Flower: A Flow-Matching Solver for Inverse Problems
Authors: Mehrsa Pourya
Bassam El Rawas
Michael Unser
Venue: ICLR 2026 Poster (2026)
Keywords: Inverse Problems
Image Reconstruction
Generative Modeling
Flow Matching
Ancestral Sampling
Primary area: generative models

Abstract: We introduce Flower, a solver for linear inverse problems. It leverages a pre-trained flow model to produce reconstructions that are consistent with the observed measurements. Flower operates through an iterative procedure over three steps: (i) a flow-consistent destination estimation, where the velocity network predicts a denoised target; (ii) a refinement step that projects the estimated destination onto a feasible set defined by the forward operator; and (iii) a time-progression step that re-projects the refined destination along the flow trajectory.  We provide a theoretical analysis that demonstrates how Flower approximates Bayesian posterior sampling, thereby unifying perspectives from plug-and-play methods and generative inverse solvers. On the practical side, Flower achieves state-of-the-art reconstruction quality while using nearly identical hyperparameters across various linear inverse problems. Our code is available at https://github.com/mehrsapo/Flower.

Published as a conference paper at ICLR 2026
FLOWER: A FLOW-MATCHING SOLVER FOR INVERSE PROBLEMS
Mehrsa Pourya Bassam El Rawas
Michael Unser
Biomedical Imaging Group, EPFL. Lausanne, Switzerland imehrsa.pouryee bassam, el elrawas, mi michael. michael..nser:@epff]
ABSTRACT
We introduce Flower, solver for lincar inverse problems. It leverages pre- trained flow model to produce reconstructions that are consistent with the ob- served measurements. Flower operates through iterative procedure steps: (i) flow-consistent destination estimation, where the velocity network predicts a denoised target; (ii) refinement step that projects the estimated des- tination onto a feasible set defined by the forward operator; and (iii) a time- progression step that re-projects the refined destination along the flow trajec- tory. We provide a theoretical analysis that demonstrates how Flower approx- imates Bayesian posterior sampling, thereby unifying perspectives from plug- and-play methods and generative inverse solvers. On the practical side, Flower achieves state-of-th thettrrrrt reconstruction quality while using nearly identical hy- perparameters across various linear inverse problems. Our code is available at attpas///gttbbb..commmaaaaappppp
three
INTRODUCTION
Inverse problems are central computational imaging and computer vision (McCann & Unser 2019 Zeng 2001 Their goal is to reconstruct an underlying signal x E RR from its observed measurements R Here, focus linear inverse problems, such that the acquisition of the measurements follows the model
Hx +
(1)
for some linear forward operator H: Rd R and additive white Gaussian noise N I From a Bayesian perspective, the simplest reconstruction approach is to obtain the maximum- likelihood estimation
XMLE arg max PY|X xx (y) arg min XERd
Hx yy1
20
xERd
However, this problem is ill-posed and yields poor-quality solutions. Another approach is to obtain the maximum a posteriori estimation (MAP)
Hx y11 logpx(x))
*MAP = arg max PXJY -y (x()) = arg min XERd
(3)
20
xER2
which requires the knowledge of the prior distribution Px of images, a quantity that is generally unknown. The minimization problem of equation 3 is consistent with the variational perspective of inverse problems, where the term (- log (x)) is replaced by a regularizer R(x) that encodes some properties of the images. From classic signal processing to the advent of deep learning, the de- sign of a good regularizer has been of interest. Classic signal processing relies on the smoothness sparsity of images introduce wavelet- or total-variation-bassee regularizers (Rudin et al. 1992
Figueiredo & Nowak 2003 Beck & Teboulle 2009). Some methods build upon classical models
and try to learn such criteria in data-driven manner (Roth & Black 2009; Goujon et al. 2024
Ducotterd et al. 2025; Pourya et al., (2025). Plug-and wog--and-lay (PnP) approaches focus on the implicit replacement of R. by its proximal operator, with learned neural network that serves denoiser

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Venkatakrishnan et al. 2013 Zhang al. 2022; Hurault al. 2022b ap. Although MAP estima- tions tend to have a good reconstruction quality, they do not necessarily provide the minimum- -ean- square estimator XMMSE that is best in terms of the peak signal-to-noise ratio (PSNR). To estimate SMMSE, one would have to compute the posterior mean XMSSE = X = y)- Moreover, for perceptual metrics, it is better generate sample from Px|Y=y instead of estimator of the distribution.
The objective of generative modeling is to sample from target distribution px. In practice, this distribution is unknown, and one typically only has access to a finite collection of its samples. Numerous approaches have been proposed to address this issue. Among them, diffusion models (Sohl-Dickstein et al. [2015] Ho et al. 2020; [Song et al.) 2020) and, more recently, flow-matching methods (Lipman et al. 2023) represent the state of the art scalable generative modeling for images.
Flow matching, introduced Lipman al. (2023). constructs a continuous-time generative process by parameterizing the velocity field of an ordinary differential equation (ODE) as neural network. takes inspiration from optimal transport and continuous normalizing flows (Ambrosio et al. 2008 initial source distribution PX. to a target distribution PX, R3 Px The choice of probability paths from Px, to px, are numerous, with Gaussian paths recovering diffusion
Hagemann et al. 2022) and transports
special case (Albergo & Vanden Vanden-Elinden 2023). However, flow matching mostly focuses on straight-line paths, which yields competitive performance and improved sampling efficiency (Liu al. 2023
Liu 2022).
The remarkable success of generative models in image generation motivates their extension to in- verse problems, where the goal shifts from the sampling of the prior distribution px, to the sampling of the posterior Px, YY=v. Several inverse solvers based on diffusion models have been introduced
(Chung et al.) 2023 2024 Kawar et al.) 2022, Song et al., 2023 Zhu et al. 2023 Zhang al., [025]
Mardani et al. 2024). Recent efforts also focus on
solvers (Pokle et al. 2024| Martin al. a 2025). Existing approaches can be broadly grouped into two categories: (i) methods that ap-
proximate the posterior score (velocity field) with gradient corrections along the generative path; and
(ii) PnP strategies that alternate between generative (diffusion or flow) updates and data- catta-conssteecyy steps.
In this work, we introduce a novel solver based on flow matching that achieves state-of-the-ee results among flow-based methods for linear inverse problems. Our approach departs from existing methods by framing the problem through a Bayesian ancestral-sane -eccstral-ammpiing perspective, which gives rise to a simple three-step procedure with natural plug-un plug-and-pl and-play interpretation. Our main contributions are as follows.
1. Flow-matching solver for inverse problems. We introduce Flower, an inverse problem solver that consists of three steps: (1) flow-ccosistent destination estimation, where the velocity network is used to predict a destination, interpretable as denoising; (2) a measurement-awwrr refinement, in which the estimated destination is projected onto the feasible set defined by the forward operator; and (3) a time progression, where the refined destination is re- re-projected along the flow path.
2. Bayesian analysis and relation to PnP. We provide a Bayesian analysis in which demonstrate how and under what considerations Flower generates approximate poste- rior samples from Px, Y=y- Specifically, we show that Step computes the condi- tional expectation E[x, X, x.). which we then use for the approximation fx1 [x= of =x Through this approximation, we show that Step 2 generates a sample x(x,,)) ~ Px, Y=y. Step then updates the trajectory given the refined desti- nation X x(x,, y) and draws a sample Xt+ ~ Pxt+ XXX+,XX which by induction and Px, Y=y- These steps rely on three as- sumptions: the velocity network is optimally trained for unconditional flow matching; the acquisition of measurements follows the forward model in equation [ and the source and target distributions are independent. To the best of our knowledge, this Bayesian construc- tion is novel within flow-based solvers for inverse problems. Although this construction is key to the derivation of our solver, the resulting procedure closely mirrors the PnP meth- ods. Thus, Bayesian justification establishes a link between the PnP approach and
ancestral sampling, produces the final sample
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approximate posterior sampling with generative models for linear inverse problems. We also discuss a possible extension of Flower to nonlinear inverse problems.
3. Numerical validation. We first examine controlled setup with Gaussian mixture mod- els and show that Flower successfully recovers posterior samples. We then evaluate our method on standard inverse problem benchmarks for flow matching. We achieve competi- tive performance, with nearly identical hyperparameters across all tasks.
The remainder of this paper is organized as follows. In Section we review the fundamentals of flow matching along with the mathematical tools required for the development of our method. We then introduce Flower in Section and present the associated theoretical analysis. In Section [ discuss related work and highlight their similarities and differences with our approach. We report numerical results in Section 5 Finally, provide potential nonlinear extension of Flower in Section
BACKGROUND
2.1
FLOW MATCHING
Let Px. be souree distribution that is easy to sample and let px, be target distribution that want to sample from. A time-dependent flow th transports px, to px, via the ODE
dv/((x)
ve((b(x)).
E 10.
(4)
for some velocity field ve : Rd + Rd The intermediate variables X, = (X.) follow a distribu- tion px.- The objective of flow matching is approximate v, with neural network vg, which will allow us to sample from px, However, the determination of the flow-matching loss
CFM = [E-22(0.11 Exxxpx,
||v]
vv((x)
(5)
challenging, it requires access to the marginal velocity field v. (x²). To address this, we focus on the conditional velocity (x, X1) and define the conditional straight-line flow and velocity
x
2/x X
x1) = (1 t) xb tx1,
Ve(Xi x1) X1 Xg.
