
FlowDPS
0.4073
0.0002
77.6
1.05
0.7391 = 0.0006
27.71 + 0.016
RSD
0.5039 + 0.0002
119.0 + 0.12
0.7217
+ 0.0001
27.08
+ 0.001
FlowChef
0.3626 0.0050
81.1 + 1.03
0.7283
+ 0.0027
26.62
+ 0.059
Ours
0.2593 + 0.0023
45.8 + 1.51
0.7582
+ 0.0252
27.81 + 0.660
A.12.1 t-Test Analysis
We further performed paired t-tests on the LPIPS scores between FlowDPS and FLAIR. The null hypothesis states that the mean LPIPS scores are the same for both methods. In all settings, we reject the null hypothesis (p < 0.001), confirming the statistical significance of our improvements
see Table 16
28

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Table 10: Statistical evaluation on FFHQ for Motion Blur. We report mean + standard deviation over reconstructions per input.
Method
LPIPS +
FID 4
SSIM
PSNR t
FLowDPS
0,4140
0.0030
83.83 + 1.00
0.7383
0.0006
27.47 0.05
RSD
0,4515 = 0.0001
108.75
0.7437 + 0.0001
27.40 0.00
FlowChef
0,4019 = 0.0007
1.69
0.7178
0.0022
25.50 0.04
Ours
0.2196 = 0.0080
38.8 = 2.25
0.7964 + 0.0319
30.10 + 0.96
Table 11: Statistical evaluation FFHQ for Inpainting. We report mean = standard deviation reconstructions per input.
Method
LPIPS |
FID 4
SSIM
PSNR 1
FIowDPS
0.3315
0.0015
74.00 + 0.58
0.7755
0.0007
19.06 0.11
RSD
0,4601 = 0.0003
103.02 0.03
0.7430
0.0000
22.19 0.01
FlowChef
0.3771 + 0.0013
102.22 0.26
0.7888 + 0.0016
18.42 + 0.25
Ours
0.1761 + 0.0012
33.23 + 2.02
0.8423 0.0172
24.07 0.80
A.13 Additional Qualitative Examples
To illustrate the visual differences behind the error metries, we present additional qualitative results for both FFHQ and DIV2k, comparing FLAIR with existing approaches. These examples complement the images in the main paper and highlight the visual fidelity, consistency, and robustness of our method across diverse scenes and different degradations. Figure features full sized version of the variance figure section subsection 5.2
A.14 Failure cases
We observe two main failure modes for FLAIR, see Figure 10 First, we find that super-resolution on DIV2K occasionally results in grainy textures, usually regions with abundant freeurryyynncy detail and complicated light transport. Potentially, this happens for images which do not have high probability under the prior. We not observe those artifacts for the FFHQ dataset. Second, observe a few instances where the strong generative prior hallucinates semantically inconsistent or misaligned structures especially facial features.
29

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Table 12: Statistical evaluation on DIV2K for x8 Super Resolution. We report mean + standard
deviation over reconstructions per input.
Method
LPIPS 4
FID 4
SSIM
PSNR f
FlowDPS
0.5517 0.0046
138.24 + 1.67
0.5207 0.0022
22.41 0.02
RSD
0.7163 = 0.0002
181.83 + 0.15
0.4892 + 0.0001
21.99 + 0.00
FlowChef
0.5726 0.0046
145.77 3.31
0.4998 + 0.0014
21.21 + 0.04
Ours
0.3716 = 0.0161
88.08 + 1.43
0.5991 + 0.0192
23.06 0.41
Table 13: Statistical evaluation on DIV2K for x x12 Super Resolution. We report mean + standard
deviation over reconstructions per input.
Method
LPIPS 4
FID
SSIM
PSNR f
FlowDPS
0.6264 + 0.0045
154.31 + 0.43
0.4866 0.0024
21.37 0.02
RSD
0.7714 = 0.0002
1 0.12
0.4683 = 0.0001
$1.15
FlowChef
0.6020 0.0060
151.32 + 2.45
0.4586 0.0020
20.10 0.03
Ours
0.4316 + 0.0151
101.12 + 4.51
0.5236 + 0.0229
21.35 0.51
Table 14: Statistical evaluation on DIV2K for Motion Deblur. We report mean + standard deviation over reconstructions per input.
Method
LPIPS 4
FID
SSIM
PSNR t
FlowDPS
0.6242 + 0.0082
161.57 + 3,90
0.4978 0.0028
21.41 0.04
RSD
0.8067 = 0.0002
216.37 0.36
0.4364 + 0.0001
20.82 0.00
FlowChef
0.6292 0.0007
158.08 + 3.03
0.4557 0.0025
19.62 0.05
Ours
0.3069 + 0.0036
77.77 + 1.89
0.6596 + 0.0316
24.46 +
Table 15: Statistical evaluation on DIV2K for Inpainting. We report mean + standard deviation over 3 reconstructions per input.
Method
LPIPS 1
FID
SSIM
PSNR t
FlowDPS
0.3738 0.0032
106.58 + 0.89
0.6579
0.0009
21.06 0.05
RSD
0.4667 + 0.0003
136.82 0.21
0.6650 + 0.0001
23.07 0.00
FlowChef
0.5111
0.0019
128.97 + 0.60
0.6355 + 0.0006
20.51 0.02
Ours
0.1729 = 0.0014
51.41 + 0.48
0.8122 + 0.0124
24.06 + 0.87
Table 16: Paired t-test p-values for LPIPS (FlowDPS vs. Ours), All comparisons are statistically significant.
Dataset
Task
p-value
SRx8
2.92 x 10
SRx 12
7.19 10
DIV2K
Motion Deblur
5.30 10
Inpainting
1.21 x 10-4
SRx8
7.05 10
SRx 12
6.56 x 10-5
FFHQ
Motion Deblur
4.29 10
Inpainting
3.00 10-5
30

