Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving
Authors: Hossein Askari
Yadan Luo
Hongfu Sun
Fred Roosta
Venue: NeurIPS 2025 poster ()
Keywords: Inverse Problems
Flow Matching
Posterior Sampling
ODE-based Inference
Training-Free Methods
Primary area: applications
TL;DR: We propose LFlow, a training-free framework for solving linear inverse problems using pretrained latent flow priors with theoretically grounded posterior guidance, achieving superior reconstruction quality over latent diffusion baselines.

Abstract: Recent advances in *inverse problem* solving have increasingly adopted flow *priors* over diffusion models due to their ability to construct straight probability paths from noise to data, thereby enhancing efficiency in both training and inference. However, current flow-based inverse solvers face two primary limitations: (i) they operate directly in pixel space, which demands heavy computational resources for training and restricts scalability to high-resolution images, and (ii) they employ guidance strategies with *prior*-agnostic posterior covariances, which can weaken alignment with the generative trajectory and degrade posterior coverage. In this paper, we propose **LFlow** (**L**atent Refinement via **Flow**s), a *training-free* framework for solving linear inverse problems via pretrained latent flow priors. LFlow leverages the efficiency of flow matching to perform ODE sampling in latent space along an optimal path. This latent formulation further allows us to introduce a theoretically grounded posterior covariance, derived from the optimal vector field, enabling effective flow guidance. Experimental results demonstrate that our proposed method outperforms state-of-the-art latent diffusion solvers in reconstruction quality across most tasks.

Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving
Hossein Askari!
Yadan Luo' Hongfu Sun'
Fred Roosta
'The University of Queensland
ARC Training Centre for Information Resilience (CIRES)
(h. askari, yadan luo, hongfu. sun, fred. roostaoosuuggeooo edu..au
Abstract
Recent advances in inverse problem solving have increasingly adopted flow priors over diffusion models due to their ability construct straight probability paths from noise to data, thereby enhancing efficiency in both training and inference. However, current flow-based inverse solvers face two primary limitations: (i) they operate directly in pixel space, which demands heavy computational resources for training and restricts scalability to high-resolution images, and (ii) they employ guidance strategies with prior-agnostic posterior covariances, which can weaken alignment with the generative trajectory and degrade posterior coverage. In this paper, we propose LFlow (Latent Refinement via Flows), training-free frame- work for solving linear inverse problems via pretrained latent flow priors. LFlow leverages the efficiency of flow matching to perform ODE sampling in latent space along an optimal path. This latent formulation further allows us to introduce a theoretically grounded posterior covariance, derived from the optimal vector field, enabling effective flow guidance. Experimental results demonstrate that our pro- posed method outperforms state-of-the-arr latent diffusion solvers in reconstruction quality across most tasks. The code will be publicly available at GitHub.
Introduction
Linear inverse problems are fundamental variety of significant image processing tasks, such as super-resolution [1], inpainting [2], deblurring [3], and denoising [4]. Solving such problems involves inferring an unknown image Xg E TR.", which is assumed follow an unknown prior distribution q(xo), from incomplete and noisy observations y E T commonly modeled as:
y Axo
where E Rmmx represents a known linear operator and denotes additive i.i.d. Gaussian noise. When the operator A is singular (e.g., if m n), the inverse problem becomes ill- ill-posed [5], hindering the unique or stable recovery of Xg from y. Consequently. accurate and plausible inferences demand strong priors that effectively integrate domain-specine knowledge constrain the solution space.
Deep generative models that perform progressive refinement via stochastic differential equations (SDEs), particularly diffusion models [6-8], have solidified their role as powerful priors for solving a broad spectrum of inverse problems [9, 10]. Specifically, these models have proven effective for zero-shot inference of images from partially acquired and noisy measurements, with extensive research focusing on the design of guidance mechanisms to inject data consistency into the generative process [1] 1-32]. Building these advancements, diffusion-basse inverse solvers have been further extended to operate in latent spaces [33-39] rather than raw pixel spaces, aiming to reduce the computational cost of training and improve generalization [40]. However, these approaches often neglect posterior variability by assuming zero covariance in likelihood- guidance, which can lead to unstable sampling and reduced coverage of the posterior distribution [21, 30).
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

Recently, flow matching [41, 42] has gained prominence as a compelling alternative for generative modeling. By parameterizing transformation dynamies with ordinary differential equations (ODEs), these models can generate arbitrary probability paths, including those grounded in optimal transport (OT) principles [41]. This flexibility enables the design of straight-line generative trajectories, leading to more efficient training and sampling compared to diffusion-based approaches [43, 44]. Motivated by these capabilities, several recent works have explored the use of flow-based priors for inverse problems, achieving faster and higher-quality solutions across diverse tasks [45-50]. Nevertheless, existing methods still suffer from two key drawbacks: (1) they operate in pixel space, which restricts scalability to high-din sigh-dimensional data and limits generalizability across different types of inverse problems; and (2) they adopt guidance techniques originally developed for diffusion models, which estimate posterior covariances independently of the learned prior. This disconnect may steer the sampling trajectory away from high-probabiliyg regions, leading to degraded sample quality and reduced fidelity, with slower convergence often observed when adaptive ODE solvers are employed.
To address these limitations, we propose LFlow (Latent Refinement via Flows), a framework that utilizes latent flow matching to solve linear inverse problems without additional training. By applying flow matching in latent space, LFlow achieves enhanced computational efficiency and enables more scalable and effective inverse solutions in a well-founded time-dependent variance for the latent identity posterior covariance, formulated using Tweedie's 's covariance formula and the optimal vector field under the assumption of Gaussian latent representation. This posterior covariance is explicitly informed by the pretrained optimal vector field, ensuring that guidance remains consistent with the generative dynamics. Our empirical evaluations demonstrate that images inferred via latent ODE sampling along conditional OT paths exhibit superior perceptual quality compared to those generated through latent diffusion- based probability paths.
domains. Additionally, we introduce
Our primary contributions are as follows:
Methodological: We propose a training-free framework based latent flow matching and casterior-guided ODE sampling for solving linear inverse problems, significantly outper- forming latent diffusion- -based approaches in both efficiency and reconstruction quality.
Analytical:
We derive a principled correction to the pretrained latent flow using the measure- ment likelihood gradient and introduce an analytically justified, time time-dependen posterior covariance to improve sampling accuracy and convergence speed.
Empirical: We validate the performance of LFlow through extensive experiments on image reconstruction tasks, including deblurring, super-resolution, and inpainting, achieving state-of-the the--eterttttt art results without requiring substantial problem-specife hyperparameter tuning.
2 Overview of Related Work
Training-free inverse problem solvers that exploit diffusion or flow priors can be generally categorized into four methodological classes: (1) Variable splitting methods decompose inference into two alternating steps: one enforces data fidelity and the other imposes regularization [51, 36, 20]: (2) Variational Bayesian methods introduce parameterized surrogate posterior distribution, typically Gaussian, whose parameters are optimized using variational objective [27, 52, 53]: (3) Asymptot- ically exact methods combine generative priors with classical samplers such as MCMC, SMC, or Gibbs sampling- to approximate the true posterior with convergence guarantees as the sample size grows [54-57]; and (4) Guidance-base methods correct the generative trajectory using an approximate likelihood gradient to steer samples toward the posterior [12, 21-23]. Our work centers on the fourth category and further elaborates on related methods built on various types of priors.
Diffusion Guidance Approximation refers estimating the likelihood score Vx, log (og((y)x))
during the reverse-time diffusion process governed by an SDE of the form:
dx, R ()) - 8(1)² (so !) + Vx, logr(y|xx [x)) di + g((() dw.,
(2)
where f(-.)) -) is the drift, g(-) the diffusion coefficient, we standard Brownian motion, and (: pre- trained score network. Notable methods such as DPS [21], JIGDM [22], TMPD [23], OPC [29], and MMPS [30] define the likelihood score as Vx, log p(y xx) Vx, log f p(y x (x))p(())))))))))x))( (xx xx)) dxp- The
main challenge lies in computing the expectation over all possible denoised states Xn given x,, which requires sampling from p(xo) xx) at every reverse step- step-posing significant computational demands. A
2