(6)
This leads to the practical conditional flow-matching loss
(CFM(0) = E2-2((0,11 E(xp:x))<< || (0v(((11)))) - t) tx1,t) (x1 x0)
where x E [[xx,, ) is a coupling over (Xo, X1), given by joint distributions on Rdd x IRd with marginals px. and px, Lipman al. (2023) have shown that the minimization of CCFM equivalent to the minimization of LFM, since their gradients with respect to 0 are equal.
The coupling æ determines how (Xn, X1) are paired. With the independent (IND) coupling Px. @ ,x,, the training is simple and scalable. However. the resulting interpolated paths can over- lap, which may slow down convergence. At the other extreme, the optimal transport (OT) cou- pling x² E arg MIDx -Px) ) E (xo,x;)>>> [||x] x,l12] produces globally aligned pairs such that straight-line flows approximate displacement interpolation along the Wasserstein-2 geodesic. If the Monge map T' satisfying T T#PX. = Px, exists and is known, then no training is needed: the sampling Xy ~ Px, and the computation of x1 T(xx) already generate sample from Px, In practice, is unknown and its approximation infeasible. practical compromise mini-batch OT, which solves an entropically regularized OT problem within each batch to compute an approx- imate coupling स. This improves alignment over independence with moderate computational over- head. For more details the mathematical background of OT, such as the definition and uniqueness of the Monge map, we refer to Peyrě (2025).
The choice of the source distribution px, is crucial for effective training and sampling. In practice, Px. is often chosen as the standard normal distribution (0, I). With the independent coupling, this simply yields Px, X, =x; Axxx,, (1 I)) However, the computation of px,xx== xx, in the OT case is challenging due to the mini- batch approach, in which is difficult to determine the batch sample x, came from, well its associated OT paths.
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Step 1: Flow-consistent destiaation estimsion
Step 2: Peessreeerttawwar destination refinement
Step 3: Movement in time
y'
e f vi(
At
(.()))
Provinal Stap
Proposition PaurailBumm Yoy
Propoctionn (1(x) E[X,XX X.)
Proposttin (x())) Ps.
Ancestral Sompling Thessem1 PR...//Yr
Figure 1: Overview of the three steps in Flower. Starting from initial sample X ~ Px. time t, the method: Step 1 predicts flow-consistent destination & (x)); Step 2 refines this destination using the measurements via a proximal step and associated uncertainty sampling obtain X1 (x,, Y); and Step 3 updates the trajectory along time by interpolating X₁ (x,,) with new noise ~ PXo The N-time repetition of these steps yields the final reconstruction X1.
2.2 PROXIMAL OPERATOR
The proximal operator of a proper, lower semi-continuous convex function f: R - R {Ufo) is defined as
(1 w x1 + [(())
prox (x) arg min
wER'
This operator can be interpreted as generalized projection of onto set associated with f. bal- ancing proximity to x and regularization by f. Proximal operators play central role in optimiza- tion algorithms that solve inverse problems and are key components of moximal-gradinn methods (Bubeck 2015).
3
METHOD
Let vg denote a velocity network trained to generate samples from px, through flow matching. Therefore, starting from Xo PXo, we get sample x1 px, if we perform N iterations of the update equation
X = x + A (x:)
with At N We aim to use the pre- pre-trained velocity network ví to generate solutions X₃ that consistent with the flow and the linear forward model y Hx for (N0,, T) as described in equation To achieve this goal, we introduce Flower which, given the measurements, y, modifies the unconditional flow path of equation 9 and outputs x by iterating N times over three steps. We first introduce these steps and then theoretically establish how and under what assumptions Flower approximates sample X₃ of the conditional posterior Px, The three steps are follows.
1. Flow-consistent destination estimation
(x,) x, (1 t)v;
(10)
2. Measurement-aware destination refinement
(x,, Me y) + yKa
(11)
for
Mx(x,, = prOx,3F, (X1 (x:)),
(0, E.),
(12)
Hx y11 and prox,eF, denotes the proximal operator of (1-1) vfFy as defined in equation 8 We have that it VP+(1-1) and that E,
where (x) =
(vr 2 2 uncertainty of the destination refinement step.
H)
The hyperparameter E {0,1) controls the consideration of the
3. Movement in time
XI+AI (1 - - A/je + (1 (x)))
(13)
where is newly sampled from px, each iteration.
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Here, At and the scheme is initialized with sample Xo PXo- In Figure we present a visual illustration of these three steps. We also summarize these steps in Algonthm of the Appendix.
We now interpret Flower through a Bayesian lens. We assume that, at each iteration, the three steps collectively draw Xt+Ai from the transition distribution PX.+ x,=x, Y=y- Under this assumption and with proper initialization, the procedure performs ancestral sampling along the conditional tra- jectory. By induction, we obtain Xr+At PX.+,./Y=y> We formalize this in Theorem [T, with proof in Appendix [8.21 which in turn implies that the final sample X produced by Flower Tollows the desired posterior Y=y We then detail how, in practice, the three steps realize draw from Px(+ar ad "X,, Y=y. Y=y which serves as an approximation of Px (x
Theorem 1. Let Xo be a sample from Px-lY-y- If x is a sample from Y-my then the sample follows px; -yy
X++AI from Px, xx.xxxxxxxxx
Remark 1. For the inductive argument to hold, Flower must be initialized with a sample from the conditional distribution Y When Xo and X₁ are assumed to be independent, this reduces to a sampling from the unconditional prior PX.. which often chosen as (0,1).
Theorem D presupposes the existence of an ancestral-sampling scheme to generate samples from Px. -Y-- This scheme requires a sampling from the transition distribution =xx+,. Y-y We now describe how to realize this transition in practice. We proceed sequentially and explain the details of each step of Flower.
First, under the assumption that vg is the optimal velocity network, show in Proposition | that the predicted &1 (x,) in Step equals the conditional expectation E E|X, X; = =x]]. The proof is provided in Appendix 8.2.2
Proposition 1. If (x,"(x,,) = v7(x) is pre-trained velocity vector field that minimizes the condi-
tional matching loss, then
& (xx) E|X1 X, xx] = xx (1- t)v (x,,
(14)
Since the distribution px |x==x, is not directly available, we propose to approximate it with
xx (xx (/?),
(15)
an isotropic Gaussian distribution centered at (x,) = E X X x.) with time- covari- ance. As t + 1, the distribution px, approaches the target px, For px, X,=x. to be consistent with this property, i4 should anneal in time. We choose the valid covariance when px, is a standard Gaussian distribution. Our approximation is indeed the IIGDM approximation proposed by Song et al. (2023) within diffusion solvers and later by Pokkc et al. (024) for flow matching. However, instead of having a score-based interpretation and using this approximation to obtain Vx, log PY =x=x> we propose to sample Xx (X,, y) from the distribu- tion Px, x "x,, Y=y that approximates Y
(1 1)/
/ (1 -1- z)z which results in
To this end, we show in Proposition 2 that px, [X.=x. is indeed Gaussian distribution, using the IIGDM approximation and the forward The proof is provided in Appendix 8.2.3
model of equation]] T
(x,((x)), (T (IIGDM approximation) and PY|x,=x; = -HHx,, I2I1) (measurement operation). Then, XX = -X(, E.),
Proposition Suppose that Px, Xx-xx.
where
Mi y) (v,21 21 on
H
H
(v, (x)) on ' HTy),
(16)
Σ, = 2 H
(17)
Proposition 2 allows us to sample from 7xx X.=x. as provided Step of Flower using
the re-parameterization trick in equation 11 However, the | in Step (see equation 12
is described using proximal operator which differs from equation [6] It is easy to verify the cquivalence between the two, through the fact that (xx.,)) of equation 16 can be written as the solution to the minimization problem
min xER
Hx
y112
xx 1 (xx)))1
(18)
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207
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This directly results in (x(x,,)) = FF (x1 (xt)) for Fx(x)
2n H yll2 under the
definition of the proximal operator in equation g Moreover, the sampling from the anisotropic
Gaussian (0,, Σ₁) is not trivial; however, if we sample two independent 6 E R and 62 E R
from standard Gaussian distributions, then we verify in Appendix 8.2.5 that Ra = [v - c1 +
On TH 62) follows N (0, ).).
Step 3 of Flower aims to sample the distribution (X for our ancestral-sampling procedure hold. We now show that if we have a sample x1 (X,, y)
which also what we required
from fx, 1X 1X==x.YY Y-y, then we could obtain sample from the distribution Px.+ad|X x.. Y=y using
XX =x, under the assumption that px, is independent of Px, in Proposition 3 which we prove in Appendix 8.2.4
equation 13 We first compute fx,
Proposition 3. from the pre-trained flow matching we have that px. =N(O,I), PX. is inde- pendent of px,, and if xx Y (x1) N (H,( Ee). then it holds that
PX.+ar|X.
N ((L + AC) (L A + (1 (1-t t- 1t)
(19)
From Proposition 8.2.4 and by the means of the re-parametrization trick, it is easy to verify that
X(+a1 = (1 (1-1- AL)E + (1 + A (x.,y)
(20)
follows |X, given that X ~~ N' (pe (X y). E,), which happens in Step 3 of Flower.