---

CHIm
Observazion
Carin Fa
RSD
Gana
ReSample
Cana
FlowDPS
Claa
FLAIR
Caaaa
Target
Figure Inpainting results on FFHQ Shown observation, reference methods, FLAIR and ground truth. FLAIR produces realistic, high-frequency details while previous works either fail to inpaint the region correctly or collapse overly smooth solutions.
31

---

Observazion
RSD
ReSample
FlowDPS
FLAIR
Target
Figure 4: x12 super-resolution results FFHQ. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend predict blurry images.
32

---

Observazion
RSD
ReSample
FlowDPS
FLAIR
Target
Figure 5: Motion de-blur results FFHQ. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend to predict blurry images.
33

---

Observation
RSD
Remampe
FlowDPS
FLAIR
Target
Figure 6: Inpainting results on DIV2k. Shown are observation, reference methods, FLAIR and ground truth. FLAIR produces realistic, high-frequency details while previous works either fail to inpaint the region correctly or collapse to overly smooth solutions. Moreover they not fit the data term (not inpainted region) very well.
34

---

Observazion
RSD
ReSample
FlowDPS
FLAIR
Target
Figure x12 super-resolution results on DIV2k. Shown are observation. reference methods, FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend predict blurry images.
35

---

Observazion
RSD
Re-Sample
Flowrro
FLAIR
Target
Figure 8: Motion de-b results results DIV2k. Shown are observation, reference methods,
FLAIR and ground truth. FLAIR produces sharp and results which still fulfill the data term, whereas the baselines tend predict blurry images.
36

---

FLAIR
FloWDPS
RSD
Figure 9: Individual samples for x12 Super Resolution with zoom and std. FLAIR produces varied samples from the posterior. For superresoltion The variance is expected to be mostly in the high frequencies, because the data term limits low frequency variations. The baselines tend to predict very similar looking images with less detail.
37

---

Target
FLAIR
Target T
FLAIR
Figure 10: Qualitative failure cases of FLAIR on DIV2k and FFHQ. Top row: grainy results from systematic error. Those errors potentially stem from weak prior for those images. For example do not observe them for the FFHQ dataset Bottom row: Semantically inconsistent failures. Sometimes the model lacks the ability incorporate globally consistent semanties into its restorations.
38