---

common remedy is a local Gaussian approximation p(xo(x+) Re N (E[xx Ke], Cov(xx/xe)). reducingg reducing the problem to estimating posterior moments. These approaches differ mainly in how they specify Cov Xo Xt]. For instance, DPS sets it to zero, and IIGDM obtains a prior-agnostic, time-scaled identity matrix derived from the forward process under Gaussian data assumption. OPC performs post hoc constant variance optimization for each step. TMPD approximates the covariance by
X [[x]|x]]] with a diagonal row- sum surrogate, whereas MMPS evaluates the full Jacobian via automatic differentiation and solves the induced linear system with Conjugate Gradients [58]. Estimating the full covariance, however, remains computationally expensive in high dimensions, with prohibitive memory and runtime costs.
replacing the Jacobian in the Tweedie relation Cov(xo |x} = axt))
a(t)²
V
Inference in Latent Space has become feasible with latent diffusion models (LDMs) [59], allowing inverse solvers to reduce training costs and improve scalability [33, 36-38]. As an initial attempt, PSLD [33] augments DPS with a "gluing" objective to enforce posterior mean consistency under the autoencoder mapping, aiming to mitigate encoder-decoder nonlinearity. However, this constraint remains empirically ineffective under noisy measurements, and reconstruction artifacts persist [34]. Other approaches avoid explicit nonlinearity correction: In particular, Resample [36] constructs a Gaussian posterior by fusing supposedly Gaussian prior on the unconditional reverse sample with Gaussian pseudo-likelihood centered at a forward-projected Similarly, DAPS [37] enhances posterior sampling by decoupling consecutive diffusion steps through a two-step procedure: first drawing Zg p(zo| y), then re-noising Z₁-Ai ~ NN of I).. which provides global corrections parrecciosspprttuullaary effective in non-linear tasks, though at the expense of weaker local guidance in low-noise or linear settings. SITCOM [60] enforces three consistency conditions- data, forward, and backward diffusion at each step, thereby enabling sampling with fewer steps. Nevertheless, these methods largely assume zero covariances, limiting posterior coverage.
sesauuurmmmt--oonsssteee posterior mean.
Flow Matching in Inverse Problems has recently proven effective for achieving fast and high- quality solutions across various tasks [45-50]. A prime example is OT-ODE [45], which adopts IIGDM [22] gradient correction within the flow regime and employs an ODE solver scheme based on the conditional OT path. C-IIGFM [48] introduces a plug-and-play framework that projects conditional flow dynamics into a more amenable space, accelerating inference. PnP-Flow [61] solves imaging inverse problems by alternating a data-fidelity gradient step, a re-projection onto the flow path via latent-noise interpolation. and a time-dependent FM denoiser, all without backpropagating through the ODE. However, each of these methods typically (1) borrows guidance strategies from diffusion models that assume either zero or aino-sggooticc posterior covariances, and (2) operates in pixel space, which limits their scalability and applicability to high-dimensional problems.
3 Preliminaries
Continuous Normalizing Flow (CNF) [62] constructs a smooth probability path (e:
that transports samples from a data distribution g(xo). with Xo E IRd to a standard Gaussian (o, I) at = 1. This evolution follows a time- -ineing vector field v RR x [0, + Rd, governed by the ODE dx, = v(x, t) dt. In practice, v(.t) is approximated by learnable field vo. trained via maximum likelihood, which requires expensive ODE simulations.
Flow Matching (FM) [41] avoids inefficient likelihood- -bsed training for CNFs by directly regress- ing a learnable vector field toward an analytically defined target field. In particular, Conditional Flow Matching (CFM) [41] introduces a time-dependent conditional vector v v(x, | xo) that governs the evolution of samples x, conditioned on an initial point Xg- This field induces a conditional probability path pe px(x;] xo)) satisfying the boundary conditions Po 8(xo) and p1 (). The training objective then becomes:
(CF E, xo x. vo v(x xo)
(3)
Using Gaussian paths, we define px(x; = [a((())xo ( with interpolation x, = a(t) xp)x
a (f)x,, where x1 ~ N0, T). Differentiating x, with respect to / gives á(t)x, - -()xx Substituting
the inversion X = [Xe ((t)xi]/a(t) yields the true vector field:
á(t)
((t)
a(t)
vx; xx)
a(t) + (1)
a(t)
a(t)
X1
(4)
where a() and (t) denote time derivatives. In particular, choosing a(l) = and o()) = recovers the OT path, which induces straight trajectories and improves efficiency.
3

---

C-OOO.D)
Flows with vector field (z,.
During pre-traingg
Conditional flow x,yy
pr(Zu)
4o (Za|No))
Ra
Unconditional flow 2,
3, (1-() E,, +
t=
0.6
t= 0.4
t 0.3
1= 0.2
= 0.1
X
Figure 1: (Top) LFlow pipeline: A VAE encoder maps the observation a latent from which a noisy start Zi, is formed. During VAE pre-training, the KL term encourages the encoder's approximate posterior go (Z Xo) approach the Gaussian prior pz(z)) (0, I). A pretrained flow field vo (Ze, t) defines the unconditional trajectory 2,. At inference, orange arrows (-) denote likelihood-
guidance that corrects the prior field, yielding the conditional latent path z, toward p(zoly). Decoding with D produces Xo. (Bottom) Gaussian deblurring snapshots along the conditional path as t decreases from 0.8 to 0.
4 Method
To address the ill-posedness of linear inverse problems, we adopt Bayesian view and target the posterior p(xo y) x p(y Xo) p(X)), where p(y Xo) the likelihood and p(xo) is a learned prior induced by a pretrained latent flow (via the decoder). Our goal is to generate samples from this posterior without retraining a task-specific model.
4.1 Latent Refinement via Flows (LFlow)
We represent the prior distribution p(Xo) implicitly via latent prior p(zo) and the decoder push- forward p(Xp) = Dyp # p(zo). Specifically, let Es Rd + R and De : R + Rd be pretrained autoencoder. The latent Zg = Es (xv) follows p(zo), modeled by a flow- now-mnaching velocity ve (zz, t) defining the ODE
dz: volz.. t) dt,
E [0.
(5)
This flow transports latent samples from Zo p(z)) to a noise distribution z₁ No I). At
inference, we integrate backward from to 2g. adding measurement-drivee guidance to follow the conditional trajectory, then decode Xg (2o)) Figure illustrates the overall inference pipeline and provides visual example of decoded reconstructions along the conditional trajectory, offering intuition for how the flow progresses in practice.
Posterior Sampling via Conditional Flows in Latent Space We aim to sample from the posterior p(zo y) by simulating reverse-time ODE characterized by conditional velocity field. The following proposition provides the theoretical foundation for this procedure, ensuring that if the conditional density evolves according continuity equation, then the reverse-time flow recovers samples from the posterior. A formal proof is deferred to Appendix A.I.
Proposition 4.1 (Posterior Sampling via Reverse-Timne Conditional Flows). Lei pelze y) denote the conditional distribution of latent variables E R time E [0, 1]. with terminal condition Pi (zı) = (0, I). Suppose this distribution evolves over time according to the reverse-time continuity equation:
O.p.(ze y) = Vz, [pe(z: Iy) velz yy],
(6)
for some conditional velocity field vz(zi y). Then, the solution the reverse-time ODE
dz, -v y) di, z ~ ~N(, yields samples from the posterior p(zo y)
0.
Conditional Vector Field Estimation Having established that posterior samples can be obtained by integrating the reverse-time ODE in Eq. (7), it remains estimate the corresponding conditional

---

vector field vz(zily). Under Gaussian latent dynamics, this field takes the form:
vz y) = a(t) a(t)
d (a(t)
Zz + A(t) 22, logpi(z; y). X(t) = at a(t))
(8)
By applying Bayes' rule, we obtain a principled decomposition of the velocity field, which leads to the following practical approximation via the pretrained flow and likelihood correction.
Ve y) = ve(zi,t)
1 V2, log zz).
where the additional term Vz. Pt Pt (y zz), commonly referred to as guidance, steers the flow toward consistency with the measurements y. A complete proof of the results in Eq. (8) and Eq. (9) in the latent domain is provided in Appendix A.2.
Likelihood Approximation A key challenge in Eq. (9) arises from approximating the gradient of the noise-conditional distribution, which we define as
22, log ply zr) Zz, log / ply Xo, (x) Ze dxx 2z, log p(xx (p(y//o)] (10)
Here, the likelihood term can be expressed as p(y xx) (y; (zn), oz which is Gaussian distribution with a nonlinear mean induced by the decoder D.p Due to the nonlinearity of D the marginal likelihood p(y zz) deviates from Gaussianity. To facilitate the computation of its gradient, we approximate Dip (Zo) via a first-order Taylor expansion around žo == E[zo zr] as:
(z( = Dzo(z)) + JD((z - Zn)-
(11)
where Jp := J/D J/((Z)) is the Jacobian of Dep at Zo. Assuming the posterior p(zol zz, N N(zo, Ez). the distribution of the image Xo = (zo) becomes approximately Gaussian as well:
Xg (F kr], Cov(xolza) (x)]) where E[xo|1z] Dup (Zo), Cov(xo 22.] Jo Eg JD. (12)
Using this approximation, we estimate the gradient of the log- lilkiiiiehodd in latent space based on the decoded mean and the propagated covariance (see Appendix A.4);
Vz, logp(y zz)= ~ (Vz, 2o) JT A (ogI+ + AJD E. bTA (y AD
(13)
which involves Jacobian-vecee product (J v), and can be efficiently computed using automatic differentiation (Appendix B). Furthermore, the Gaussian posterior assumption enables the application of Tweedic's s formula [63], which connects the pretrained vector field ve (zz, t) to the posterior moments [Z]) and Cov /2 (2)) Specifically, have (refer to Appendix A.3):
E E(zu
t (zo,ttt)) t),
(14)
t2
Cov(zo
2:]
(I V ()(
(15)
Latent Posterior Covariance Computing the posterior covariance Cov |2] in diw-dsmensional latent spaces is more tractable and can be efficiently estimated via automatic differentiation. However, inverting the matrix (a I + AJp Eg JAAA) in Eq. (13) remains a significant computational bottle- may solve the corresponding linear system using the Conjugate
neck. To avoid explicit inversion, Gradient (CG) method. Yet, CG's convergence crucially depends on the symmetry and positive definiteness of the system in the pretrained velocity field Va. potentially leading to instability or divergence. To address this, we further analyze the structure of the posterior covariance and the Jacobian Va, vo t), under certain regularity assumptions on the latent prior p(zo).
conditions that may be violated in practice due imperfections
be the latent prior over Rd, where & is a twice continuously differentiable potential function. We assume that p(2)) is y-stronlly log-concuve for some " > 0; that is, <² Ž 7 for all E Rdd
Assumption 4.2 (Strong Log-Concavity of the Latent Prior [64]). Let p(zo) =
(1) Log-concave Prior: This assumption imposes uniform lower bound on the Hessian of the prior's potential function, which translates to lower bound on the Jacobian of the vector field.
5