In our Bayesian justification, we assumed that px, and px, independent. This assumption ex- cludes, for example, the mini-batch optimal-transpon coupling. Nevertheless, in practice, Flower can still be applied in such settings and interpreted in PnP manner. The iterative structure of Flower closely resembles PnP methods: Step acts as a denoising step, while Step 2 enforces data consistency. From this viewpoint, Step 3 can be seen discussed in Martin et al. (025). However, rather than relying solely on this interpretation, we pro- vide Bayesian justification of the procedure. This perspective highlights a conceptual link between PnP methods and posterior sampling. We also address this empirically in our numerical results. For exact posterior sampling to hold under our approximations, y should be set to one. Interestingly, in practice we find that the choice = 0 (i.e., ignoring the uncertainty in the destination refine- ment step) yields better reconstruction quality. We provide further discussion on this effect with our numerical results in Section 5 Moreover, our framework remains valid for more general noise distributions n ~ (O, R.,) where R., is any symmetric positive-definite matrix, as detailed in Appendix 8.3
re-projection onto the flow trajectory, as
RELATED WORKS
An extensive body of work adapts pre-trained diffusion or flow priors to inverse problems by modi- fying the dynamics to approximate the conditional posterior. We first review diffusion-based solvers, then flow-based ones. Throughout this section, we highlight how Flower differs from similar meth- ods. We use the flow-matching notation with source X, and target X1.
Among diffusion solvers, DPS (Chung et al. 2023) approximates x by P where x (xx) is the diffusion-based denoised version of Xt, which leads a gradient correction to the dif- fusion dynamics. IIGDM (Song et al. 2023) approximates px, x,=x, = (x (x+), 1/21) for some time-annealing Fn and replaces the gradient correction of DPS with pseudoinverse-bbsed update. Flower adopts the same approximation as IIGDM, but the subsequent steps differ. DDS (Chung al. 2024) shares the same perspective DPS but replaces the gradient with a proximal step mo- tivated by a manifold-preservvvn similar structure through quaf--uuaratic splitting, alternating proximal data updates with diffusion denoising. Both DDS and DiffPIR structurally close Flower but, unlike Flower, they lack the Bayesian justification that interprets the updates as posterior sampling. DAPS (Zhang et al. 2025) also uses ancestral sampling similar Flower, but instead of directly computing Px, [X.=x., Y=y- applies Langevin updates for its evaluation, which is more computationally demanding but extends naturally to nonlinear inverse problems.
gradient perspective. DifiPIR (Zhu et al. 2023) arrives at a very
In the flow-me www.hinthh domain, OT-ODE (Pokle et al. 2024) employs a IIGDM- -baeed approximation, similar in spirit to that of Flower. However, instead of adopting our ancestral sampling scheme, they,

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similar the approach of IIGDM, approximate the score of the conditional distribution PY x, xx in order to construct the new velocity field. Therefore, while our method, Flower. relies on the methods act in different ways. FlowDPS (Kim et al. 2025)
same approximation for "x,,x,=x0 extends diffusion posterior sampling to the flow models by decomposing the ODE via a Hlw--version of Tweedie's formula, which differs from the ancestral sampling approach of Flower. Flow-Priors
Zhang et al., 2024) tackle inverse problems by reformulating the MAP objective as a sequence of time-dependent MAP subproblems with closed-form evaluations by taking advantage of the velocity network. However, unlike Flower, their approach relies on the computation of Tir Vvg, which is costly. D-Flow (Ben-Hamu et al.) 2024) adopts an implicit regularization strategy, replacing the data-fidelity objective x - ||Hx yll with latent loss solution of the flow ODE. The latent loss is non-convex with an implicit regularization effect that prevents convergence to trivial solutions. The optimization is performed by back-prope prop-propagttion through ODE solutions, which is more computationally demanding compared to the steps of Flower. FLAIR (Erbach et al. (225) optimizes decoupled variational objective with time-dependent calibration scheme, rather than relying on the Bayesian perspective of Flower. PnP-Flow (Martin et al. 2025) introduces PnP framework that employs the velocity network as denoiser. Its update steps are similar to Flower, but we replace their gradient update with a proximal operation, which leads improved reconstruction quality. Moreover, Flower offers a Bayesian justification of the process while PnP-Flow is purely blug-and-play.
(H(((()))) - yll where f is
NUMERICAL RESULTS
Here, we benchmark Flower against state-of-the-art flow-based inverse solvers across a range of lin- ear inverse problems. In Appendix [.4.1] we validate the Bayesian interpretation of Flower through a toy experiment with Gaussian mixtures, where ground-trun posterior samples are computable. We also present further numerical results for Fourier sampling and non-isotropic Gaussian noise in Appendix 8.5.7
We implement Flower as described in Algorithm I of the Appendix. As a practical note, the com- putation Σ,b for any b E Bd requires inversion (2/ 'I o " - Wherever this inversion is required, we instead solve the corresponding linear system (v-21++ -I+o,,2 H)z = b using con- jugate gradients (CG) with maximum of 50 iterations and an f2-residual tolerance of 10-3 and return 2 as the solution of the operation. We found the CG implementation sufficiently efficient in practice due to the positive-definite structure of v 2 o, 2H H.
5.1 BENCHMARK EXPERIMENTS
The goal of this section is to benchmark our method against other Now-matching-baat solvers for inverse problems. For fair comparisons, we adopt the benchmark introduced Marttin et al.) (2025), which also includes otate-of-the-art PnP and diffusion models. We describe the datasets and experi- mental setup for completeness, present quantitative results in Tables [ [nd and provide qualitative examples in Figure Finally, we diseuss the key observations, highlighting the performance of Flower and its empirical considerations.
We use two datasets for our numerical comparisons. First, use (128 128) human-face images from Yang et al. (2015), denoted by CelebA. Second, we use resized (256 x 256) cat images from Choi et al. (2020), denoted by AFHQ-Cat. We normalize all images to the range [-1, 1]. We train on the full training sets of both datasets. We tune the hyperparameters of different methods using a validation set. For CelebA, we use 32 images from the dataset's validation split. AFHQ-Cat has no validation split, so following Martin et al. (2025), we construct one by selecting 32 images from the test set and removing them from that set. For reporting the metrics, we use 100 test images from each dataset, limit imposed by the computational cost of baseline methods (D-Flow and Flow Priors). For faster methods (including Flower), we also report metrics on larger datasets consisting of 1000 images for CelebA and 400 for AFHQ, which are presented in Appendix 8.5.6 These extended results show the same performance trends as the 100-image evaluations presented in this section.
We compare Flower against flow-matching solvers OT-ODE (Pokle et al.) 2024), D-Flow (Ben- Hamu et al., 2024), Flow-Priors (Zhang et al. 2024). and PnP-Flow (Martin et al. 2025). as well as two other baselines: PnP-GS (Hurault al. 2022b). state-of-the-art plug-and-play method, and
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DiffPIR (Zhu et al.
2023 J. diffusion-based inverse solver. All models (except DiffPIR) use the
same U-Net backbone (Ronneberger et al. 2015) trained with Mimi-Batch OT Flow Matching (Tong et al. 2024) and Gaussian latent prior. Pre-trained weights for the flow models and PnP-GS are taken from Martin et al. 2025). trained with learning rate 10-4 on CelebA for 200 epochs (hatch size 128) and on AFHQ- for 400 epochs (batch size 64). For Flower, additionally train variant without target coupling (Flower-IND) using the same hyperparameters, which corre- sponds to our theoretical setting. While Flower-IND achieves higher performance (see Appendix 8.5.2.. we primarily report Flower-OT for consistency with other flo--matching baselines. Training DifiPIR with this backbone proved ineffective due limited capacity, so following |Martin et al. 2025), we adopt pretrained model Choi et al. (2021) from the Deeplnv library (Tachella et al.)
2023 originally trained on FFHQ (Karras et al., 2019). This introduces some mismatch but pro- vides the fairest diffusion-based baseline. Note that we use the latest checkpoints from Martin et al.) (2025), but our averaging strategy differs from theirs. In Martin ct al. (2025), results are reported by grouping four images into one batch and then averaging across 25 such batches. In contrast, we recomputed the results using 100 independent averages over the images themselves. Consequently, our reported numbers differ from those in Martin al. (022)).
We evaluate performance on five restoration tasks: (i) denoising with Gaussian noise with Gn = 0.2; (ii) deblurring with 61 61 Gaussian kemnel (o% 1.0 for CelebA, σb = 3.0 for AFHQ-Cat) and additive noise on = 0.05; (iii) super-resolution (2x downsampling for CelebA and 4x for AFHQ- Cat, with on = 0.05); (iv) random inpainting with 70% of pixels removed (a,, = 0.01); and (v) box inpainting with centered 40 x 40 mask for CelebA and 80 x 80 mask for AFHQ-Cat (a,, = 0.05). To report quantitative results, we use peak hignal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and learned perceptual image patch similarity (LPIPS). Note that for PSNR and SSIM, higher values indicate better performance, while for LPIPS, lower values are better.