Reviews:
[{"review_id": "gQEM3hityl", "reviewer": "Reviewer_ybuv", "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.", "questions": "See Weakness.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths: \n- The paper is well written, with a clear structure and smooth presentation.\n- It introduces flow matching into a variational inference framework.\n- The reconstructed images exhibit fine details with strong perceptual quality. \n- The experimental section is thorough with comprehensive baseline comparisons and well-designed ablation studies.\n\nWeaknesses: \n- 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. \n- 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. \n- 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. \n- L179 should be subsection A.4, not A.3. L222 should be subsection A.9, not A.5.", "was_revised": true, "final_justification": "The authors have addressed my concerns regarding [W1] the novelty of the work, [W2] the hyperparameter settings, and [W3] the evaluation of runtime and memory usage. I appreciate the clarifications and will maintain my current score as borderline accept.", "rebuttal": "We thank the reviewer for acknowledging the novelty of integrating flow matching into a variational framework, the high quality of our results, and the comprehensiveness of our experiments. Your feedback is much appreciated and we we incorporated it into our manuscript.\n\nIf our answers are satisfying, we kindly ask to consider raising your score.\n\n**1. Core components are an assembly of known ideas**\n\nWe would like to clarify that our work introduces several novel contributions beyond assembling existing techniques:\n\n- **Variational posterior sampling for flow matching:**\n    \n    We are the first to introduce a variational approach for posterior sampling with flow matching models. The derivation of this method is non-trivial and is provided in detail in Appendix A.1–A.3.\n    \n- **Deterministic trajectory adjustments:**\n    \n    Our proposed deterministic trajectory adjustment is a crucial and completley novel component of the method, and our ablation studies show it yields the largest improvements in LPIPS. This adjustment is theoretically motivated and has not been previously explored in the literature.\n    \n- **Model accuracy-weighted regularization (CRW):**\n    \n    We are the first to propose using model accuracy as a weight for the regularization term (CRW), which is feasible only for variational posterior sampling approaches. This is not possible for non-variational methods.\n    \n- **Hard data consistency in a variational framework:**\n    \n    While hard data consistency has been used in other works, our framework (FLAIR) is, to our knowledge, the first to apply it within a variational setting.\n    \n\nOur work is positioned within the field of *training-free posterior sampling using generative priors*. Given that flow matching models represent the current state-of-the-art in image generation, we believe it is both timely and important to develop posterior sampling methods that are specifically compatible with these powerful generative priors.\n\n**2. Non-trivial hyperparameters**\n\nThe hyperparameters we introduce are $\\lambda_{\\mathcal{R}(t)}$ and $\\alpha$, both of which are fixed across all our experiments and datasets. The gradient descent learning rate for the data term must be selected depending on the chosen task, which is also the case for all baseline methods.\n\nRegarding the CRW, we explore alternative weightings in Appendix A.6. Additionally, our ablation study without CRW corresponds to setting $\\lambda_{\\mathcal{R}(t)} = t$.\n\nWe also note that, in variational approaches, a function for $\\lambda_{\\mathcal{R}(t)}$ must always be specified (e.g., RSD and RED-diff both use $\\lambda_{\\mathcal{R}(t)} = t$). Our contribution is to make this choice dependent on the model error.\n\n**3. Runtime Memory Analysys**\n\nRuntime optimization was not the scope of our project. Our baseline methods are evaluated with the same number of neural function evaluations. The methods may differ in the number of gradient steps taken on the data term, which depends on the early stopping constant chosen.\n\nOur method can also be deployed with a fixed number of data term steps (e.g., 5), if a deterministic runtime is required. If runtime or memory footprint are of importance one can switch to a \"tinyVAE\" version of SD3.\n\nTo validate that the usage of the tinyVAE or less steps does not degrades the performance noticeably we also provide a metrics for x12 Super Resolution on 100 samples of FFHQ:\n\n| Method | Runtime (s) ↓ | Memory (MB) ↓ | LPIPS ↓ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 88.02 | 19009.24 | 0.4606 | 25.31 |\n| FlowDPS | 34.15 | 12228.63 | 0.4036 | 27.74 |\n| RSD (no repulsion) | 21.19 | 12399.96 | 0.4624 | 28.11 |\n| FlowChef | 15.23 | 12227.72 | 0.3609 | 26.56 |\n| Ours (HDC, largeVAE) | 172.34 | 12389.37 | 0.259 | 27.42 |\n| Ours (HDC, tiny VAE) | 40.77 | 5960.22 | 0.256 | 27.59 |\n| Ours (5 data term steps, tiny VAE) | 22.46 | 5960.22 | 0.264 | 27.61 |"}, {"review_id": "v1us68Ts2w", "reviewer": "Reviewer_Krp7", "summary": "The 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", "questions": "1. 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?", "limitations": "Authors 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.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "Thank you for the follow-up clarifications and for completing the requested CRW ablation study across all tasks. The results provide helpful empirical evidence that CRW outperforms both linear and constant weighting strategies, and the inclusion of perceptual observations regarding inpainting quality is appreciated.