---

Proposition 4.3 (Bound on the Jacobian of the Vector Field). Let vo(z,,t) 1) denote the velocity field of the interpolant Z; z + (t)), between a standard Gaussian prior and target distribution p(zo), defined via coefficients a(t), a(t) E R. Under Assumption 4.2, the Jacobian of the vector field satisfies the following bound:
d
d loga(1) Ia, dt
dt (2 log (a(1)² + +o(()))) Ia. 3 Vn.
VIE ZER"
(16)
The proof is provided in Appendix A.5. The resulting sandwich bound ensures the Jacobian remains well-behaved the upper bound guarantees valid and stable posterior covariance, while the lower bound prevents overestimated uncertainty during posterior-guided inference.
(2) Gaussian Special Case: For practical inference and efficient computation, we now consider tractable special case where the latent prior is Gaussian, Zu ~ NO T, a choice that is both theoretically justified and widely used in practice for instance, in variational autoencoders (VAEs), where the latent prior is regularized toward a standard Gaussian via KL divergence [65]. This enables us derive the optimal velocity field and its Jacobian in closed form, which is characterized in the next proposition, whose proof can be found Appendix A.6.
Proposition 4.4 (Optimal Vector Field). Let Za (O, La. and zı ~N Ia_) be indepen- dent random variables. Define 2 = (1 - 1)ze + (z) for E [0, 1]. The optimal vector field v (Zr, !)
that minimizes the expected squared error arg min, [l, - (zo- 21)1 is given by
(1 1)
v" (2( 1) (1 - t)3 + 12 21
(17)
Remark 4.5 (Tightness of Bound). When Zg NoO, o Inr the optimal vector field v² (zz, t) achieves the Jacobian lower bound in Proposition 4.3. This follows from the potential function
((z)) = . zoll? yielding 7 - and interpolation coefficients a(t) 1 t. (t) t.
Substituting into the bound confirms it matches the exact Jacobian of'", making the bound tight.
Plugging the Jacobian of the optimal vector field into Eq. (15) yields:
Cov [oolzol 2x] = (1) Ia.; with r (1) = f2 (1 t) (0(1-2)) 28) 26
(18)
(1 t) [11 2)2 t2]
where we assumed Jattr = 1. As result, the propagated covariance can also be simplified as
Cov(xo 2 sz Jp- Cov/zo z,] J. = (t) JDJD.
(19)
Computing the full Jacobian Jp dd is often infeasible in practice. To simplify, we assume that the decoder acts approximately as a local isometry near 2o [66], such that JDJD = -P, where P is the orthogonal projector onto the image of Jp. For computational convenience, approximate this behavior as isotropic in the full space, resulting in:
Cov|xo 23] ~ ((t) In_.
(20)
Comparison with IIGMM and OT-ODE The posterior covariance in IIGM is derived solely from the forward process under the strong assumption of Gaussian data space, resulting in
variance for the identity covariance as (t) +[o)
a(t)²
I. Please see Appendix A.7 for details.
Additionally, JIGDM leverages the inverse of this posterior covari- ance as Fisher information for natural gradient [67] updates on sam- ples during the guidance step, thereby enhancing performance. OT- ODE can be regarded as a flow-based extension of ITGDM, where adopts the same covariance expression but replaces the diffusion forward process with a flow-specific noise scheduler, specifically f and (tt) an identity posterior covariance that is independent of the learned score or vector field. In contrast, our proposed covariance explicitly incorporates information from the pre-trained vector field. This in turn allows for more effective guidance during flow-based ODE sam- pling, ultimately improving reconstruction quality. Figure 2 visually illustrates how our time-dependent variance differs in magnitude and effect from the simpler identity-base based covariance used in OT-ODE.
Liive
setting att)
Therefore, both methods rely on
Figure Posterior covariance values across i E [0, 0.9] for our method and OT-ODE, along with reconstructed images for sin- gle FFHQ sample the super- resolution task.

---

Initiating the Flow Sampling Process Inspired by prior works in pixel spaces [68, 45], we propose initializing the reverse ODE ODE-across y in the latent space. Specifically, rather than starting from pure noise 2 (0, 1) at L = 1, we
across all tasks-from from a partially corrupted version of the encoding of
initialize at l. 1:
2, = (1 (1-t,) t.) Ex(y) +t, Z1 ~0,
(21)
This initialization ensures that 2t, is closer to the posterior mode Zo y. making the subsequent backward integration more stable and likely to remain plausible manifold.
We summarize the complete sampling algorithm via latent ODE flows in Algorithm (Appendix C).
5
Experiments
Datasets and Tasks We evaluate our method on three datasets: FFHQ [69], ImageNet [70], and CelebA-HQ [71], each containing images with resolution of 256 x 256 x 3 pixels. We use 200 randomly selected validation samples per dataset. All images are normalized to the range [-1, 1]. We present our findings several linear inverse problem tasks, including Gaussian deblurring, motion deblurring, super-resolution, and box inpainting. The measurement operators are configured as follows: (i) Gaussian deblurring convolves images with Gaussian blurring kernel of size 61 x 61 and a standard deviation of 3.0 [29]; (ii) Motion Deblurring uses motion blur kernels that are randomly generated with a size of 61 x 61 and an intensity value of 0.5 [29]; (iii) Super-Resolution (SR) involves bicubic downsampling by a factor of 4; and (iv) Box Inpainting simulates missing data by masking a 128 128 pixel box, which is randomly positioned around the center with margin of [16, 16] pixels, following the methodology of [21]. All measurements in these experiments are corrupted by Gaussian noise with standard deviation of ay = 0.01, ensuring that the reconstruction methods are evaluated under realistic noisy conditions.
Baselines and Metrics Since the primary goal of this study is improve solving inverse problems in the latent space, we focus on comparing our proposed method against state-of tf-thettt latent diffusion solvers--specifically, PSLD [33], MPGD [38], Resample [36], DMplug [51], DAPS [37], and SITCOM [60] through both quantitative and qualitative analyses. Additionally, we include results from the pixel-based IIGDM [22], OT-ODE [45], and C-IIGFM [48], when reported, for a more comprehensive comparison. Following the evaluation protocols of prior works, we report quantitative metries including Learned Perceptual Image Patch Similarity (LPIPS) [72], Peak Signal- to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Fréchet Inception Distance (FID) [73] (Appendix D.3).
Architecture, Training, and Sampling In contrast to previous latent diffusion inverse solvers, which rely on the LDM-VQ-4 or Stable Diffusion (SD) v1.5 models [59] built on U-Net [74], adopt the LFM-VAE framework [75] based on DIT transformer architectures [76] for all datasets. Following the methodology outlined in [75], we leverage their pre-trained model. It is important to note that, despite our architectural differences, the quality of our prior does not exceed that of SDs or LDMs. This limitation may partly arise from using lower-dimensional latent space of 32 x 32. After evaluating various numerical ODE integration methods, we selected adaptive_hen _eeu as the default solver, utilizing its reliable implementation from the open-source torchdiffeg library [62]. All experiments were conducted on a single NVIDIA 3090 GPU with a batch size of 1. Further details on baselines, model configurations, and other typeer-arametert are provided in Appendix C.
5.1 Results
For clarity, the guantitative results for the four tasks on both FFHQ and ImageNet are presented in Tables I and 2, while qualitative comparisons are shown in Figures 3 and 4. The following subsections provide a detailed discussion of each task. Due to space constraints, results for CelebA-HQ are included in Table 5 and Figure 7 in Appendix D. Additional results are also presented in this section.
Gaussian Deblurring LFlow achieves the best perceptual quality across both datasets, attaining the lowest LPIPS and highest SSIM-surpasii surpassing the second- best method by approximately 8.8% in LPIPS on FFHQ While PSLD slightly leads in PSNR, LFlow closes the gap within 1.18 dB and avoids the over-smoothing and loss of fine detail often observed in PSLD outputs (e.g., blurred backgrounds and softened skin textures). On ImageNet, L.llw similarly excels in perceptual quality, restoring structures like the dog's snout, fur, and facial wrinkles more faithfully. The reconstructions
7

---

Table I: Quantitative evaluation of inverse problem solving on FFHQ samples of the validation dataset. Bold
and underline indicates the hest and second-best respectively. The methods shaded in gray in pixel space.
Debturring (Gaussian)
Deblurring (Motion)
SR (x4)
Inpainting (Box)
Method
PSNRT
SSIM LPIPS)
PSNRT
SSIM LPIPS)
PSNRT
SSIM LPIPS)
PSNRT SSIM LPIPS.
LFlow (ours)
29.10
0.837
0.166
30.04
0.849
0.168
29.12
0.841
0.176
23.85
0.867
0.132
SITCOM [60]
30.42
0.828
0.237
28.78
0.828
0.183
29.26
0.833
0.191
24.12
0.839
0.198
DAPS [37]
28.52
0.789
0.231
29.00
0.831
0.252
29.38
0.826
0.197
24.83
0.819
0.191
DMplug [51]
27.43
0.784
0.240
27.95
0,817
0.243
29.45
0.838
0.183
22.55
0.807
0.220
Resumple [36]
28.73
0.801
0.201
29.19
0,828
0.184
28.90
0.804
0.189
20.40
0.825
0.243
MPGD [38]
29.34
0.815
0,308
27.98
0,803
0.324
27.49
0,788
0.295
20.58
0.806
0.324
PSLD [33]
30.28
0.836
0.281
29.21
0.812
0.303
29.07
0.834
0.270
24.21
0.847
0.169
OT-ODE [45]
29.73
0.819
0.198
28.15
0.792
0.238
28.56
0.823
0.198
25.77
0.751
0.225
IIGDM (22)
28.62
0.809
0.182
27.18
0.773
0.223
27.78
0.815
0.201
26.82
0.767
0.214
Measurements
Referances
MPGD
PSLD
Resample
DAPS
SITCOM
LFlow (ours)
Figure 3: Qualitative results FFHQ test set. Row I: Deblur (gaussian), Row 2: Deblur (motion), Row 3:
SRx4, Row 4: Inpainting. Our approach better preserves fine image details than latent-based diffusion methods.
remain sharp and natural, free from the spurious high-frequency artifacts or texture distortions that affect other approaches.
Motion Deblurring LFlow reconstructs aation-bluured scenes with strong structural consistency and minimal artifacts, effectively preserving sharp transitions along motion boundaries. Fine de- suls- hair contours, facial edges, and accessories and natural appearances. Without introducing ringing or over-enhancement, LFlow balances fidelity and realism across both FFHQ and ImageNet, achieving -ate-of-the-ar performance on all met- rics. Notably, LFlow improves LPIPS by a clear margin of 0.015 over the second- best method, SITCOM, on FFHQ, reflecting its ability to retain high-frequency content while maintaining spatial coherence coherence-qualitiee qualities that are visually evident in the restored hair and facial structures.
recovered with smooth gradients
Super-Resolution
High-frequency structures such as hair strands, floral patterns, and facial details are reconstructed with remarkable clarity by our method, without the over-sharpening or softness often observed in competing approaches. On FFHQ, it achieves perceptual gain of 0.021 LPPIS over the next-best method, while also maintaining competitive PSNR. On ImageNet, establishes clear lead across all metrics, with LPIPS margin of 0.018. These improvements translate to visually cleaner textures and more coherent spatial gradients. Unlike MPGD, which may produce overly synthetic details, PSLD and Resample, which occasionally yield hazy regions, our approach preserves the natural appearance fine structures while avoiding haloing and texture overshoot.
Inpainting When completing missing regions, LFlow excels in producing semantically coherent and perceptually consistent content that blends naturally with surrounding areas. On FFHQ, surpasses the second-best method in LPIPS by a margin of 0.037 while also achieving the highest SSIM, reflecting both perceptual sharpness and structural accuracy. On ImageNet, continues to lead in LPIPS and SSIM, recovering fine textures and edges with minimal boundary artifacts.