To ensure fair comparisons, we adopt the optimal hyperparameters reported in Martin et al. (2025) for each method, obtained via grid search on the validation set to maximize PSNR. For PnP-Flow, we report two variants: PnP-Flowl and PnP-Flow5, which apply one and five evaluations of the velocity network per denoising step, respectively. For Flower, we follow the same procedure, re- porting in Tables and 2 the output of a single evaluation (Flowerl-ON) or the average of five evaluations (Flower5-OT) -OT) .. A key property of Flower is that, apart from the number N of iterations and the knowledge of the noise level on. it uses the same hyperparameters across different inverse problems, unlike other flow models. In particular, aside from N, the only hyperparameter of Flower 7, which controls the uncertainty of the destination refinement. Across all setups, 7 = 0 yields higher reconstruction quality. As discussed in Appendix [.5.3] the choice 7 = produces sam- ples that appear realistic but requires the averaging of multiple runs to achieve competitive PSNR, whereas = 0 attains better quality with fewer averages. This observation is consistent with our toy experiments, where ? encouraged sampling from match the number of steps used by our main competitor, PnP-Flow. We provide further ablation studies on the effect of the number of evaluations for the averaging and different time discretiza- tions within Flower in Appendix 885.4 and 8.5.5 respectively. The full hyperparameter details are reported in Appendix (8.588
regions. For N, we always
Key Observations. On CelebA (Table D]. Flower achieves the best or near-best results across all tasks, with clear gains in deblurring and box inpainting. The five-step averaging further improves the results. On AFHQ-Cat (Table 2. Flower remains highly competitive and outperforms baselines in deblurring, box inpainting, and random inpainting, while PnP-GS is strongest in denoising. In these tables, bold numbers indicate the best results among single-average results of methods. Underlined numbers indicate the second best. Blue numbers highlight the overall best across all methods. We illustrate in Figure 2 representative reconstructions across denoising. deblurring, super-resolution. and inpainting tasks. Compared to OT-ODE, D-Flow, and Flow-Priors, Flower consistently pro- duces fewer artifacts, while also avoiding the over-smoothing often observed in PnP-Flow. These visual trends align with the quantitative results and highlight the robustness of Flower across diverse degradations. In F we illustrate the solution path of Flower for the box inpainting task shown in Figure 2 As expected, Step 1 produces flow-based denoised images, while Step 2 enforces con- sistency with the measurements. In this specific box-inpainting setup, Step 2 primarily aligns the region outside the box with the measurements and preserves the result of Step I inside the box. Step 3 then mixes the refined destination with fresh source noise; as t increases, the noise decreases and the reconstruction emerges. The injected noise is essential to prevent the velocity network from
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Table 1: Results on 100 test images of the dataset CelebA.
Denoising
Deblurring
Super-resolution
Random inpainting
Box inpainting
Method
PSNR
SSIM
LPIPS
PSNR
SSIM LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
Degraded
20.00
0.348
0.372
27.83
0.740
0.126
10.26
0.183
0.827
11.95
0.196
1.041
22.27
0.742 0.214
PuP-GS
32.64
0.910
0.035
34.03
0.924
0.041
31.31
0,892
0.064
29.22
0.875
0.070
DiffPIR
31.20
0.885
0.060
32.77
0.912
31.52
0.895
0.033
31.74
0.917
0.025
OT-ODE
30.54
0.859
0.032
33.01
0.921
0.029
31,46
0.907
0.025
28.68
0.871
0.051
29.40
0.920
0.038
D D-Flow
26.04
0.607
0.092
31.25
0.854
0.038
30.47
0.843
0.026
33.67
0.943
0.015
30.70
0.899
0.026
Flow-Priors
0.134
31.54
0.858
28.35
0.713
0.102
32.88
0.871
0.019
30.07
PnP-Flow
31,80
0.905
0.044
34.48
0.936
0.040
31.09
0.902
0.045
33.05
0.944
0.018
30.47
0.933
0.037
Flowerl OT (ours)
32.28
0.914
0.034
34,98
0.947
0.026
32.36
0.923
0.034
33.08
0.944
0.018
31.19
0.945
0.022
PaP-Flows
32.30
0.911
0.056
34.80
0.940
0.047
31.49
0.906
0.056
33.98
0.953
0.022
31.09
0.940
0.043
Flower5 OT (ours)
33.14
0.926
0.038
35.67
0.954
0.032
33.09
0.932
0.040
33.95
0.953
0.020
31.87
0.952
0.023
Table 2: Results on 100 test images of the dataset AFHQ-Cat.
Denoising
Deblurring
Super-resoluton
Random inpainting
Box inpainting
Method
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR SSIM
LPIPS
PSNR
SSIM
LPIPS
Degraded
20.00
0.314
0.509
23.94
0.517
0.444
11.70
0.208
0.873
13.36
0,223
1.081
21.80
0.740
0.198
PnP-GS
32.58
0.894
0.072
27.91
0.753
0,349
24.15
0.632
0.362
29.42
0.836
0.126
DitIPIR
30.58
0.835
0.189
27.56
0.728
0.342
23.65
0.624
0,402
31,70
0.881
0.062
OT-ODE
30.03
0.815
0.076
27.06
0.713
0.123
25.91
0.716
0.108
29.40
0.839
0.090
24.62
0.875
0.085
D-Flow
26.13
0.574
0.175
27.82
0.721
0.164
24.64
0.601
0.190
32.20
0.894
0.040
26.26
0.842
0.077
Flow-Priors
29.41
0.763
0.153
26.47
0.700
0.181
23.51
0,570
0.272
32,37
0.906
0.047
26.20
0.818
0.118
PnP-Flow
31.18
0.863
0.135
27.87
0.760
0,304
26.94
0.763
0.171
33.00
0,918
0.037
26.00
0.897
0.103
Flowerl OT (ours)
31.69
0.879
0.102
28.64
0.775
0,255
26.23
0.741
0.272
32.97
0.918
0.040
26.19
0.915
0.063
PaP-FlowS
0.168
28.19
0.766
27.37
0.774
0.183
33.75
0.929
0.048
26.68
Flowr5- OT (ours)
32.35
0.891
0.116
28.97
0.784
0,283
26.57
0.750
0,282
33.70
0.927
0.045
26.88
0.922
0.066
getting stuck at the previous iterate and to allow it to predict improved destinations. As shown in Table [3 of Appendix [8.3.T] Flower has a runtime that is similar to PnP-Flow and OT-ODE, with only slight overhead relative to PnP-Flow due to the proximal-projection step replacing simple gradient update, while requiring the same minimal memory. In contrast, D-Flow and Flow-Priors substantially slower and more memory-intensiee
6 POTENTIAL EXTENSION TO NONLINEAR INVERSE PROBLEMS
Flower as it is can handle different linear forward operators. To extend it to nonlinear inverse problems, Proposition 2 needs to be revisited. When the measurement operator H is linear, the likelihood P (Hx,, T is Gaussian, and, combined with the TIDM Gaussian prior Xx, N (x (xx), vi)), the approximate posterior remains Gaussian: px, (x=x,==> (X,, E.). This yields the closed-form mean μ4 and covariance Ej in Proposition 2
If the measurement model is nonlinear, i.c..
PY|x= = [h((xx)) o
(21)
for some nonlinear function h: Rdd + R then still holds that
xx,Ix.xx,==((x)) & exp 20 lly h2
1 [x] x1 (x)))
(22)
20p
which is no longer Gaussian due to the nonlinearity of h. Nevertheless, sampling from this distri- bution can be done using iterative sampling schemes, since the score (gradient of the log density) is available in closed form:
Vxj log Px Ixxxx. = o, (x()) T h(x)) - -x)x(xx))x)x xx()))
(23)
where Jn denotes the Jacobian of h. This score function could be used to generate samples of Px,Ix, =x, Y= via schemes such Langevin dynamics which provides a valid substitute for Step 2 of Flower, which enables the handling of nonlinear cases without requiring modifications to the remaining steps.

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Published as a conference paper at ICLR 2026
Orcard Truth
Doy PSR 25.02
OFFCCE, PINE 27.30
D-FON, PARR: 24.35
Flo PLATS 20.03 PEP-FON22 PSNT: 25-02
Riwword PSV::
Ocnsiiio
A710-CC
Day wind 20W 27181
DIAEE PARH
D170N PAR 17.41
NEPA06 FSS 11.31 110001 PADE 27
Debtar
Supc Rest-rrtttnn
Dogiadet PaW:: 1527
DFOCE Paht:
DFOR, PAMI:
MSP-3633 POWeY3.01
Pardimm Hipk mirg
OPOCE PARl
DVEON 2012
Box Inpottcnng
Figure 2: Visual comparison for flw-matching inverse solvers.
= 0.0
t= 0.19
0.39
0.30
0.79
U.99
Scpp (80))
Stop x
Step 3: Xe-ax
Time
Figure 3: Solution path of Flower for box inpainting.
7 CONCLUSION
We introduced Flower, a method that leverages pre-trained flow-matching models to solve linear inverse problems through simple three-step iterative procedure. By combining flow-consistent pre- dictions, mcasurement- refinement, and time evolution, Flower provides principled Bayesian interpretation while retaining the plug-and-play flexibility of existing approaches. Our analysis tablished the conditions under which the method recovers approximate samples from the conditional posterior. Our experiments demonstrated both validity on toy data and state-of-the the-artr performance across diverse inverse problems.