\n\nI will maintain my current score but acknowledge that this is a strong paper with clear technical contributions and well-documented experiments. I thank the authors for the constructive exchange throughout the rebuttal phase.", "rebuttal": "We thank the reviewer for acknowledging the novelty of our unified inverse-problem solver, the strong generalization of our flow prior across tasks, and the high quality of our results. We also appreciate your recognition of the importance of our ablations. Your feedback is much appreciated and has been incorporated into the manuscript.\n\nIf our answers are satisfying, we kindly ask to consider raising your score.\n\n**1. DTA resembles classifier-free guidance; CRW is a heuristic weighting fitted offline hence incremental novelty**\n\nWe want to thoroughly highlight, that the proposed deterministic trajectory adjustment does not share any resemblance to CFG. However, CFG can be used to increase the prompt following and therefore condition the model more strongly on the given text prompt. In the inverse problem setting we usually do not have good text prompts or no text prompts at all because the input is degraded and potentially we do not know what the original image depicts. With the deterministic trajectory adjustment we propose to use a \"special noise vector\" $\\hat{x}_1$, which additionaly conditions the generative model on the current optimization estimate $\\mu$. This ensures that the optimization does not jump between possible modes and instead stabilizes the regularization term.\nOther variational approaches like RED-diff or RSD also have to chose a weighting for the regularizer. We are the first to propose one which is motivated by the accuracy of the regularizer.\n\n**2. Runtime Concerns**\n\nThis is not a new drawback of our method but one that all training free posterior sampling methods share. We decided to run all inverse problems using the same amount of neural functions evaluations. For certain problems the required number could be higher or lower. The amount of NFEs depends also on the generative model used. \n\nSee a runtime and memory analysis below. Our hard data consistency can strongly influence the runtime and we also provide measurements with number of data term steps<=5 and a fast version using a \"tinyVAE\" of SD3.\n\nTo validate that the usage of the tinyVAE or less steps does not degrades the performance noticeably we also provide a metrics for x12 Super Resolution on 100 samples of FFHQ:\n\n| Method | Runtime (s) ↓ | Memory (MB) ↓ | LPIPS ↓ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 88.02 | 19009.24 | 0.4606 | 25.31 |\n| FlowDPS | 34.15 | 12228.63 | 0.4036 | 27.74 |\n| RSD (no repulsion) | 21.19 | 12399.96 | 0.4624 | 28.11 |\n| FlowChef | 15.23 | 12227.72 | 0.3609 | 26.56 |\n| Ours (HDC, largeVAE) | 172.34 | 12389.37 | 0.259 | 27.42 |\n| Ours (HDC, tiny VAE) | 40.77 | 5960.22 | 0.256 | 27.59 |\n| Ours (5 data term steps, tiny VAE) | 22.46 | 5960.22 | 0.264 | 27.61 |\n\n**3. Pixel Space experiments**\n\nWe thank the reviewer for the suggestion. Our focus was on high-resolution tasks and arbitrary scenes, where latent models such as SD3 offer clear benefits in terms of dataset scale and fidelity. Therefore, we did not include pixel-space experiments in the manuscript.\n\nHowever, we agree that such experiments are valuable for completeness, and we now provide them in the table below. s shown, our method outperforms previous works also in pixel space, demonstrating its broader applicability.\n\n| Method (Inpainting) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| DDNM | 0.158 | 26.9 | 0.732 | 18.31 |\n| DPS | 0.195 | 30.2 | 0.689 | 20.49 |\n| Moment Matching | 0.161 | 28.8 | 0.728 | 20.59 |\n| $\\Pi$GDM | 0.195 | 30.2 | 0.689 | 20.49 |\n| Ours | **0.097** | **14.2** | **0.831** | **21.87** |\n\n| Method (SR8) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| DDNM | 0.199 | 31.9 | 0.635 | 23.59 |\n| DPS | 0.172 | 27.8 | 0.658 | 24.59 |\n| Moment Matching | 0.172 | 29.1 | 0.669 | 24.65 |\n| $\\Pi$GDM | 0.157 | 26.5 | 0.677 | 24.98 |\n| Ours | **0.143** | **22.9** | **0.718** | **25.93** |\n\n**4. Statistical Relevance**\n\nWe want to highlight that our method is training free and that variance in the reconstructed images is intentional. We test all methods with the same random seeds and evaluate over 1000 samples for FFHQ and 800 for DIV2K. pFID is evaluated on patches of 256x256 which results in 9000 samples and 7200 samples for evaluation. In table 5 in A.8 shows the means and standard deviations over different samples.\n\nFor the rebuttal we also conducted experiments on 100 samples of FFHQ and 80 of DIV2K where we sampled 3 reconstructions per input for each method and computed the mean for each metric over all samples and the std of the means.\n\n| Method (FFHQ, SRx8) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| FlowDPS | 0.3701 +- 0.0012 | 70.7 +- 1.45 | 0.7551 +- 0.0010 | 28.98 +- 0.008 |\n| RSD | 0.4678 +- 0.0001 | 102.9 +- 0.05 | 0.7362 +- 0.0001 | 28.45 +- 0.001 |\n| FlowChef | 0.3316 +- 0.0055 | 63.5 +- 1.45 | 0.7593 +- 0.0027 | 28.12 +- 0.072 |\n| Ours | 0.2039 +- 0.0048 | 40.5 +- 0.94 | 0.7970  +- 0.0228 | 29.74 +- 0.668 |\n\n| Method (FFHQ SRx12) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| FlowDPS | 0.4073 +- 0.0002 | 77.6 +- 1.05 | 0.7391 +- 0.0006 | 27.71 +- 0.016 |\n| RSD | 0.5039 +- 0.0002 | 119.0 +- 0.12 | 0.7217 +- 0.0001 | 27.08 +- 0.001 |\n| FlowChef | 0.3626 +- 0.0050 | 81.1 +- 1.03 | 0.7283 +- 0.0027 | 26.62 +- 0.059 |\n| Ours | 0.2593  +- 0.0023 | 45.8 +- 1.51 | 0.7582 +- 0.0252 | 27.81 +- 0.660 |\n\n**5. User studies**\n\nA user study is the gold standard for image quality assessment, but was out of scope for this project. However, in a training free approach there are limited parameters to tune and metric overfitting is usually not an issue. Specifically, we conducted a small grid for FLAIR and all baselines methods to select optimal learning rates. All other parameters were kept fixed.\nFurthermore, LPIPS is already a distance metric which aligns very well with the human perception. The \"realness\" of the images on the other hand is usually evaluated with the FID metric.