---

Table 2: Quantitative results of inverse problem solving ImageNet samples of the validation dataset. Bold and underline indicates the hest and second-best respectively. The methods shaded in gray in pixel space.
Deblurring (Gaussian)
Deblurring (Motion)
SR (x4)
Inpainting (Box)
Method
PSNRT
SSIM
LPIPS.
PSNR1
SSIMT
L.IPS
PSNRT
SSIMT
LPIPS.
PSNRT SSIM I.PIS.
LFlow (ours)
0.328
0.711
0.344
0.338
21.92
0.772
0.227
SITCOM [60]
25.38
0.672
0.388
0.382
25.62
0.687
0.374
20.34
0.698
DAPS [37]
0.413
25.97
0.706
0.362
25.18
0.356
21.13
0.701
Resample [36]
25.04
0.665
0.408
24.32
0.623
0.390
24.81
0.683
0.404
19.42
0.663
0.305
MPGD [38]
24.27
0.695
0,397
24.81
0.662
0.404
25.50
0.648
0.398
17.05
0.672
0.324
PSLD [33]
26.79
0.721
0.372
25.45
0.692
0.351
26.16
0.692
0.363
20.58
0.687
0.274
IIGDM [22]
25.27
0.636
0.332
23.03
0.617
0.347
24.73
0.629
0.359
22.13
0.589
0,361
Measurements
References
MPGD
PSLD
Resample
DAPS
SITCOM
LFlow (ours)
Figure 4: Qualitative results on Image Net test set. Row I: Deblur (gaussian), Row 2: Deblur (motion), Row 3: S Row 4: Inpainting. Our method reconstructs fine image details more faithfully than the basclines.
In contrast to methods that may introduce visible seams, blotchy patterns, or inconsistent colors, LFlow reconstructs faces, objects, and natural scenes with smoother transitions and well-aligned local details in reconstructions that appear complete and visually seamless.
Perceptual-Fidelity Trade-off While LFlow occasionally reports slightly lower PSNR than pixel- addlitty-oieented baselines such PSLD, this reflects the well-known trade-off between distortion and perception. Our approach consistently achieves substantially lower LPIPS and sharper, more natural reconstructions, indicating that it prioritizes perceptual realism and fine-detail preservation over pixel-wise averaging effects that often inflate PSNR.
5.2 Ablation Study
To examine how the time-dependent posterior covariance influences overall performance, we conducted series of ablation studies. As shown in Table 3, adapting the posterior covariance (labeled Cov_LFlow) provides systematic gains over the baseline "Cov "TGV_I Across both FFHQ and ImageNet, Cov_ yields higher PSNR and lower LPIPS for deblurring and x SR, indicating that L.Flow helps reduce perceptual artifacts while preserving fine details. Figure 6 presents qualitative comparisons between deblurring and inpainting, Cov_LFlow yields sharper facial structures, cleaner textures, and fewer artifacts. For motion deblurring, it better restores contours and eye details without ringing, while inpainting results show improved shading consistency and reduced boundary errors. These visual observations align with the quantitative improvements reported in Table 3.
Posterior Covariance
and Cov fov_IIGDM on FFHQ. Across both motion
Starting Time t2 We also investigated the impact of the starting time t, on the results. Figure illustrates the impact of varying the start time l, for the flow process (SR task). We see that overly large 4, values tend to slightly degrade perceptual quality, whereas overly small L, values can cause excessive smoothing or artifacts. The plot indicates that an intermediate choice of L, (around 0.7-0.8) strikes favorable balance, leading to consistently lower LPIPS on both FFHQ and ImageNet.

---

Superressoltion
Table 3: Ablations on effect Cov (Zo z.]. Bold indicates hest.
FFHQ
ImageNet
040
Dataset
Deblur (G)
Inpainting
Deblur (M)
SR (x4)
PSNRT
LPIPS)
PSNRT
LPIPS)
PSNRT
LPIPS)
PSNRT
LPIPS.
CFv_LFlow
29.10
0.166
23.85
0.132
26.10
0.344
25.29
0.338
6.6
Cov_ ITGDM
29.04
0.179
22.69
0.151
25.22
0.363
24,80
0.351
Figure 5: Ablation study on the
start time 1..
Meaturements
References
Cov. NGOM
Cov LFlow
Table 4: Average inference time (seconds image) for latent-based inverse solvers ImageNet samples. Timings are measured on an NVIDIA 3090 GPU.
Inceinting
Method
Gaussian Deblur
SR
LPIPS 0.1384
LPIPS 9.1073
Resample [36]
550.26s
410.68s
MPGD [38]
Eeekaer leootion)
705.31s
675.85s
SITCOM [60]
345.60s
328.12s
LFllww
267.76s
227.92,
(CDV_TICDM)
LPIPS 0.1311
LPISI 0.1807
Figure Visual results on the effect Cov |zo |zs].
Inference Time To highlight the practical advantages of posterior covariance modeling beyond standard evaluation metrics, we report the average inference time per image in Table 4. Despite using an adaptive ODE solver with potentially higher NFE, LFlow achieves significantly faster inference than prior latent-based solvers. This efficiency stems from faster convergence and more accurate trajectory estimation. The variant BDv_IIGMM, which employs less accurate covariance, converges more demons--mmonssratinn the importance of proper covariance modeling. Notably, PSLD runs for 1000 iterations and Resample for 500, yet both are slower than LFlow.
6 Discussions
In this paper, we present LFlow, which efficiently addresses linear inverse problems by utilizing flow matching in the latent space of pre-trained autoencoders without additional training. Based on justified latent Gaussian representation assumption, our approach introduces theoretically sound, time-dependent latent posterior covariance that enhances gradient-basee inference. Experimental results across such tasks deblurring, super-resolution, and inpainting demonstrate that LFlow outperforms current latent diffusion models reconstruction quality.
Limitation. One limitation of LFlow lies in its runtime: the current implementation can require approximately 3:30-10 minutes solve inverse problems on an NVIDIA RTX 3090 GPU, with Gaussian deblurring and super-resolution averaging 2:30-6 minutes, and motion deblurring and inpainting taking around 3:30-10 minutes. While this more efficient than existing latent diffusion solvers, as demonstrated in our ablation studies, it may still pose challenges in time-sensitive applications. Nonetheless, the improved reconstruction quality and the efficiency gains achieved during training help mitigate this drawback.
Future work. Future work will focus on optimizing the solver and investigating alternative numerical integration schemes to further reduce inference time. In addition, we plan to extend LFlow to improve its robustness under distributional shifts, enabling broader applicability real-worlh scenarios and downstream tasks.
Acknowledgement
This research was partially supported by the Australian Research Council through an Industrial Transformation Training Centre for Information Resilience (IC200100022).
10