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Acknowledgements The authors acknowledge support from the European Research Council (ERC Project FunLearn, Grant 101020573) and the Swiss National Science Foundation (Grant 200020 219356). We thank Anne Gagneux and Ségolène Martin for their assistance in reproducing results, and Pakshal Bohra for insightful discussions on sampling schemes.
Ethics Statement. This work proposes a methodology for solving inverse problems in imaging us- ing pre-trained generative models. Our method designed as general-purpose solver and does not target specific sensitive domains. Our experiments are conducted exclusively on publicly available datasets (CelebA and AFHQ-Cat) that are commonly used in the literature. No private or otherwise sensitive data were collected or used. Our method has potential positive applications in areas such as medical imaging, but, as with other generative techniques, should be applied responsibly to avoid misuse in creating misleading content.
Reproducibility Statement. We aim ensure that our work is fully reproducible. We pro- vide detailed descriptions of the algorithm (Section 3 and its pseudo-code (Algorithm [ the theoretical analysis (Appendix [.2), and the experimental setup (Section [ and Appendix 8.5) including datasets, hyperparameters, training procedures, and evaluation metrics. All datasets are publicly available and cited in the text, and our method relies on standard architectures and benchmarks, allowing independent verification of our results. Our implementation is available at https: / wwwp raapo// rsapo/Flowrr
meh
Use of LLMs. The authors of this manuscript acknowledge the use of large language models (LLM) for grammatical polishing and typographic corrections.
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8 APPENDIX
8.1 FLOWER ALGORITHM
We outline the steps of Flower in Algorithm [
Algorithm Flower: Flow Matching Solver for Inverse Problems
Require: Measurements y, forward operator H, noise level on pretrained velocity vy, steps N,
uncertainty flag 7 {0, 1}
1: Set At 1/N; sample Xy ~ PXa
P be.g.
2: for k
N do
3:
= k
4: (Step 1) Destination estimate: (x:) Xt (1 vxxi)
V: Ve+(1-122" Fx(x) - Hx y112
5:
6:
(Step 2) Refinement mean: 14; = ,F (x1 (xx))
7:
(Step 2) Optional uncertainty: Σ, (v, II + 0 2
8:
sample E ~N Ia), ~20~ Im),
K₁ = Ex (vi - + on (HHHT)
9:
E (X;, y) = μr + K0
10:
(Step 3) Time progression: sample px, and set
11:
Xt+ = (1 t At) c (t+ At) x1 (x,,
12: end for
13: return X
8.2
PROOFS
8.2.1 PROOF OF THEOREM 1
To establish this result, we first recall ancestral sampling. is a procedure that enables us to draw samples from marginal distribution Pžx when the full joint distribution over sequence of vari- ables Z (Z, Zz,..... Zx) is defined a chain of conditional densities and when the marginal distribution of Zg can be written
Pzx (zk)
Pzx (ZK- (zz) ... PZ. =2 (Zz) pz, (2)) dzj ... ...dz---- (24)
Jzk-1
Ancestral sampling offers practical way to generate samples from pax without direct evaluation of this integral. The process samples sequentially from the distributions
2 Pz:. 22 P2,|Z
ZK Pzx 2
(25)
By following this sequence, we obtain a valid sample from the marginal ZK ~ PZx
Proof of Theorem bility, we obtain
By using the marginal distributions and the general chain rule for joint proba-
PX Pxy
X.aai (X, X(+ dx..
(XX (xyxx
(26)
then suffices to follow the ancestral sampling procedure with K = 2, Zi = X, = y. and
Zn X,+4/ y complete the proof.
8.2.2 PROOF OF PROPOSITION 1
Proof. Lipman et al. (2023) have shown that the velocity vector field that minimizes the conditional
flow- matching loss is
(x)) E [X
X, xx),
(27)
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which then yields
&1 (xx) = X = =x] = + (1-t)(X, - Xx)) = x = + (1 (x) (28)
This is the desired result under the assumption that v((x)) vi (x).
8.2.3 PROOF OF PROPOSITION 2
Proof. Using Bayes' rule, write
Xxx P (x) (x))
xYix xx (y)px, x, x,=x; (x))
XX (y)
PY|X,=X, (y)
(29)
where we used the conditional independence, given 1,, of X, and the measurement Y. We assumed that Px, [X,=x, is approximated with -x,, = [N{ (x,), af)a), and we have pY]x;=x; = N (Hx, I) by construction. Put together, we obtain the approximate density
Xx|IX = PY|X,=x, (y) [x=:x (XI)
(30)
Pr|x,=x, (y)
Taking the logarithm of equation
30
2og (Xx (x1)) is given by
(y Hx,) T o o Hx1) + (x1 - {xx)) 1/2 (x1 xx(x))) +
(31)
2x] (vi 2x (xx)
o H y) + x (v; "I on " H) x1 +
(32)
= 2x] (M 2 (x+) + の, H y) + x Σ, x1 + C²,
(33)
where C, O' are independent of x1 and considered constants and E, = I H which is well-defined because " + - H is positive-definite Completing the square with term independent of X1, we get
log (ix, Yx, (x1) (x1 me) = E (x1 -xe)
pe) + +C"
(34)
2H y) and C" is again constant independent of x1. This yields where p4, E, 21 (x:) to "
px,|x, x (x1) = c- exp (-5(x, (4) -.) (1.)
(35)
As =xx is a probability density function, e Q0"112 corresponds to its normalization factor, which therefore proves that Y=xx = (xxx; pc, E.).
8.2.4 PROOF OF PROPOSITION 3
Proof. By using the marginal distributions, the general chain rule for joint probability, and indepen- dence, we obtain the expression of PX. Xxxx. Y=y (X++Ar) as
J0 Px Yxy
Y=y (x++ dx) dxi
(36)
Rd Xx x. (x1
(x(+Ac) dx
(37)
Yxy ((())))(())))))))) (t At)xi, (1 - 1-At) dx1,
(38)
+ where used the fact that, conditioned X, X1, X,+A: (1 AI) +
At)xi N N(t At)x; (1 - t Lt) By inserting the expression of the approximation Xx (x1). the approximate density Xx (x is given by
N Nxi;m E Rd
-1 t -1)2 dxi
(39)
(xiiHe E (x+4A- - (t+ (1- - At)² Lo) dxi.
(40)
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This integral can be rewritten convolution of two Gaussian distributions, which also yields Gaussian distribution. Explicitly, we have that
J/L N (x1: N (xe+ai At) 0, (1 -1) Iz) dx,
(41)
= =fe N
( (t+ Ee) N (3L+A) -; (1 - A (t+ (t)d
dz
(42)
- N (z: (t At) (t 2)) NV (xetse-2;0, 2; - --)==//00 Is) dz.
(43)
Jud
Using the Gaussian convolution identity
f
E,) 2: =z) 12, E1 + +2),
(44)
the result simplifies to
(xx, N (x++ar (t (t) A1) (1-1- (1))-
(45)
which completes the proof.
8.2.5 SAMPLING FROM THE NON-ISOTROPIC GAUSSIAN
We want to show that
Re = Es (V, E1 - o HHH E2)
(46)
has distribution E.), where 61 ~ (a) and 62 ~ NO0, Las) are independent, and
E. (v 2H HH
(47)
Observe that K2 is Gaussian random vector because it is linear transform of the independent Gaussians é and 62. Moreover, since E and 62 are zero-mean, we have that [[Ke] 0. Next, we compute the covariance matrix of K2 as
Cov(Kr) E, Cov(u, a + T 'H (2) Ex.
(48)
By independence of e1 and E2, obtain that
Cov(v, E + o
'H "H' Te) t V1 2I on H H"H. H.
(49)
which implies that
Cov(ne:) = E, (yj 2 I o. H H)E. But, by definition, Σ. = (vr "I or
(50)
HH 2 H) which leads to the desired result,
Cov(Ne)) = E, (vj I + o. HH H E. Σ,
(51)
8.3 EXTENSION TO MORE GENERAL NOISE TYPES
Throughout this manuscript, isotropic Gaussian noise ~ of was considered. Our frame- work remains valid in the more general setting n ~ (.,).. where R,, is a symmetric positive definite M M covariance matrix, up the modification of some formulas detailed below.
The results of Proposition become
y) = (1/ "I. + R H)
(v, (x,) + H R,,yy)
(52)
Σ, = (Nj II +
(53)
since the measurement operation with general noise type implies that PY|x,=x] = N ((Hx, R.,.
Next, sampling from (0, E,) is achieved with a method similar to the one of Appendix 8.2.5 with the updated formula
K₁ = Ee ( + H R 22))
(54)
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1- 0.0
-- 0.166
(- 0.333
(- 0.5
t- 0.006
f- 0.833
1.0
Uaconditioal Thrgtt Xs
Curstrsnt mp
Putxtic
Proterier Tran Peeternc X
Time
Figure 4: Temporal evolution of 2D Flower and comparison with true posterior for noise variance
an 0.25.
This formula uses the standard definition of the square root of symmetric positive-defintt matrix A. A PDP is its cigenvalue decomposition. then A PD.PT where D is the diagonal matrix whose entries are the square roots of the nonnegative cigenvalues of A.