\nAt this point we also want to emphasize that PSNR and SSIM are not ideal metrics to access the quality of the reconstructions, because they favor a MMSE prediction which is overly smooth and unrealistic. Consequently, any plausible texture shift, while perceptually correct, incurs a large $\\ell_2$ penalty and reduces PSNR/SSIM. \n\nTo highlight that our model can also be used to obtain good MMSE estimates, we also conducted ensemble predictions by running posterior sampling 8 times and averaging the result. The results show that ensembling increases PSNR values, but reduces LPIPS and confirms that our samples are indeed distributed around the posterior mean and that results very close to the posterior mean like the baseline methods are perceptually further away (LPIPS) from the ground truth compared to our samples.  We included the ensembling experiments in our supplementary material.\n\n100 samples for FFHQ and 80 for DIV2K\n\n| Method (FFHQ) | x8 PSNR | x8 LPIPS | x12 LPIPS | x12LPIPS |\n| --- | --- | --- | --- | --- |\n| Resample | 26.26 | 0.3860 | 25.31 | 0.4606 |\n| FlowDPS | 29.02 | 0.3659 | 27.74 | 0.4036 |\n| FlowDPS (8x ensemble) | 30.12 | 0.3267 | 28.65 | 0.3749 |\n| RSD | 29.69 | 0.3915 | 28.11 | 0.4624 |\n| FlowChef | 28.05 | 0.3303 | 26.56 | 0.3609 |\n| FlowChef (8x ensemble) | 29.54 | 0.3267 | 28.15 | 0.3602 |\n| Ours | 29.36 | 0.2028 | 27.42 | 0.2594 |\n| Ours (8x ensemble) | 30.94 | 0.2457 | 29.00 | 0.2999 |\n\n| Method (DIV2K) | x8 PSNR | x8 LPIPS | x12 PSNR | x12 LPIPS |\n| --- | --- | --- | --- | --- |\n| Resample | 22.08 | 0.5480 | 20.60 | 0.6576 |\n| FlowDPS | 22.53 | 0.4837 | 21.47 | 0.5524 |\n| FlowDPS (8x ensemble) | 23.28 | 0.5157 | 22.06 | 0.5995 |\n| RSD | 23.45 | 0.5389 | 21.96 | 0.6836 |\n| FlowChef | 21.44 | 0.4898 | 20.20 | 0.5228 |\n| FlowChef (8x ensemble) | 22.60 | 0.5502 | 21.60 | 0.5931 |\n| Ours | 22.83 | 0.3627 | 21.05 | 0.4270 |\n| Ours (8x ensemble) | 23.79 | 0.4244 | 22.27 | 0.4930 |\n\nFurthermore a high PSNR does not mean that the results accurately follow the measurement. For x12 super resolution on DIV2K with FLAIR achieves 22.83 dB vs RSD with 23.45 dB. However, if we evaluate the PNSR in the measurement space to access how well the prediction follows the measurement y: PSNR(y, A($\\hat{x}$)). FLAIR achieves 45.45dB in PSNR and RSD 40.28 dB. That means our prediction follows the measurement more closely even though the PSNR of the prediction compared to the ground truth is lower.\n\n**6. Non Gaussian noise and non-linear forward operators**\n\nThe data term can also be adapted to non-Gaussian noise sources. E.g. Chung et al. Diffusion posterior sampling for general noisy inverse problems, ICLR 2023 elaborate on how to adapt the data term for Poisson distributed nose. We adapted the manuscript to clarify that our approach is not limited to only Gaussian noise models.\n\nBecause SD3 is a latent model, the non-linear decoder of the SD3-VAE is part of our forward operator. Therefore, all tasks we evaluated on are non-linear. We also conducted experiments on non-linear motion deblurring (non-linearity through saturation effects and gamma correction) see tables below:\n\nNon-linear Motion Deblurring:\n\n| Method (FFHQ) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 0.446 | 118.1 | 0.792 | 25.21 |\n| FlowDPS | 0.413  | 83.0 | 0.738 | 27.27 |\n| RSD | 0.448 | 109.87 | 0.7441 | 27.27 |\n| FlowChef | 0.402 | 74.5 | 0.716 | 25.29 |\n| Ours  | 0.201 | 38.0 | 0.7851 | 29.37 |\n\n| Method (DIV2K) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 0.551 | 149.1 | 0.607 | 21.79 |\n| FlowDPS | 0.612 | 157.8 | 0.500 | 21.30 |\n| RSD | 0.626 | 177.3 | 0.542 | 21.81 |\n| FlowChef | 0.632 | 160.4 | 0.455 | 19.57 |\n| Ours  | 0.303 | 78.2 | 0.644 | 23.99 |\n\n**6. CRW compared to a linear $t$ weighting or a constant $\\lambda$**\n\nOur CRW ablation is a comparison to $t$ weighting."}, {"review_id": "0032XjTpNI", "reviewer": "Reviewer_5yNE", "summary": "The 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.", "questions": "1. 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?", "limitations": "In the Discussion section, the authors should also provide future optimization directions regarding the limitations.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The 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.", "was_revised": true, "final_justification": "I have read the authors' rebuttals. The authors have addressed my concerns, including clarifying the innovation of this research relative to existing work and adding discussions on quantitative evaluation results and on non-Gaussian cases. Taking into account the innovation and contribution of this study, I have updated rating in the system.", "rebuttal": "We thank the reviewer for acknowledging the strong perceptual performance of our method. Your feedback is much appreciated and we incorporated it into our manuscript.\nIf our answers are satisfying, we kindly ask to consider raising your score.\n\nOur work is positioned within the field of training-free posterior sampling using generative priors. Given that flow matching models represent the current state-of-the-art in image generation, we believe it is both timely and important to develop posterior sampling methods that are specifically compatible with these powerful generative priors.\n\n**1. Highlight insights and innovations**\n\nWe have revised the manuscript to put more emphasis on clearly highlighting the core innovation of our approach.\n\n- **Variational posterior sampling for flow matching:**\n    \n    To the best of our knowledge, we are the first to introduce a variational approach for posterior sampling with flow matching models. The derivation of this method is non-trivial and is provided in detail in Appendix A.1–A.3. We also compare to Repulsive Score Distillation (RSD), which adapts and extends RED-diff to latent diffusion models. We outperform RSD on all relevant metrics, which shows how valuable our contributions are.