---

References
[1] Brian B Moser, Arundhati S Shanbhag, Federico Raue, Stanislav Frolov, Sebastian Palacio, and Andreas Dengel. Diffusion models, image super-resolution, and everything: A survey. IEEE Transactions on Neural Networks and Learning Systems, 2024.
[2] Andreas Lugmayr, Andres Danelljan, Fisher Yu, Radu Timofte, and Luc Van Gool. Repaint: Inpainting using denoising diffusion probabilistic models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 11661-11771, 2022.
[3] Kaihao Zhang, Wenqi Ren, Wenhan Luo, Wei-Sheng Lai, Björn Stenger, Ming- Yang, and Hongdong Li. Deep image deblurring; A survey, International Journal of Computer Vision, 13009):2103-21300 2022.
[4] Peyman Milanfar and Mauricio Delbracio. Denoising: A powerful building- blockk for imaging, inverse problems, and machine learning. arXiv preprint arXiv:2409.06219, 2024.
[5] Finbarr O' O'Sullivan. A statistical perspective on ill- ill-posed inverse problems. Statistical science, pages 502-518, 1986.
[6] Jaseha Doch-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsuper- vised learning using nonequilibrium thermodynamics. In International conference on machine learning, pages
PMLR, 2015.
[7] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances neural information processing systems, 33:6840-6851, 2020.
[8] Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.
[9] Giannis Daras, Hyungjin Chung, Chich-Hsin Lai, Yuki Mitsufuji. Jong Chul Ye, Peyman Milanfar, Alexandros G Dimakis, and Mauricio Delbracio. A survey diffusion models for inverse problems. arXiv preprint arXiv:2410.0008; 2024.
[10] Hongkai Zheng, Wenda Chu, Bingliang Zhang, Zihui Wu, Austin Wang. Berthy Feng, Caifeng Zou, Yu Sun, Nikola Borislavov Kovachki, Zachary E Ross, Katherine Bouman, and Yisong Yue. Inversebench: Benchmarking plug-and-plea diffusion priors for inverse problems in physical sciences. In The Thirteenth International Conference Learning Representations, 2025.
[11] Jooyoung Choi, Sungwon Kim, Yonghyun Jeong, Youngjune Gwon, and Sungroh Yoon. IIvr: Conditioning method for denoising diffusion probabilistic models. arXiv preprins arXiv:2108.02938, 2021.
[12] Ajil Jalal, Marius Arvinte, Giannis Daras, Eric Price, Alexandros G Dimakis, and Jon Tamir. Robust compressed sensing mri with deep generative priors. Advances in Neural Information Processing Systems, 34:14938-14954, 2021.
[13] Bahjat Kawar, Gregory Vaksman, and Michael Elad. Snips: Solving noisy inverse problems stochastically. Advances in Neural Information Processing Systems, 34:21757-21769, 2021.
[14] Yang Song, Liyue Shen, Lei Xing, and Stefano Ermon. Solving inverse problems in medical imaging with score- based generative models. In NeurIPS 202/ Workshop on Deep Learning and Inverse Problems, 2021.
[15] Bahjat Kawar, Michael Elad, Stefano Ermon, and Jiaming Song. Denoising diffusion restoration 2022.
models. Advances in Neural Information Processing Systems,
[16] Yang Song, Liyue Shen, Lei Xing, and Stefano Ermon. Solving inverse problems in medi- cal imaging with based generative models. In International Conference on Learning Representations, 2022.
[17] Hyungjin Chung and Jong Chul Ye. Score-based diffusion models for accelerated mri. Medical image analysis, 80: 302497479 2022.

---

[18] Mare Anton Finzi, Anudhyan Boral, Andrew Gordon Wilson, Fei Sha, and Leonardo Zepeda- Núñez. User-defined event sampling and uncertainty quantification in diffusion models for physical dynamical systems. In International Conference on Machine Learning, pages 10136- 10152. PMLR, 2023.
[19] Arpit Bansal, Hong-Min Chu, Avi Schwarzschild, Soumyadip Sengupta, Micah Goldblum, Jonas Geiping, and Tom Goldstein. Universal guidance for diffusion models. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 843- 03-85,, 2023.
[20] Yuanzhi Zhu, Kai Zhang, Jingyun Liang, Bihan Cao, Radu Timofte, and Luc Van Gool. De- noising diffusion models for plug-and and-play image restoration. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1219-12299 2023.
[21] Hyungjin Chung, Jeongsol Kim, Michael Thompson Mccann, Mare Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. In The Eleventh Interna- tional Conference on Learning Representations, 2023.
[22] Jiaming Song, Arash Vahdat, Morteza Mardani, and Jan Kautz. Pseudoinverse-guide diffusion models for inverse problems. In International Conference on Learning Representations, 2023.
[23] Benjamin Boys, Mark Girolami, Jakiw Pidstrigach, Sebastian Reich, Alan Mosca, and Omer Deniz Akyildiz. Tweedie moment projected diffusions for inverse problems. Transactions on Machine Learning Research, 2024. Featured Certification.
[24] Yinhuai Wang, Jiwen Yu, and Jian Zhang. Zero-shot image restoration using denoising diffusion null-space model. In The Eleventh International Conference on Learning Representations, 2023.
[25] Jiaming Song, Qinsheng Zhang, Hongxu Yin, Morteza Mardani, Ming- Yu Liu, Jan Kautz, Yongxin Chen, and Arash Vahdat. Loss-guided diffusion models for plug-and-pla play controllable generation. In International Conference on Machine Learning, pages 2023.
PMLR,
[26] Jiwen Yu, Yinhuai Wang. Chen Zhao, Bernard Ghanem, and Jian Zhang. Freedom: Training- free mnergy-guided conditional diffusion model. In 2023 IEEE/CVF International Conference on Computer Vision (ICCV), pages 231 17-12223. IEEE Computer Society, 2023.
[27] Berthy T. Feng, Jamie Smith, Michael Rubinstein, Huiwen Chang. Katherine L. Bouman, and William Freeman. basee diffusion models as principled priors for inverse imaging. In P Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pages 10520-10531, October 2023.
[28] Hyungjin Chung. Suhyeon Lec, and Jong Chul Ye. Decomposed diffusion sampler for accel- crating large-scale inverse problems. In The Twelfth International Conference on Learning Representations, 2024.
[29] Xinyu Peng, Ziyang Zheng, Wenrui Dai, Nuoqian Xiao, Chenglin Li, Junni Zou, and Hongkai Xiong, Improving diffusion models for inverse problems using optimal posterior covariance. In forty-first International Conference on Machine Learning, 2024.
[30] François Rozet, Gérôme Andry, Francois Lanusse, and Gilles Louppe. Learning diffusion priors from observations by expectation maximization. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
[31] Hossein Askari, Fred Roosta, and Hongfu Sun. Training-free medical image inverses via bi-level guided diffusion models. In 2025 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pages 75 84. 2025.
[32] Severi Rissanen, Markus Heinonen, and Arno Solin, Free hunch: Denoiser covariance estimation for diffusion models without extra costs. The Thirteenth International Conference on Learning Representations, 2025.
12

---

[33] Litu Rout, Negin Raoof, Giannis Daras, Constantine Caramanis, Alex Dimakis, and Sanjay Shakkottai. Solving linear inverse problems provably via posterior sampling with latent diffusion models. In Thirty-seventh Conference on Neural Information Processing Systems, 2023.
[34] Hyungjin Chung, Jong Chul Ye, Peyman Milanfar, and Mauricio Delbracio. Prompt- tuning latent diffusion models for inverse problems. In Forty-first International Conference on Machine Learning, 2024.
[35] Litu Rout, Yujia Chen, Abhishek Kumar, Constantine Caramanis, Sanjay Shakkottai, and Wen- Sheng Chu. Beyond first-order tweedie: Solving inverse problems using latent diffusion. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9472-9481, 2024.
[36] Bowen Song, Soo Min Kwon, Zecheng Zhang, Xinyu Hu, Qing Qu, and Liyue Shen. Solving inverse problems with latent diffusion models via hard data consistency. In The Twelfth International Conference on Learning Representations, 2024.
[37] Bingliang Zhang, Wenda Chu, Julius Berner, Chenlin Meng, Anima Anandkumar, and Yang Song. Improving diffusion inverse problem solving with decoupled noise annealing. In Proceedings of the Computer Vision and Pattern Recognition Conference, pages 20895-20905, 2025.
[38] Yutong He, Naoki Murata, Chieh-Hsin Lai, Yuhta Takida, Toshimitsu Uesaka, Dongjun Kim, Wei-Hsiang Liao, Yuki Mitsufuji, Zico Kolter, Ruslan Salakhutdinov, and Stefano Ermon. Manifold preserving guided diffusion. In The Twelfth International Conference on Learning Representations, 2024.
[39] Rayhan Zirvi, Bahareh Tolooshams, and Anima Anandkumar. Diffusion state-guided pro- jected gradient for inverse problems. In The Thirteenth International Conference on Learning Representations, 2025.
[40] Nataniel Ruiz, Yuanzhen Li, Varun Jampani, Yael Pritch, Michael Rubinstein, and Kfir Aberman. Dreambooth: Fine tuning text-to-image diffussonn models for subject-driven generation. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 22500-22510, 2023.
[41] Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations, 2023.
[42] Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. In The Eleventh International Conference on Learning Representations, 2023.
[43] Neta Shaul, Ricky TQ Chen, Maximilian Nickel, Matthew Le, and Yaron Lipman. On kinetic optimal probability paths for generative models. In International Conference on Machine Learning, pages 30883 30933- PMLR, 2023.
[44] Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling rectified flow transform- ers for high-resolution image synthesis. In Forty-first International Conference on Machine Learning, 2024.
[45] Ashwini Pokle, Matthew J. Muckley, Ricky T. Q. Chen, and Brian Karrer. Training- free linear image inversion via flows, 2024.
[46] Heli Ben-Hamu, Omri Puny, Itai Gat, Brian Karrer, Uriel Singer, and Yaron Lipman. D-flow: Differrnntiaiing through flows for controlled generation. Forty-first International Conference on Machine Learning. 2024.
[47] Yasi Zhang, Peiyu Yu, Yaxuan Zhu, Yingshan Chang, Feng Gao, Ying Nian Wu, and Oscar Leong. Flow priors for linear inverse problems via iterative corrupted trajectory matching. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
13