As result, in the measurement-awwer destination refinement step of Flower described in Section equation 12 now computes pr with
1 (x 5(y Hx) R (y Hx).
(55)
In the same step, the matrix E, redefined be in equation 53
In most imaging models, the noise is effectively isotropic. Moreover, if we assume that R,, is diagonal, the resulting expressions simplify and become straightforward compute.
8.4
NUMERICAL RESULTS EXTENSION
8.4. TOY EXPERIMENT
The goal of this experiment is to validate our proposed sampling perspective in a setting where samples from the posterior available. To this end, consider Gaussian mixture model (GMM) as the target distribution of the data X E R²2 The forward measurement model consists of single measurement vector RRd (i.e., the forward operator is H hT) corrupted by additive white Gaussian noise n ~ (O, o²)), which results in y h x + n. In this setup, the posterior distribution Fx Y=y itself GMM with known parameters, whose analytical expression given in equation 57 Geometrically, the noiseless measurement = h T x defines a line in the two- dimensional plane with normal vector h. Consequently, when sampling from the posterior Px Y=y- we expect the samples to concentrate on the portions of this line that intersect regions where the prior distribution px has high density.
We trained the unconditional flow-matching vector field using the source px, = -0 I) and target Px, = px, with further details provided below. We ran Flower with N 1000 iterations, which corresponds to the step size Al 0.001. The results are shown in Figure 4 where we used h [1.5, 1.5). os 0.25, and the observation y = and where we report the solution paths of Flower with 7 E {0. }} alongside true posterior samples. We observe, that when = 1 (the setting required by our theory), Flower successfully recovers the samples at true posterior. When 0 (a configuration in which the uncertainty in the destination estimation step ignored), Flower fails to capture samples from the tails of the distribution. present another example of Flower posterior sampling with h [1.5, 1.5], on = 0.75, and the observation y = 1. Once again, we observe that Flower successfully generates samples that closely match the true posterior for y In contrast, for y 0, samples from the tails of the true posterior are missing. In Figures 6 and 7 the solution path is illustrated across the successive steps of Flower for 7 = 0 and 7 respectively, which allows to visualize the dynamics of each step directly.
1, which closely resemble the
Figure 5 we
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Target Prior. The target distribution of our data X explicitly by
R² is a GMM with uniform mixtures given
Px
K N 2)
Σ
(56)
for some K E N, PA E R2, and Σ E 8 Specifically, we use K = 3 with 14
(-0.25, 0.25), P2 = (-0.25, 0.25), M3 (0.25, -0.25), and covariance matrix E = 0.25
Target Posterior. The advantage of this setup is that the posterior distribution Px|Y=y can be computed exactly using Bayes' rule. It is given by
PXIY=y Σ (pR. Exost)
(57)
where, for all 1, ..... K, wk 2 0 some weights and
pospost (
hh
) ( + --- +
(58)
E E
Epost
hh
(59)
Training Details. The underlying unconditional velocity network is fully connected network that takes as input 2D vector and scalar time, concatenated into a 3D input. consists of two hidden layers of size 256 with SiLU activations, followed by final linear layer that outputs 2D vector. For training, we use batch size of 2048 with 20000 training steps and learning rate of 10-3
18

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==
t= 0,106
0,333
1=
0.008
0.833
1.0
Unrendricnal Targrt X,
Constrairr
Pasterice I
The Posterin X
Time
Figure 5: Temporal evolution of 2D Flower and comparison with true posterior for noise variance on = 0.75.
0.0
1- 0.1:2
[= 0.285
1 0.228
1l57
0.713
= 0.856
0,000
Sup 1: xxxe
Stop 3:
Tine Samples
Unoonctioral Target
hTs
Figure 6: The three steps of 2D Flower with temporal evolution for 7 = 0.
0.0
1= 0.122
0.25
<= 0.128
0.57
0.713
t= 0.830
== 0.999
Step I: x:bo)
sls..g!
Goco 3
Tine
Unconcttoaal Target
Samples
Figure 7: The three steps of 2D Flower with temporal evolution for 7 = 1.
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8.5 BENCHMARK EXPERIMENTS
8.5.1 1 COMPUTATIONAL EFFICIENCY
We report in Table 3 the computational time and memory usage for several methods. Each entry corresponds to the average runtime for the deblurring inverse problem, averaged over 10 CelebA test images of size 128 x 128. All experiments were conducted on a Tesla V100-SXM2-32GB GPU.
Table 3: Computation times and memory usage for various methods.
Method
OT-ODE
D-Flow
Flow Priors
PnP-Flow1
Flowerl
Time (s)
6,549
142.18
63.771 3.807
3.020
5.622
Memory (GB)
1.183
11.125
0.216
0.217
8.5.2 EFFECT OF SOURCE- TARGET COUPLING
Our goal in this section is to benchmark the effect of source-target coupling in the training of the underlying (unconditional) velocity network. To this end, we consider two variants of coupling: one based on mini-batch optimal transport (OT) coupling and the other on independent (IND) coupling. For each variant, we report two settings: single evaluation of Flower; and average over five eval- uations. This results in four cases, summarized in Table H where we provide results for all inverse problems discussed in the main text, evaluated 100 test images from the CelebA dataset. For this table, we fix 7 = 0 and N 100.
Table 4: Effect of source-target coupling of the underlying velocity network of Flower on different inverse problems on 100 test images of the dataset CelebA.
Denoising
Deblurring
Super-resolation
Random inpainting
Box inpainting
Method
PSNR
SSIM LPIPS
PSNR
SSIM LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
Degraded
20.00
0.348
0.372
27.83
0.740
0.126
10.26
0.183
0.827
11.95
0.196
1.041
22.27
0.742
0.214
Howerl-OT (ours)
32.28
0,914
0.034
34.98
0.947
0.026
32.36
0,923
0.034
33.08
0.944
0.018
31.19
0,945
0.022
FlowerS-OT (ours)
33.14
0.926
0.038
35.67
0.954
0.032
33.09
0.932
0.040
33,95
0.953
0.020
31.87
0.952
0.023
Flowerl LND (ours)
32.60
0.918
0.032
35.22
0,950
0.026
32.65
0.927
0.034
33.23
0.947
0.017
31.90
0.950
0.021
FlowerS-IND (ours)
33.48
0.930
0.037
35.90
0.957
0.031
33.41
0.935
0.039
34.24
0.955
0.020
32.78
0.958
0.022
We observe that independent coupling improves the results, which is consistent with our theoretical requirements. Nevertheless, the OT-based variant remains highly competitive, as also illustrated in we show an example from the deblurring task
the main paper. For visual comparison, in Figure X on an image from the CelebA dataset.
8.5.3 EFFECT OF
The hyperparameter E {00 in Flower controls whether the uncertainty of the refinement step (Step 2) taken into account. While 7 is required for a Bayesian interpretation of our method, in practice we find that 0 yields more favorable image-reconstruution metrics. This observation consistent with the toy experiments, where 7 = 0 led Flower to generate samples concentrated in higher- regions, while failing to capture the tails of the posterior.
To illustrate this effect, we provide a visual example of the deblurring task on a CelebA image, using the velocity network trained with mini-batch OT coupling for consistency with the numerical results reported in this paper. single reconstruction with 7 = achieves a PSNR of only 30.84, compared to 33.01 for single reconstruction with y 0 (Figure 8). Moreover, with 7 1, it is necessary to average over 100 reconstructions to reach a PSNR comparable to that of y = 0, where we average only over 5 reconstructions (Figure
To further illustrate this behavior, we show in Figures 6 and the solution paths of the three Flower steps over time for y 0 and 7 respectively. We observe that 7 leads to noisier refine- ment step. Consequently, we adopt 7 = 0 for all inverse problems, as this provides a more stable refinement step and consistently yields better quality of reconstruction despite fewer averages.
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Grexal Trath
Degnske, PENT: 24:6
PlowrkOOT PSMP salb. 0
Flenn--O PSMte 31.0 7-0
Cheen--ND PSMP 36.1
ElweesOND PSMle 20
Doottrinng Coocba
Figure 8: Effect of the source-target coupling for the underlying flow (7 0),
Daguiel
FoweoBoTT Flooe33000 PROEE
Teblering
Figure 9: Deblurring results with y = 1, obtained by averaging over 1, 5, 50, 100, and 200 runs of Flower.
0.00
<= 0.19
= 0.30
[= 0.59
0.79
0.99
Stap +) Slep X(x)))
St Xo-at
Time
Figure 10: Solution path of Flower for deblurring with O.
==
Slep X(x))
Stap
Slep Xe-st
Figure 11: Solution path of Flower for deblurring with 7 1.