\n    \n- **Deterministic trajectory adjustments:**\n    \n    Our proposed deterministic trajectory adjustment is a crucial and novel component of the method, and our ablation studies show it yields the largest improvements in LPIPS. This adjustment is theoretically motivated and, to our knowledge, has not been previously explored in the literature.\n    \n- **Model accuracy-weighted regularization (CRW):**\n    \n    We are the first to propose using model accuracy as a weight for the regularization term (CRW), which is feasible only for variational posterior sampling approaches. This is not possible for non-variational methods.\n    \n- **Hard data consistency in a variational framework:**\n    \n    While hard data consistency has been used in other works, our framework (FLAIR) is, to our knowledge, the first to apply it within a variational setting.\n    \n\n**2. Challenges from the abstract**\n\nOur method addresses non-linear mappings by operating in the latent space while explicitly backpropagating through the decoder to ensure consistency with the observed data in pixel space. This allows us to handle the non-linear encoder–decoder structure effectively, while still being able to initialze with the pseudo inverse.\nWe observed that FLAIR reconstructs diverse samples (see Fig. 3) which also look realistic. As shown in Fig. 3 baseline approaches collapse to a single solution which is close to the mean, but does not look realistic. Under the unconditional prior all modes are unlikely. One has to condition the model very strictly to increase the likelihood for certain modes e.g. with very precise text prompts which is not feasible for degraded images. Our solution is to compute “special noise vectors” $\\hat{x}$ to condition the model and increase the likelihood of modes which match the measurement. \n\n**3. CRW weighting**\n\nThe CRW weighting is determined through a one-time accuracycalibration procedure, independent of the task or any hyperparameter. Specifically, we compute the expected model error using a small set of arbitrary calibration images. This error is derived from the conditional flow-matching loss, which directly corresponds to the regularization term in our framework. Additional details can be found in the Appendix A.6.\n\n**4. PSNR/SSIM metrics**\n\nWe want to **highly emphasize** that our goal is to obtain diverse and visually convincing reconstructions that remain consistent with the observed data (i.e., sampling from the posterior modes), rather than pixel-wise accurate ones (i.e., estimating the posterior mean). This inherently leads to lower PSNR and SSIM scores, since there are many possible samples that can explain the observed data. However, our method achieves significantly closer reconstruction in perceptual distance as shown by the LPIPS evaluations. Furthermore, the strong FID scores show that our results are highly realistic while baselines fail to reconstruct images which look realistic.\n\nDue to the perception-distortion tradeoff, it is fundamentally not possible to achieve both high perceptual quality and high pixel-level fidelity simultaneously, especially under severe degradations. Reconstructions that appear realistic often deviate from the pixel-wise mean and therefore yield lower PSNR and SSIM, despite being perceptually closer to a human observer. \n\nTo highlight that our model can also be used to obtain good MMSE estimates, we also conducted ensemble predictions by running posterior sampling 8 times and averaging the result. The results show that ensembling increases PSNR values, but reduces LPIPS and confirms that our samples are indeed distributed around the posterior mean and that results very close to the posterior mean like the baseline methods are perceptually further away (LPIPS) from the ground truth compared to our samples. We included the ensembling experiments in our supplementary material.\n\n100 samples for FFHQ and 80 for DIV2K\n\nDIV2K Results\n\n| Method (DIV2K) | x8 PSNR | x8 LPIPS | x12 PSNR | x12 LPIPS |\n| --- | --- | --- | --- | --- |\n| Resample | 22.08 | 0.5480 | 20.60 | 0.6576 |\n| FlowDPS | 22.53 | 0.4837 | 21.47 | 0.5524 |\n| FlowDPS (8x ensemble) | 23.28 | 0.5157 | 22.06 | 0.5995 |\n| RSD | 23.45 | 0.5389 | 21.96 | 0.6836 |\n| FlowChef | 21.44 | 0.4898 | 20.20 | 0.5228 |\n| FlowChef (8x ensemble) | 22.60 | 0.5502 | 21.60 | 0.5931 |\n| Ours | 22.83 | 0.3627 | 21.05 | 0.4270 |\n| Ours (8x ensemble) | 23.79 | 0.4244 | 22.27 | 0.4930 |\n\nFFHQ Results\n\n| Method (FFHQ) | x8 PSNR | x8 LPIPS | x12 PSNR | x12 LPIPS |\n| --- | --- | --- | --- | --- |\n| Resample | 26.26 | 0.3860 | 25.31 | 0.4606 |\n| FlowDPS | 29.02 | 0.3659 | 27.74 | 0.4036 |\n| FlowDPS (8x ensemble) | 30.12 | 0.3267 | 28.65 | 0.3749 |\n| RSD | 29.69 | 0.3915 | 28.11 | 0.4624 |\n| FlowChef | 28.05 | 0.3303 | 26.56 | 0.3609 |\n| FlowChef (8x ensemble) | 29.54 | 0.3267 | 28.15 | 0.3602 |\n| Ours | 29.36 | 0.2028 | 27.42 | 0.2594 |\n| Ours (8x ensemble) | 30.94 | 0.2457 | 29.00 | 0.2999 |\n\nWe therefore want the reviewers to be aware that our method is explicitly designed for posterior sampling with a generative prior, not for producing a single point estimate such as the MMSE solution. Consequently, **PSNR is not an appropriate metric for evaluating such generative approaches** (especially under heavily ill posed problems), as it penalizes diversity and favors averaging, which is contrary to the goal of realistic posterior sampling.\n\n**5. Real world non Gaussian noise**\n\nThis goes beyond the scope of the paper. However, we want to emphasize that our method as well as other methods in the field are not limited to Gaussian noise. The data term can. also be adapted to other noise distributions (e.g. Poisson noise).\n\n**6. Training and Inference Time**\n\nWe want to emphasize that our model is completely training free. We simply use the SD3 weights as they are. A runtime and memory consumption analysis for x12 super resolution can be seen below. \nOur hard data consistency can strongly influence the runtime and we also provide measurements with number of data term steps<=5 and a fast version using a \"tinyVAE\" of SD3.