---

[48] Kushagra Pandey, Ruihan Yang, and Stephan Mandt. Fast samplers for inverse problems in iterative refinement models. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
[49] Yici Yan, Yichi Zhang, Xiangming Meng, and Zhizhen Zhao. FIG: Flow with interpolant guidance for linear inverse problems. In The Thirteenth International Conference on Learning Representations, 2025.
[50] Ségolène Tiffany Martin, Anne Gagneux, Paul Hagemann, and Gabriele Steidl. Pnp-flow: Plug- and-play image restoration with flow matching. In The Thirteenth International Conference on Learning Representations, 2025.
[51] Hengkang Wang, Xu Zhang, Taihui Li, Yuxiang Wan, Tiancong Chen, and Ju Sun. DMPlug: plug-in method for solving inverse problems with diffusion models. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
[52] Berthy Feng and Katherine Bouman. Efficient bayesian computational imaging with a surrogate score-based prior. In NeurlPS 2023 Workshop on Deep Learning and Inverse Problems, 2023.
[53] Morteza Mardani, Jiaming Song, Jan Kautz, and Arash Vahdat. A variational perspective solving inverse problems with diffusion models. In The Twelfth International Conference on Learning Representations, 2024.
[54] Luhuan Wu, Brian L. Trippe, Christian A Naesseth, John Patrick Cunningham, and David Blei. Practical and asymptotically exact conditional sampling in diffusion models. In Thirty-seventh Conference Neural Information Processing Systems, 2023.
[55] Brian L. Trippe, Jason Yim, Doug Tischer, David Baker, Tamara Broderick, Regina Barzilay, and Tommi S. Jaakkola. Diffusion probabilistic modeling of protein backbones 3d for the motif- scaffolding problem. In The Eleventh International Conference on Learning Representations, 2023.
[56] Gabriel Cardoso, Yazid Janati el idrissi, Sylvain Le Corff, and Eric Moulines. Monte carlo guided denoising diffusion models for bayesian linear inverse problems. In The Twelfth Interna- tional Conference on Learning Representations. 2024.
[57] Zehao Dou and Yang Song. Diffusion posterior sampling for linear inverse problem solving: A filtering perspective. In The Twelfth International Conference on Learning Representations, 2024.
[58] Åke Björek. Numerical methods matrix computations, volume 59, Springer, 2015.
[59] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjöm Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10684-10695, 2022.
[60] Ismail Alkhouri, Shijun Liang, Cheng-Han Huang, Jimmy Dai, Qing Qu, Saiprasad Ravishankar, and Rongrong Wang. SITCOM: Step-wise triple-consistent diffusion sampling for inverse problems. In International Conference on Machine Learning (ICML), 2025.
[61] Ségolène Martin, Anne Gagneux, Paul Hagemann, and Gabriele Steidl. Pnp-flow: Plug-and-play image restoration with flow matching. arXiv preprint arXiv:2410.024233 2024.
[62] Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018
[63] Bradley Efron. Tweedie's ' formula and selection bias. Journal of the American Statistical Association, 106(496):100--166)) 2011.
[64] Patrick Cattiaux and Arnaud Guillin. Semi log-concave markov diffusions. Séminaire de probabilités XLVI, pages 231 231-292, 2014.
[65] Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. In A. Beygelzimer, Y. Dauphin, P. Liang, and Wortman Vaughan, editors, Advances in Neural Information Processing Systems, 2021.
14

---

[66] Bin Dai, Ziyu Wang, and David Wipf. The usual suspects? reassessing blame for vae posterior collapse. In International conference on machine learning, pages 2312-2322. PMLR, 2020.
[67] Shun-Ichi Amari. Natural gradient works efficiently in learning. Neural computation, 10(2):251-
276, 1998.
[68] Hyungjin Chung, Byeongsu Sim, and Jong Chul Ye. chae---oooo---sss----eeeeee--eee Accelerating conditional diffusion models for inverse problems through stochastic contraction. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 12413-12422, 2022.
[69] Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 4401-4410, 2019.
[70] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large- scale hierarchical image database. 2009 IEEE conference on computer vision and pattern recognition, pages 248-255. leee, 2009.
[71] Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of the IEEE international conference computer vision, pages 3730 3730,-333,2220 2015.
[72] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unrea- sonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 586-595, 2018.
[73] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems, 30, 2017.
[74] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Medical image computing and computer-assisted MMMervention-MICCAA 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, pages 234-241. Springer, 2015.
[75] Quan Dao, Hao Phung, Binh Nguyen, and Anh Tran. Flow matching in latent space. arXiv preprint 2.1.::2307.06699 2023.
[76] William Peebles and Saining Xie. Scalable diffusion models with transformers. In Proceedings
of the IEEE/CVF International Conference on Computer Vision, pages 4195-4205, 2023.
[77) Sinho Chewi and Aram-Alexandre Pooladian. An entropic generalization of caffarelli's contrac- tion theorem via covariance inequalities. Comptes Rendus. Mathématiiuue 361((9):1171-1448 2023.
[78] Nanye Ma, Mark Goldstein, Michael S Albergo, Nicholas Boffi, Eric Vanden-Eijnden and Saining Xie. Sit: Exploring flow and diffusion-based generative models with scalable interpolant transformers. In European Conference on Computer Vision, pages 23-40. Springer, 2024.
[79] Kai Zhang, Lue Van Gool, and Radu Timofte. Deep unfolding network for image super- resolution. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 3217-3226, 2020.
[80] Su Wang, Chitwan Saharia, Ceslee Montgomery, Jordi Pont- Cont-Tuset, Shai Noy, Stefano Pellegrini, Yasumasa Onoe, Sarah Laszlo, David Fleet, Radu Soricut, al. Imagen editor and editbench: Advancing and evaluating text-guided image inpainting. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 18359-18369, 2023.
[81] Chitwan Saharia, Jonathan Ho. William Chan, Tim Salimans, David Fleet, and Mohammad Norouzi. Image super-resolution via iterative refinement. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4)::713-477) 2022.
[82] Noam Elata, Hyungjin Chung, Jong Chul Ye, Tomer Michaeli, and Michael Elad. Inv- fussion: Bridging supervised and zero-shot diffusion for inverse problems. arXiv preprint arXiv:2504.0 000.2990..069,... 2025.
15

---

A Proofs
Lemma A.1 (Tweedie's Mean Formula ). Suppose the joint distribution of Zo and z, factors as
Pe(zo, ze) p((z)) (z(zz) 20)
with
pz(z:
2o)
Ni(z aa()) zo: o(t)2
Then
Zu
zt]
Z + o(t)² Vz, log logp)(zi).
(22)
a(t)
Proof. Starting from the definition of the score,
Vz, Pog))(Zz)) = 2. (z()
(z() pz())) Vz. / Pe(zo, Ze) dzo
(z() / Ve, (pe(zo; 21)) dzo pe(z)) / pe Pe 21) Vz, log Pe (zo, ze) dzo
1
f "eleo 22) Vzi logpo(ze 20) dzo / pi(zo zi) a(1)² (a(l) 20 2.) dzo
1
a(t) (a(t) zi] - 21)
Rearranging completes the proof.
Lemma A.2 (Tweedie's Covariance Formula). For any distribution p(zo) and
Pe(ze 2n) = N(zz ((t) Zo. ott) I),
the posterior (zo 2z) is also Gaussian with mean E[zo zi] and covariance Cov(zo w.). These are connected the score function Vz, lz, via
o(1)²
Cov(za 2:]
I a(1)2 log pr(p))
(23)
a(t)
Proof. We start with the Hessian of log (o
3 log (o
V2 log))(zz)= = Vz. ( log (og:)())))
=
Oziz
Oz.1
((l) E 20
2i] 2z )
T
Vz
= a(t) Va, (a((())
21
(t))
at)
1
2z, (B|zo (x]
o1)2
o(1)2
a(t)
/ Pe(zo z() Vz. log
Pi(ze Zu)
z dzo
I
ot)2
Pe (z()))
o(t)²
(0())) / pe((o 2z:)
(zo-
Elzo
E3
2 zel]
dzo
1
I
a(t)
all) (000))
1
E|zo zo
xx]
E Iz0
E[zo
E2
2.]
I.
att)
Cov' (mo [as]
Solving for Cov(zo zı] yields the stated formula.
16

---

Lemma A.3 (Connection between Posterior Mean and Vector Field). Let a(t) Zo + o(t) 21 be a one-sided interpolant. where Zo Paata and z NN(O, I). Then the vector field vizr. t) satisfies
o(t) Z1 (á(1) ((t)a(t)) o()
vize, 1)
Elzo [r]
(24)
o(1)
Proof. Starting from the definition,
v(z [(d(t) zo + ((t) 21 zi] á(t) E'zu z + ((t) E|z1 zr). where (t). a(t) represent the first-order time derivatives of or (t) and att), respectively, Noting that
Elz, [ Ze a(1) Eizo zr]
(tt)
we obtain
Zt -
a(t)
E/zo
+ (d(t) a(t) ((t))(t) o(t)
o(t)
t(z,,t) = á(t) E(zu
z + att)
E
zz)-
(t))
o(t)
Lemma A.4 (Cramér-Rao inequality). Let j((x) exp(-x(x))dss be probability measure on R., where 4 : R + IR is twice continuously differentiable Then for any C³ (RR)).
Var,()) (RLlV (E, "B
(25)
Proof. For comprehensive proofs of the Cramér- Rao inequality, we refer readers to [77], and the references cited therein.
Lemma A.5 (Posterior Covariance Lower Bound via Cramér Cramér-Rao), Let a(t)zo + t()) where Zo ~ p(z)) & (xp(-((zo)) with E C22 C²(TR))) and 2 ~ NO Ia) are independent. Suppose:
(i) The likelihood is Gaussian: p(z: zo) -N(a(t)zo.
(ii) & is systrongly convex, ie., (2n( ž TIa.
Then the posterior satisfies:
-1 a(t)² (1)3
Cov(zo | 2x)) 7
La-
(26)
Proof. Consider assumption (i). By applying Bayes' rule, the negative log- peg-posterior is:
p(zo 2x)) = - log p(z,
2o) logp(zo) + const.
Taking second derivatives yields:
a(t)² a(1)
2. - log 2:)
La + (20(
Using assumption (ii), (2 2 rad we obtain:
a(1)
| (zo 22) 7
La
o(1)3
Finally, applying Lemma A.4, the Cramér-Rao bound gives:
a(t)2 at))
Cov(zo zz] 2 (Ezgizz
[V2 (- log zz)))) 2 +
Ia.
We write E (2(R denote that & has continuous first second derivatives on Rd Likewise. E (Rd) means that f has continuous gradient.
17