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8.5.4 EFFECT OF THE NUMBER OF EVALUATIONS FOR AVERAGING
In Tables[and22a the main paper, we reported results for Flower-I and Flower-5, which correspond to using single evaluation and the average over five evaluations of Flower, respectively. Here, we ablate the effect of this averaging on both the reconstruction metrics and the computational cost. We focus on the deblurring and random inpainting tasks described in Section [5.] for the CelebA dataset. Specifically, we vary the number of Flower averaging NAvg from average PSNR, SSIM, LPIPS, and runtime (in seconds) over 100 CelebA images, measured on Tesla V100-SXM2-32GB GPU. For the hyperparameters of Flower, we use 7 = 0, 100.
and report the
As shown in Figure 12 PSNR and SSIM improve with steeper gain for smaller numbers of averag- ings (up to around five) and tend to saturate afterward. LPIPS also increases with averaging, which is undesirable since higher LPIPS indicates worse perceptual quality. The runtime scales linearly with the number of averagings, as expected. Finally, in Figure 13 we present a visual comparison that confirms these trends: averaged reconstructions (e.g., using 5 10 evaluations) appear smoother than the single-evaluation result, while the difference between averaging over and 10 evaluations marginal.
PSNR
SSIX
LPIP8
Time
PSNH (91))
LET1''
Tiaer 5:1
600
0100
Ruudon hegetezkng
pece (00))
TTue 66
6089
Nunbara Avtnngagg Nar
Numler Aeenging Nag
Xumber of Annmgginyy Nast
Yumbersk of Aancagigg Muve
Figure 12: Effect of averaging over different numbers of evaluations of Flower on quantitative metrics and computational time.
Degridee PSNR, 25.501, ESIM o 7458 LPIPS. 0.2802
Flonert OT PENR. 30.76, SSIM, c.927 LPIPS, DU3777
Pipentó OT PSNK, 31.30, SalM. 0.5053 .PIPS C.00
Floneri0 OT
PSNR, 31.44, ESIM 31.4, 0.9406 LPIPS 0.006
Grouas. Touth
The drrinng
Dagradee
Flowsr OT PENR: 26.78 SSTM: c,00ss LPIS% BJSTT
FlomecE OT PSKR: 27,62, SSTM: nu9u LPPS CJ0058
Fower20 PSNR: 27.58, 5STM: 0.1184 LPIPS 0,6771
PSNR: 12,68, SSIM 2753 LPIPS 0,3015
Grornd Truth
Random teyainting
Figure 13: Visual comparison of the effect of averaging over different numbers of evaluations of Flower.
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8.5.5 EFFECT OF ADAPTIVE TIME STEPS
Our theoretical framework does not tie Flower particular time discretization. This raises the question of whether non-uniform time steps could improve practical performance. To explore this, we compare the uniform (Euler) time discretization with several alternatives. Specifically, we con- sider a power-law schedule & = ( where a 1 recovers the uniform grid, a > 1 concentrates steps near the start of the trajectory, and a < allocates more steps near the end. In our experiments, 2. We further evaluate cosine schedule ty 1-cos(rk/M) which yields finer resolution at both the beginning and end of the trajectory. For clarity, Figure 14 visualizes these time grids for N 100. We vary the total number of steps N E (10, 20, 50, 1 and validate these schedules on two restoration tasks (deblurring and random inpainting) shown in Figures [5 and [6] For the hyperparameters of Flower, we use 7 0, and NAve = 1. Our results indicate that the a = 0.5 schedule achieves noticeably better reconstruction quality with fewer total steps N, while a 2 tends underperform relative the uniform case. The cosine schedule sometimes provides improvements at low step counts. For larger numbers of steps, all schedules converge to similar performance. Our findings highlight the potential of adaptive time discretizations (with more resolution toward the end of trajectory) to improve quality-compute trade-offs.
0.5 and
Offferent Dicretiations
06
Conrse
210
Figure 14: Time lg vs step l for different time discretizations.
8.5.6 INCREASING THE SIZE OF TEST SETS
In the main paper, we evaluated all tasks for each method using 100 CelebA images and 100 AFHQ- Cat images. We chose a test size of 100 because methods such as Flow Priors and D-Flow require backpropagation during inference and are therefore slow. In this appendix, we increase the test set to 1000 images for CelebA and 400 images for AFHQ-Cat in order to further validate Flower compared existing (efficient) flow-matching methods. The results, presented in Tables 5 and 6 show the same trend as in the main paper as Flower achieves competitive reconstruction quality.
Table 5: Results on 1000 test images of the dataset CelebA.
Denoising
Deblurring
Super-resolution
Random inpainting
Box inpainting
Method
PSNR
SSIM
LPIPS
PSNR SSIM LPIPS
PSNR
SSIM LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
Degraded
20.00
0.351
0.368
27.83
0.741
0.123
10.38
0.185
0.828
12.10
0.197
1.038
22.29
0.745
0.211
OT-ODE
30.53
0.857
0.032
33.06
0.920
0.029
31.54
0.905
0.024
28.74
0.870
0.052
29.65
0.911
0.035
PaP Flow
31.84
0.904
0.044
34.56
0.935
0.038
31.16
0.901
0.044
33.16
0.944
0.019
30.47
0.935
0.036
Howerl-OT (ours)
32.31
0.913
0.033
35.03
0.946
0.026
32.45
0.922
0.034
33.19
0.944
0.017
31.22
0.946
0.021
Table 6: Results on 400 test images of the dataset AFHQ-Cat
Denoising
Deblurring
Super-resolution
Random inpainting
Box inpainting
Method
PSNR
SSIM LPIPS
PSNR SSIM LPIPS
PSNR
SSIM LPIPS
PSNR
SSIM
LPIPS
PSNR
SSIM
LPIPS
Degraded
20.00
0.315
0.517
24.03
0.516
0.452
11.88
0.217
0.881
13.55
0.231
1.071
21.80
0.741
0.203
OT ODE
30.01
0.814
0.077
27.06
0.711
0.126
25.92
0.715
0.109
29.38
0.839
0.091
24.77
0.875
0.085
PaP-Flow1
31.17
0.862
0.136
27.94
0.759
0.306
26.96
0.762
0.170
33.01
0.918
0.037
26.46
0.102
F-O (ours)
31,66
0,878
0.104
28.63
0.773
0.255
26.24
0.740
0.273
32.98
0.918
0.040
26.68
0.915
0.062
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Ground Truth
PShR-23.41
PS1R=32.11 Timb=0.33s
PSNRw244.6. Timo=1044
P3MR-34.85 Time=2.1s
FSUR=24.59 Time-
Unlomm
Degratted
Strps
Steps
Power C.
Degrades
Steps
Shaps
Poware
aegrades
Hower1-OT
NRoor3-01:
Howeer..D No.30 Steps
Flown Sheps
Figure 15: Deblurring example for uniform versus adaptive solvers.
Degraded
Rower1-OT:
Flaver3-OT: N-50 Steps
Flowerd 0.00 Steps
Degrados
Homer1-oT:
Floow3-07 Me20 Staps
Hower:.-D Nm:30 steps
Picwnr 3.DT
Steps
Powirs as
Degraded
Rower1 OT N-10 Stene
Hlnerr) Ma Steps
Rlwwer OT Nu30 Steps PSKR=33.01
Flowe 1G0 Steps
Powe =20
Degrades
Rower1.on OT. Ne10 Stone
Rever1.Or 20 Stops
Rower No50 Steps
Fowe 3.0T 1o Stops
Colire
Figure 16: Random inpainting example for uniform versus adaptive solvers.
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8.5.7 APPLICATION TO FURTHER INVERSE PROBLEMS
We aim to validate the generality of our framework by evaluating it on wider range of inverse problems.
Compressed Sensing Fourier Sampling. We choose forward operator inspired by magnetic res- onance imaging (MRI) reconstruction: the Fourier transform followed by binary sampling mask. We use two masks: (i) Cartesian mask with a sampling ratio of 0.2188, and (ii) radial mask with sampling ratio of 0.2990. The measurements standard deviation o = 0.002. For our examples, we use two images from the AFHQ-Cat dataset. For the hyperparameters of Flower, are RGB, we apply the forward operator channel by channel. We summarize our results in Figure 17 In both cases, observe that Flower successfully handles this inverse problem and produces reconstructions that improve on the zero-filled fillefiledd baseline.
corrupted with additive white Gaussian noise of
y 0, N 100, and NAvg 1. Since the images
Gread Teed
PANE 2 oun
THE BE
Copodaa Vaal
Caad Tead
HOLE 11241
Figure 17: Reconstruction results for compressed sensing with Cartesian and radial masks.
Non-Isotropic Gaussian Noise. Here, we aim to verify the theoretical results from Appendix 8.3 for non-isotropic additive Gaussian noise using numerical experiments. For demonstration. we focus on two tasks: image denoising and random inpainting using the same setup described in Section S for the CelebA images, except for the noise dynamies. Here, add non-isotropic Gaussian noise to the image by applying Gaussian noise with
3 to the central box of size (64 x 64) of the (128 128) image and on outside this box. For the Flower hyperparameters, we use y O, N = 100, and NAvg 1. We observe that, consistent with our theory, are able to recover good-quality reconstructions given a non-isotropic Gaussian noise.
Degraded
Flowerl OT
Ground Trath
PSNR 21
PSNR: 31,07 dB
Demoising sinu
lack
lack
Hack
Degraded PSNR: 12:66 dD
Flo
Ground Truth
PSND: 25.18 AB
Randon lapainiing
Figure 18: Visual results for inverse problems with non-isotropic Gaussian noise.
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8.5.8 HYPERPARAMETER, FOR ALL METHODS
In Tables 7 and 8
report the hyperparameters that we used for all methods. Most of the hyper-
parameters are adapted from Martin et al. (2025).