\n\nTo validate that the usage of the tinyVAE or less steps does not degrades the performance noticeably we also provide a metrics for x12 Super Resolution on 100 samples of FFHQ:\n\n| Method | Runtime (s) ↓ | Memory (MB) ↓ | LPIPS ↓ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 88.02 | 19009.24 | 0.4606 | 25.31 |\n| FlowDPS | 34.15 | 12228.63 | 0.4036 | 27.74 |\n| RSD (no repulsion) | 21.19 | 12399.96 | 0.4624 | 28.11 |\n| FlowChef | 15.23 | 12227.72 | 0.3609 | 26.56 |\n| Ours (HDC, largeVAE) | 172.34 | 12389.37 | 0.259 | 27.42 |\n| Ours (HDC, tiny VAE) | 40.77 | 5960.22 | 0.256 | 27.59 |\n| Ours (5 data term steps, tiny VAE) | 22.46 | 5960.22 | 0.264 | 27.61 |\n\n**7. Novelty of Hard Data Consistency (HDC)**\n\nThe decoupling of the data term is a novelty for variational posterior sampling approaches based on stochastic interpolants such as diffusion or flow matching models. A major drawback of most posterior sampling approach with stochastic interpolants is the need to approximate the likelihood term, because it is not tractable. Applying hard data consistency to the approximation does not necessarily bring an advantage and could event be harmful. However, in the variational formulation no approximation for the likelihood term has to be applied and therefore decoupling the data term and applying hard consistency can bring a large advantage as is shown in our ablations. We are the first to analyze and leverage this advantage."}, {"review_id": "HZvT17mUCN", "reviewer": "Reviewer_fMnY", "summary": "The 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.", "questions": "I 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.", "limitations": "The authors address limitations.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "### 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)", "was_revised": true, "final_justification": "Given the increasing use of large-scale flow-based models, extending inverse problem-solving techniques to this class is both relevant and important. The authors provided a thorough rebuttal, particularly w.r.t. computational efficiency (reducing the number of data term steps and using tiny VAE). Considering the rebuttal, as well as the responses to the other reviewers, I have raised my score to 4.", "rebuttal": "We thank the reviewer for acknowledging the novelty of our variational formulation, the relevance of our work in light of recent developments in large-scale latent diffusion models, the high quality of our results and the comprehensiveness of our experiments . Your feedback is much appreciated and we we incorporated it into our manuscript.\nIf our answers are satisfying, we kindly ask to consider raising your score.\n\n**1. Similarity to RED-diff**\n\nThe cited equations are part of our background section. We do not claim them as contribution, but they are important to understand the method. One of our contributions lies in the derivations of the variational objective for flow matching models stated in Propostion 1. The derivations can be found in the appendix A.1-A.3. We also compare to Repulsive Score Distillation (RSD), which adapts and extends RED-diff to latent diffusion models. We outperform RSD on all relevant metrics, which shows how valuable our contributions are.\nWe will emphasize our contributions to the variational objective for flow matching models more clearly in the final version.\n\n**2. Zero variance of variational distribution**\n\nOne can think of the new variational distribution as particle approximation of the posterior. In our experiments we only used one particle. With the particle approximation the gradients of the regularizer can be approximated using Wasserstein Gradient Flow and equation (7) is not required. However, eq. (7) motivates the weighting of the regularizer with the conditional flow matching loss for the CRW.\n\n**3. Source of Posterior Variance**\n\nWhile the variational objective remains fixed, noise vectors are drawn randomly during the optimization which leads to a different optimization trajectory. This is analogous to how different random seeds with the same text prompt yield diverse outputs in generative models like Stable Diffusion. In our case, these different trajectories correspond to different posterior modes.\n\n**4. Non linear forward operators**\n\nAs you said the decoder, which is part of the forward operator for latent models makes all the tested tasks non-linear forward operators. We added the task of non-linear deblurring. The non-linearity lies in the non-linear conversion from sRGB to linear color space and back. The blur opoerator is applied in the linear color space. Additionally, values are clipped after blurring simulating saturation effects.\n\n| Method (FFHQ) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 0.446 | 118.1 | **0.792** | 25.21 |\n| FlowDPS | 0.413 | 83.0 | 0.738 | 27.27 |\n| RSD (no repulsion) | 0.448 | 109.87 | 0.7441 | 27.27 |\n| FlowChef | 0.402 | 74.5 | 0.716 | 25.29 |\n| Ours | **0.201** | **38.0** | 0.7851 | **29.37** |\n\n| Method (DIV2K) | LPIPS ↓ | FID ↓ | SSIM ↑ | PSNR ↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 0.551 | 149.1 | 0.607 | 21.79 |\n| FlowDPS | 0.612 | 157.8 | 0.500 | 21.30 |\n| RSD (no repulsion) | 0.626 | 177.3 | 0.542 | 21.81 |\n| FlowChef | 0.632 | 160.4 | 0.455 | 19.57 |\n| Ours | **0.303** | **78.2** | **0.644** | **23.99** |\n\n**5. Stochasticity of HDC**\nThank you for spotting this error. The optimization of the data term is not stochastic. The optimization of the variational objective however is due to the random sampling of noise vectors.\n\n**6. Overfitting to noise in HDC**\n\nThe optimization of the data term can be stopped early for noisy observations as it is the case for our experiments. As mentioned in A.4 we stop at an error of less than $1\\times 10^{-4} \\cdot \\text{len}(y)$. However, for tasks like inpainting, absence of measurement noise is realistic.