---

A.1 Proof of Proposition 4.1
Proof. The proof follows from Appendix C of [78]. We assume that the time-dependent latent variable z; Fd is generated by a known forward process of the form:
2, (1)) 20 + (()) e,
E~ &~,(0,
where Zo is the initial (clean) latent sample and a(t), (t) scalar functions of time.
This process induces iimeinddeedd family of conditional distributions pz y), where the expecta- tion is taken over the joint distribution of Zo y and the noise E.
We define the characteristic function of the marginal at time t as:
(.(k) = / 1k pz y)
[e
20
(27)
where E denotes expectation over the conditional joint distribution of zo y and E.
We now differentiate this characteristic function with respect to time:
(kpr 0 E ik xi
R
ik
R
ik
24 E x
ee Txx
Since z₁ is deterministic function of Zo and E, we may condition on z and write:
22 z) cekk² s]
Oep: (k)
(x)
E
iik
ik
2 z: z) ekk pz y) dz.
(z)
On the other hand, by the definition of the characteristic function:
(kp.(k) = 0, / c pc(z y) dz
= =f E 0,pil = yy)d. dz.
Equating the two expressions for Ok)r(k), we obtain:
/ eik' 0,p 1y) dz ik S (((e) pr(z y)) dz.
We now use the identity Veeik ik cik and integration by parts to move the gradient from eik = to the density term:
f e (, y) dz dz= cik²vv, ( y)) dz.
Finally, since both expressions match for all k, their inverse Fourier transforms must be equal. Hence, we conclude:
(zp.(z V (Vr(2) y))
This is the continuity (Fokker-Planck, equation with drift field v,. Therefore, integrating the ODE:
dz.
dt (z(e))
backward in time from Z ~ (O yields samples zo Po(z y), desired.
18

---

A.2 Conditional Vector Field
Proof. From Lemma A.3, the unconditional vector field is
d(1) o(t)
2 + (a(t) {(1)((1) Elzu ze).
v(zi, t)
a(t)
By Lemma A. we have
1
2: o(t)² Uz, log po(z))
B 20
a(l)
Substituting this in,
a(t)
alI) (o(t) a(t)a(t)) Vn. logPe(z)). a(t)
v(z,,) t)
a(t) 2s
+
Conditioning on the score becomes
Zz log y) Uz logpi(z)) + Uz logpr(y z.).
Hence, the conditional vector field is
V: v(z,,) (t) ((())(()) a(1) (())(()) Vz, log 22). att)
Finally, choosing the linear schedule a(l) I and (1)) simplifies this
v(z.. = v(zi,t) 1 t Z lo z:).
Approximating the unconditional velocity v(ze, 1) by parametric estimator ve (z, t) gives y,t) = volzi,t) 1-t
t
Vz, lo zz).
A.3 Derivation of Posterior Mean (Eq. (14)) and Posterior Covariance (Eq. (15))
Proof. From Lemma A.3, we know
v(2,, t) (t) 21 (a(1) +(1)a(1) a(t)
o(1)
E(zo mz]]-
With (l) = 1 - and (1) = l,, one obtains
Elzo 2r] 2, Lv(z,,))
Meanwhile, from Lemmas A.I and A.2, one show
Cov(zo 21] = o1)2
V Elzo 2e].
a(t)
Thus,
t2
Cov(zo 1z = (I t tV 1.-VVV..ve vo(z,,t)
A.4 Proof of Noise Conditional Score Approximation in Latent Space
Proof.
We assume the likelihood and posterior over X are Gaussian:
ply xo) = =A(xx Im), p(xo xx) N(m. E)
with:
Xo. X R
YER" AE M := E[xo xe). Σ = Cov(xo x1]
19

---

Define:
Ig *EAA Rm r y Am R"
Then:
ply xx1) = / ply (x))p(x)) xi) dxo = ((Am, S)
The log-likelihood becomes:
1 r -r logply xx) log log det S
We compute its gradient w.r.t. X E IR"" in two parts:
(I) Log-determinant term:
8S 0
A OΣ A Org
Xx, log det S = Tr Vv)
So,
Tr(AAS S S-IA
OΣ
Vx, log det S
E TR
TrA S A
(2) Quadratic term:
Tr
S -r S--1
88
Xx (r (T) = -2((x, T
R"
Tr S Tr 1
Dee.
Putting both together:
Tr (ATS-IA
aΣ 0r,11
+ Tr (S- 1-11 as
Xx, logr(y xx)) AA
Tr (ATS-1A
+ +TTr (S 'r s
88 Bar.
Assuming Cov(xo | Xr] is slowly varying w.r.t. Xx, we neglect its gradient, yielding the approximation:
Xx, logply xx) (Vx, m) A S
xxI) AT (oz AEAT)- (y AE xx)))
Now, we aim to derive principled extension of this formula to the latent space. Let Zo and Zr be latent variables, and Du be a (generally nonlinear) decoder such that xo (zo). Then the likelihood can be written as:
ply 20) = AD, 1)
which is Gaussian with nonlinear mean. We further assume:
p(zo 2.) A (Zo, Σz), where Zn = E|zo zz].
Σ = Cov 2i]
Since D.p is nonlinear, p(y 22)) is not Gaussian. However, approximate D (2p) via first-order Taylor expansion around Zg:
(z0 (Zo (Do))) Zu)
where (z()) is the Jacobian of Dop at Zo.
20

---

Using this approximation, the distribution over X becomes approximately Gaussian:
E X
z ~ Dup (Zn): Cov
zz] ~ Jp (z)) Σg JD
We now apply the same posterior score approximation used in pixel space, but using the decoded mean and propagated covariance:
Vx, log 2+) ~ (Vz, 20) JD(2)) T A (o + AJp(2z) Σ, (Zo)) A (y A (2))
This expression is now space, based on first-order decoder linearization and Gaussian propagation.
and justified approximation to the posterior score in latent
A.5 Proof of Proposition 4.3
Proof. We aim to compute the Jacobian 22 v(zi,) t). From Lemma A.3, we have:
o(t)
Zt + ((() ((t)a(t)
v(z,, t) o(t)
E zo zr].
o(t)
Taking the gradient with respect to Ze, we obtain:
a(t)
(((() - ((1)a(1))
Ve, v(z, t)
I+
Ve, Eizo 2z].
(t)
(t)
From Lemma A.1, the posterior mean is given by:
E 2o
z = a(t)
(zs + (t) log))(z)))
and therefore its gradient is:
V 2o
Zt
(I + o logne(z)))
a(I)
Substituting this into the expression for the Jacobian, get:
(tt)
o(t)a(t)) 1 (I + ( log))(z)))
V2, v(z, 1) ott)
I
((()) -(0))
a(t)
a(t)
Applying Lemma A.2, the Hessian of the log-density can be expressed as:
a(t) o(1)²
a Vt) lo:))(z:)=
Covizo 2r] - I,
which implies:
a(t)² ot))
I + (t) logp((z))
Cov(zu zr].
Therefore, the Jacobian becomes:
x(t)
I (a())
((t)a(t)
a(t) Cov(zo 2r].
Vx, v(z, 1)
o(t)
a(t)
(t)
By Lemma A.4, if p(zo) is c-semi-log-convex, then the Cramér Rao inequality yields:
a(1)2)
Cov [zo
2. 2 7
la-
a(t)
Substituting this bound into the Jacobian expression gives:
((t))
(a(1) a(ta((t)
a(t)
(7+ a(t)2)
Vz, (2), 2
((1) -00)
La.
(28)
a(1)
(1)²
a(1}
21

---

This expression simplifies to:
+
Vz. (z,,t) =
+a()) 1r-
(29)
Finally, observe that the right-hand side can be written as:
d ( log (a(1)² +xo(1)9) (0 o ((t)
r(() di
-0(1)2+ +
V2, v(z,, t) ł (t))
Now, assume the posterior covariance is positive semidefinite:
Covizu 2 ž O.
Define the scalar coefficient
c(t) = a(t) b(t)a(t) a(t) o(1)
o(t)²
Then the Jacobian simplifies
o(t) V2, (z,,t)
I + c(t) Cov|zo
zr]-
a(1)
Since Cov(zo ze] ž O, the sign of e(t) determines the direction of the inequality:
cIf)()) 2 then c(t) Covizz
2,1 0, and
a(t) o(1)
Vz, v 2
I.
If c(t) < 0, then c(t) Cov [20 21] 1 0, and
o(t) o(t)
V2, v(2, 1) 1
In particular, for the common case where a+() and (() have c(t) < for all
t E (0, 1J. and thus the Jacobian is upper bounded. This completes the proof.
A.6 Proof of Proposition 4.4
Proposition A.6 (Optimal Vector Field). Let Zy ~ NO, of I) and 2₁ ~ NO, I) be independent. Define 2, (1 t) 2o + tz for t [0, 1]. The optimal vector field v" v= that minimizes
arg mig [[Ilv(z,, - -(z -
is
(z,,)) - ( oir -t
(1 t)2 oh + t2
Ze-
Proof. Since the loss function is quadratic and v* (2z, t) depends only z, and the optimal vector field is the conditional expectation:
(z. t) Zz z [z.]
Given that Zo and are independent Gaussian random variables, and z, is a linear combination of Zo and z the joint distribution of 2o. 21, and z, is multivariate Gaussian. We will compute |zo 2,] and E iz [2:] using the properties of multivariate normal distributions.
First, we identify the covariance matrices:
Exors I Ezizi I, Σzazi 0,
22