Table 7: Hyperparameters for all methods on the CelebA dataset.
Method
Hyperparameters
Denoising
Debloriing
Supar-resolution
Random inpainting
Box inpainting
DiaPIR
(blending)
1.0
1.0
1.0
1.0
N/A
tregularicaion)
1.0
1000.0
100.0
1.0
N/A
PaP-GS
(carinn rate)
2.0
2.0
1.0
N/A
(inertia puram.)
0.5
(factor For noise input)
1.0
1.8
3.0
1.0
N/A
Pla (number iter.)
35
20
23
N/A
OTODE
to (initial time)
0.3
0.4
0.1
0.1
0.1
time- dependennt
lime-dependent
constant
depedeeenddent
Prw--rroos
(regularizaion)
100
1,00
10,000
10,000
10,000
(leaming rute)
0.01
0.01
0.1
0.01
0.01
D.How
fregularization
0.0011
0.001
0.001
(blcading)
0.1
0,1
0.1
0.1
0.1
Plur (number
10
20
PaP-Flow
o (leariing-rte -aainnng--eeeee--o factor)
0.8
0.01
0.3
0.01
0.5
(Number time steps)
100
100
100
100
100
Navg (Number averagings)
PaP-FlowS
(leariing-re raili factor)
0.8
0.01
0.3
0.01
0.5
(Number
stops)
100
100
100
Naug (Number of averagings)
5
5
5
5
Flowert-OT
trefinement uncertainty)
(Number time steps)
100
100
100
100
100
NAug (Number of averagings)
Flower5-OT
7 (refinement uncertainty)
0
N (Number of time steps)
100
100
100
100
100
NAvg (Number avoragings)
5
Table Hyperparameters for all methods on the AFHQ- dataset.
Method
Denoising
Deblurring
Super-resolutio
inpainting
inpainting
DiAPIR
(blending)
1.0
1.0
1.0
1.0
N/A
Arcgularizatin)
1000.0
100.0
PaP-GS
(learning ratc)
2.0
2.0
1.0
N/A
(inertia param.)
1.0
0,3
1.0
0.5
N/A
øy (factor for noise imput)
5.0
1,0
N/A
(number of iter.)
60
50
23
N/A
OT-ODE
(initial
dime-dependent
lime-dopendent
constant
constant
time-dependent
Flow-Priors
fregularization) 7 (learning rate))
10,000
10,000
10,000
0.01
0.01
0.1
0.01
0.01
D-Flow
(regularization)
0.001
0.01
0.001
0.001
0.01
(bleading)
0.1
0.5
0.1
0.1
(number iter.)
3
20
20
20
PaP-Floo1 fleaming-ra
(Number of time stops)
100
500
500
200
100
Navg (Number averagings)
1
FlowS
a fleariingrra rale factor)
0.8
0,91
0.01
0.01
0.5
(Number time stops)
100
500
500
200
100
Navg (Number averugings)
Flwwerl-OT trefinement uncertuinty)
0
0
0
0
N (Number time sleps)
100
500
200
100
Naug (Number averugings)
Flowers-Or (tefinement uncertainty)
0
0
(Number steps)
200
100
Navg (Number of averugings)
5
5
5
26


Reviews:
[{"review_id": "gTFR40nlQA", "reviewer": "Reviewer_kQup", "summary": "The paper proposes a solver for inverse problems by leveraging pre-trained flow models to produce reconstructions of observed measurements. The proposed solver use pretrained network to predict the target, which is then refined by a proximal step and noise-injection step. The paper provides theoretical analysis showing that the process aproximates bayesian posterior sampling, and provides empirical results on CelebA and AFHQ-cat datasets.", "questions": "Please refer to the weaknesses section for questions if any. Thanks.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "+ The paper proposes a new inverse problem solver that unifies PnP intuition with a Bayesian posterior-sampling narrative.\n+ Although I didn't check all the math details presented in the paper, it seems to provide a decent theoretical connection to bayesian posterior sampling using \\piGDM-style gaussian approximation.\n+ The practicality of this method seems good, with runtime comparable to PnP-Flow and OT-ODE; memory light (tab. 4).", "weaknesses": "- The empirical contribution of this submission, though, seems relatively weak. The main numerical experiments are conducted on two small scale datasets, 100 samples from CelebA and AFHQ-Cat, respectively.\n- The overall improvements upon previous methods seem not very significant (in tab. 1 & 2).\n- In terms of the limitation of assumptions, i) the paper focuses only on linear problems, and ii) if I understand correctly the proximal step further assumes isotropic prior and measurement operation, which could be oversimplified.\n- The most stand-out distinctions from previous flow-based inverse problem solvers are not very clear to me.\n- Admittely, I am not very familiar with the current progress of this particular sub-domain. I would therefore also like to hear other colleague reviewers' opinions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bqaGvkbOod", "reviewer": "Reviewer_rzV5", "summary": "Flower is a plug-and-play solver for linear inverse problems with pre-trained flow matching models. Given a pre-trained velocity model, Flower solves a linear inverse problem by iterating over three steps: \n1. Estimation of clean image $x_1$ given the current noisy image and velocity model. \n2. Projection of the estimated image $x_1$ into a feasible set with a proximal operator \n3. Estimation of the next noisy image along the flow using the projected $x_1$.  \n\nThe correctness of the proposed solver has been explained through Bayesian analysis and the key result shows that Flower samples from an approximate posterior distribution.", "questions": "1. Estimating $\\Sigma_t$ involves computing an inverse of a matrix. How do you efficiently solve this such that the solver itself is light weight?\n2. Line 342 states that the validation set for AFHQ-Cat was constructed with 32 images. However, the caption of Table 2 suggests that 100 test images were used. One of these statements need to be updated. \n3. What is $m_t$ in line 715? Is it $\\hat{x}_1(x_t)$? If that is indeed the case, it’s more readable to use $\\hat{x}_1(x_t)$ instead.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The core method has been explained intuitively and seems easy to implement. \n2. Computational efficiency: Runtime of Flower is less compared to many other baselines as indicated in Table 4. \n3. Robustness: The method uses same hyper parameters for all the linear inverse problems included in the paper. This simplifies hyper parameter tuning.", "weaknesses": "1. Writing should be more specific. \n    1. Contributions should indicate that this is a solver for linear inverse problems. Statements such as “Thus, Flower can be seen as a plug-and-play inverse solver with general application, with our Bayesian justification establishing a novel link between posterior sampling and the PnP frameworks in generative models.” need to be updated to clarify that this is a solver for linear inverse problems. \n    2. The paper should also indicate that the Bayesian analysis is only valid for sampling from the approximate posterior and not the true posterior. \n2. Only 100 images have been used for quantitative results which is a really small test dataset.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ufol4LDu4l", "reviewer": "Reviewer_Kgmh", "summary": "The paper “FLOWER: A Flow-Matching Solver for Inverse Problems” introduces Flower, a general-purpose solver that leverages a pre-trained flow model to reconstruct signals consistent with given measurements. Flower operates through a three-step iterative process: (1) flow-consistent destination estimation, where the velocity network predicts a denoised target; (2) measurement-aware refinement, projecting the estimate onto the feasible set defined by the forward operator; and (3) time progression, updating the trajectory along the flow path. The authors provide a Bayesian interpretation, showing that Flower approximates posterior sampling, thereby linking plug-and-play (PnP) and generative flow-based solvers within a unified framework. Empirically, Flower achieves state-of-the-art reconstruction quality across diverse inverse problems using nearly identical hyperparameters. Its main contribution lies in bridging Bayesian posterior sampling and PnP methods through a theoretically grounded and practically efficient flow-matching formulation.", "questions": "- Could the authors provide an ablation plot showing the effect of varying the number of evaluations per step (e.g., Flower1-10?) to better illustrate the trade-off between reconstruction quality and computational cost?\n\n- The current method is compatible with the standard Euler sampler used in flow matching. Could the authors discuss whether other ODE or higher-order samplers (e.g., Heun or adaptive solvers) could be integrated into Flower, and what potential advantages or challenges might arise?\n\n- Could the authors elaborate on possible extensions of Flower to nonlinear or real-world inverse problems (e.g., MRI reconstruction, phase retrieval, or compressed sensing)? In particular, what modifications or additional components would be required to handle these more complex forward models?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The paper:\n-  Introduces a novel three-step flow-matching framework (Flower) for inverse problems, bridging generative flow models and plug-and-play (PnP) methods under a unified Bayesian view.\n\n-  Provides solid theoretical grounding with a clear Bayesian justification and strong empirical results showing state-of-the-art/competitive reconstructions across diverse tasks.\n\n- The paper is well-structured and accessible, clearly explaining algorithmic steps and visualizing the process through intuitive figures.", "weaknesses": "-  **More ablation** could be conducted on the number of evaluations performed at each step. The paper currently presents results for Flower1 and Flower5, where Flower5 shows steady improvement over Flower1. An ablation plot illustrating this trend would strengthen the analysis.\n\n-  Although the method claims hyperparameter robustness, the validation is limited to standard imaging tasks. Demonstrating applications to **nonlinear** or **real-world** inverse problems (e.g., MRI, or compressed sensing) would further validate its generality and impact.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)