\n\n**7. Claim that FLAIR is able to recover atypical modes** \n\nThis is a conclusion we draw from the observation that FLAIR reconstructs diverse samples (see Fig. 3) which also look realistic. As shown in Fig. 3 baseline approaches collapse to a single solution which is close to the mean, but does not look realistic. Under the unconditional prior all modes are unlikely. One has to condition the model very strictly to increase the likelihood for certain modes e.g. with very precise text prompts which is not feasible for degraded images. Our solution is to compute “special noise vectors” $\\hat{x}$ to condition the model and increase the likelihood of modes which match the measurement.\n\n**8. Expectation over time t**\n\nYou are correct. Thank you very much for pointing out that error. We will corrected it in the manuscript.\n\n**9. Why does the model perform better without HDC, DTA, CRW?** \n\nAt this point we also want to emphasize that PSNR and SSIM are not ideal metrics to access the quality of the reconstructions, because they favor a MMSE prediction which is overly smooth and unrealistic. Consequently, any plausible texture shift, while perceptually correct, incurs a large $\\ell_2$ penalty and reduces PSNR/SSIM. LPIPS on the other hand is able to measure distances between images in a way that correlate with the human perception\n\n**When deactivating our contributions the method severely degrades terms of LPIPS**. The explanations is that the results collapse to a posterior mean similar to the baseline methods. This mean-prediction achieves a higher PSNR, but is perceptually farther from the ground truth and achieves worse LPIPS, as the respective qualitative results in the ablation study show.\n\nTo highlight that our model can also be used to obtain good MMSE estimates, we also conducted ensemble predictions by running posterior sampling 8 times and averaging the result. The results show that ensembling increases PSNR values, but reduces LPIPS and confirms that our samples are indeed distributed around the posterior mean and that results very close to the posterior mean like the baseline methods are perceptually further away (LPIPS) from the ground truth compared to our samples. We included the ensembling experiments in our supplementary material.\n\n80 samples of DIV2K\n\n| Method (DIV2K) | x8 PSNR | x8 LPIPS | x12 PSNR | x12 LPIPS |\n| --- | --- | --- | --- | --- |\n| Resample | 22.08 | 0.5480 | 20.60 | 0.6576 |\n| FlowDPS | 22.53 | 0.4837 | 21.47 | 0.5524 |\n| FlowDPS (8x ensemble) | 23.28 | 0.5157 | 22.06 | 0.5995 |\n| RSD | 23.45 | 0.5389 | 21.96 | 0.6836 |\n| FlowChef | 21.44 | 0.4898 | 20.20 | 0.5228 |\n| FlowChef (8x ensemble) | 22.60 | 0.5502 | 21.60 | 0.5931 |\n| Ours | 22.83 | 0.3627 | 21.05 | 0.4270 |\n| Ours (8x ensemble) | 23.79 | 0.4244 | 22.27 | 0.4930 |\n\n100 samples of FFHQ\n\n| Method (FFHQ) | x8 PSNR | x8 LPIPS | x12 PSNR | x12 LPIPS |\n| --- | --- | --- | --- | --- |\n| Resample | 26.26 | 0.3860 | 25.31 | 0.4606 |\n| FlowDPS | 29.02 | 0.3659 | 27.74 | 0.4036 |\n| FlowDPS (8x ensemble) | 30.12 | 0.3267 | 28.65 | 0.3749 |\n| RSD | 29.69 | 0.3915 | 28.11 | 0.4624 |\n| FlowChef | 28.05 | 0.3303 | 26.56 | 0.3609 |\n| FlowChef (8x ensemble) | 29.54 | 0.3267 | 28.15 | 0.3602 |\n| Ours | 29.36 | 0.2028 | 27.42 | 0.2594 |\n| Ours (8x ensemble) | 30.94 | 0.2457 | 29.00 | 0.2999 |\n\n**10. HDC ablations settings**\n\nWithout HDC we perform only 1 step on the data term similar to some of the baselines. We will clarify this in the manuscript.\n\n**11. What is $\\nu=0.01$ in Table 1? In Line 204 you write that you use an observation noise of 0.5\\%**\n\nMany methods specify the noise in absolute terms without mentioning the data range. We use 0.5\\% of the data range as noise which corresponds to 0.01 on a range from -1 to 1. We will be consistent in the final version.\n\n**12. Reduction of NFEs**\n\nOur method can also be deployed with a reduced number of data term steps (e.g., maximum of 5), if a faster runtime is required. Furthermore, one can switch to a \"tinyVAE\" version of SD3.\n\nTo validate that the usage of the tinyVAE or less steps does not degrades the performance noticeably we also provide a metrics for x12 Super Resolution on 100 samples of FFHQ:\n\nIn our manuscript we set the number of NFEs to a fixed value for all methods and baselines for an easier comparison. For different tasks the required number might be lower or higher. While runtime optimization is out of scope for this project we want to highlight that the number of NFEs needed does not only depend on the posterior sampling method used but more so on the stochastic interpolant. For example RED-diff used 1000 NFEs for most of their experiments on a different model. \n\n| Method | Runtime (s) ↓ | Memory (MB) ↓ | LPIPS ↓ | PSNR↑ |\n| --- | --- | --- | --- | --- |\n| Resample | 88.02 | 19009.24 | 0.4606 | 25.31 |\n| FlowDPS | 34.15 | 12228.63 | 0.4036 | 27.74 |\n| RSD (no repulsion) | 21.19 | 12399.96 | 0.4624 | 28.11 |\n| FlowChef | 15.23 | 12227.72 | 0.3609 | 26.56 |\n| Ours (HDC, largeVAE) | 172.34 | 12389.37 | 0.259 | 27.42 |\n| Ours (HDC, tiny VAE) | 40.77 | 5960.22 | 0.256 | 27.59 |\n| Ours (5 data term steps, tiny VAE) | 22.46 | 5960.22 | 0.264 | 27.61 |"}]

Author rebuttal:

Meta-review: This paper presents FLAIR, a novel and well-motivated variational framework for solving inverse problems using powerful flow-based generative priors like Stable Diffusion 3. Through innovations like Deterministic Trajectory Adjustment and Calibrated Regularizer Weighting, the method achieves state-of-the-art results with excellent perceptual quality across a range of challenging tasks. After a comprehensive rebuttal that included new experiments addressing all initial concerns about novelty, runtime, and design choices, all four reviewers converged on a positive recommendation for acceptance.
Decision: Accept (poster)