---

since Zo and are independent.
Next, compute the covariance between 2g and Zz:
Σ Eoz
E [202]] (1 1)T [2000]
+ M... [202]] (1 -1)ooanr
Similarly, the covariance between 21 and Z+:
Σz,z, E [212,]] = = (1- [2)[(1) + +1 [z|2]]] = tl.
The variance of 2, is:
Ex,z. (1 - 1)2 Exnz 20 + 1 E x Ex11s
(1 2)) Tanr I 2 ((1 =) ( I =S,
where = - (1 + t.
The joint covariance matrix of Zo. 21, and Z, is:
Ezazi Ez,zi Ezazi
Dnoze
har
t
Ew =
Dz,ze
0
I
tl
Ezeze
(1 t)o
t1
$²1
Using the properties of conditional expectations for multivariate normals, we compute the conditional expectations:
E [ [2:] Dzore (Eeiz)) (1 1)
2²
E | [2,] Exize (Erize)
2, s2 Zz.
Therefore, the optimal vector field is:
(1 1)
(1 (1- 110 1
(z,,, t) = E (20 21 [22]
E Iz
ze]
- E 21 [2:]
latr
2t
har
$2
(1 2) lair + t2
A.7 Proof of Covariance in ITGDM
(LIGDM [22]). Let Xo ~ NO, da 1), and consider the forward process
X a(t) Xo + (())xı,
~ (0,
Then the conditional covariance Cov [xo xx] is
2 (t)2 Cov[xo xx] (11) o
a1)2 I.
(30)
Proof. Again Xg and x₃ are independent Gaussians, and x2 a linear combination. Thus Xg. X2 are jointly Gaussian with
Cov[x,] = att)² oz I + I
and
Cov[Xe; Xol = E (x, (x E[xo]) a<t) Cov[xo] = a(t) 2 I.
Using the standard formula for conditional covariances in Gaussian,
Cov(xo xx] Cov[xo] Cov[xo. xx] Cov[x]] Cov(xx.
Since Cov = 222m I, one obtains
Cov(xo xx] = 2 I (aa()) 1) (a(1) an + =(1)2) (a(t) oamm I).
which simplifies to
200 (t)
Cov(xo xx]
2 + o(1)²
23

---

+ n, B Closed-form Solutions for computing vector in Eq. (13)
In this section, we derive efficient closed-form expressions for computing the vector v under the assumption of isotropic posterior covariance, i.e. Cov 2o 2z] = r(1)2 I. We begin by introducing essential notation.
Notations.
Let m E (0,1)4t 13 represent the sampling positions in an image or signal.
The downsampling operator associated with m is Dm E {0, 1| selects only those rows (i.e. entries) of vector matrix corresponding the non-zero entries of For example, s-fold downsampling with evenly spaced ones is denoted Dis-
Dij represents distinct block downsampler, which averages blocks (each of size d/s) from a vector.
the (unitary) Fourier transform matrix of dimension d d, and F4s the analogous transform matrix for signals of dimension d/s.
v denotes the Fourier transform of the vector v, and v denotes its complex conjugate.
The notation refers element-wise (Hadamard) multiplication. Divisions such '/" or ' also apply element-wise when the vectors/matrices match in dimension.
Lemma B.1 (Downsampling Equivalence). Standard s-fold downsampling in the spatial domain is equivalent to s-fold block downsampling in the frequency domain. Concretely,
Dis D.a
Proof. Please see [79, 29] for details.
B.1 Image Inpainting
The observation model for image inpainting can be written as
m D
Xn
(31)
A
where n is noise. A convenient zero-filling version of can be defined
y D y
(xo n), ñ I).
The closed- fored-fommmaoo solution for v in image inpainting is then
(m @ (Pioo 2x)))
oZ r())2
Proof. Starting with the more general form,
== DJ oz 1 + r(t)² Dm m
(y y DD
Dm Di (E'z0 kull).
Since Dm DT on the support of and (o I ++((1)²))) = 1/ +r()))), it simplifies to
D (y Dm Di (E|zo 2zl))
3 (E'z zz!)
& + r(t)2
0 + r
Recalling that ý DT @ (xo + ii), arrive at the stated closed-fon solution.

---

B.2 Image Deblurring
For image deblurring, the observation model is
X0 *
n,
(32)
where k is the blurring kernel and * is the circular convolution operator. Using the Fourier transform, this can be written as
F- diag( k) F xo
n,
A
where is the DFT of k. Under isotropic-covariance assumption, the closed-form solution for
Fly AD E El 2.1)
o3 r(t)2 k|2
Proof. Because A is real linear operator of convolution type, we have AT AA Thus,
A AT (o r(t) AAT) A A zil)]
Substituting A digg (k) F and simplifying in the Fourier domain (using the diagonal structure in frequency space), one obtains
Fly - (D) oy r(1) kk/2
F
k
B.3 Super-Resolution
Following [79], the super-resolution observation model is approximately
(xo k) n,
(33)
which, in "canonical form," can be written as
DJ. F- d k) F xa
Hence, A = D4s F diag(k) 3 F. The closed-form solution under the isotropic assumption
F1s y AD 2o zil)
of rt)2 (k o k)
where O, denotes block-wise Hadamard multiplication.
Proof. Since AT = AH and
A Dis F-1 diag(())F F,
we get A = FT diag k (F- (F-I)TD D.. Applying Lemma B.1, namely D1s F-1 =-1== D Dua, and AAT)
its conjugate-transpose version, reduces the inverse (o]I+ ((t)AAAAATT)
to diagonal form in the
"downsampled" Fourier domain. One obtains
FI.ly - AD F F- O.
o3 r(t)2 (k k)) 38
25

---

C
Implementation details
Algorithm 1 LFlow Sampling: Posterior- Latent ODE Inference for Linear Inverse Problems
1: Input: measurements y. encoder Eg. decoder D.. forward operator A. pre-trained vector field
vo(.,t) for t E [ts, 0j, to = 0.8, and K = 2.
2: Initialize. ze, +- (1 - 1.) (y) + +4, 21, 21 ((0,
3: for t = ts down to do
4: Zp - 2, (o,
b Posterior mean prediction Eq. (14)
5: (1) - I- (1 (z,,)(
_
Posterior covariance estimation Eq. (18)
2. log p(y zz) - (V Dep (Zo)) AT (D=ADx(Eo))
6:
Eq. (13)
(22++1(1))AAT)
7:
for k = 1 to K do
V. 5 (o(2,,1) f Uz logply zi) 9: end for
8:
K-step update of vector field Eq. (9)
10: Ar & ODESolverStep (z,, Volz,,
11: end for
12: Return: + (20)
C.1
LFlow
For the solver parameters, we set the absolute and relative tolerances (atol and rtol) to 10-3 for inpainting and motion deblurring tasks, and to 10-5 for Gaussian deblurring and super-resolution tasks.
We set the hyperparameters to = 2 and lg 0.8 via ablation validation performance and were found to balance guidance strength and reconstruction fidelity across tasks.
tasks. These values were selected
To ensure dimensional compatibility, the measurement y is upsampled (e.g., via bicubic interpolation) to match the input size of the latent encoder. For consistency, we adopt the same super-resolution and deblurring operators in [29] across both method and the relevant baselines.
For inpainting tasks, incorporate the strategy proposed in the PSLD method [33] into our LFlow algorithm. This strategy reconstructs missing regions that align seamlessly with the known parts of the image, expressed xg- A Axn + (I-A"" (x))) Unlike the DPS sampler, which generates the entire image and may lead to inconsistencies with the observed data, this approach ensures that observations are directly applied the corresponding parts of the generated image, leaving unmasked areas unchanged [80]. For other tasks, such as motion deblurring, Gaussian deblurring, and super-resolution, this extra step is unnecessary since no box inpainting is involved, i.e., Xo = (zo).)
C.2 Comparison methods
PSLD [33] applies orthogonal projection onto the subspace of A between decoding and encoding to enforce fidelity:
Z DDIM(Z)) - VVz. [|y AD
[z z + 7 B[zo 2z] Es (Dip
(EE Zo
zz zz)))
- Ess (A + -I-
EEl 2o
(34)
We use fixed step size of p and select 7 as recommended in [36]. For our experiments, we rely on the official PSLD implementation with its default configurations. Specifically, we conduct ImageNet experiments using Stable Diffusion v1.5, which is generally considered more robust compared the LDM-VQ4 models.
PSLD aims to ensure that latent variables remain close the natural manifold by enforcing them to be fixed points after autoencoding. While this approach seems to be theoretically justified, it has proven empirically ineffective [34].
Phttps://githas http://iccabb.coo///ttttttt/////
26

---

Resample [36] estimates first clean latent prediction zn) (z++1) from the previous sample Z(+1 using Tweedie's formula, as described in Eq. (22). This prediction then used to update the latent state via DDIM:
z = DDIM cat (Z++1) (
(35)
The updated sample z is then projected back to measurementt latent variable z!!ed via:
N proj of Va.zz cond + +1- ā,)2% of aa.) of + (1-aa))
Ix
(36)
of + (1 x+)
where zimd is latent vector that satisfies the measurement constraint, obtained by solving the following optimization problem:
E arg min 1 ly - AP initialized at zū" (Z(+1).
cond 2
(37)
x 2
Here, of is a tunable hyperparameter controlling the trade-off between the prior and the data fidelity, and ā, is predefined DDIM noise schedule parameter. For experiments, adopt the publicly available implementation provided by the authors 3, using the pre-trained I.DM-V VQQ models on FFHQ and ImageNet [59], along with their default hyperparameters and 500-step DDIM sampler.
Resample refines latent diffusion sampling by balancing the reverse-time prior from the unconditional model with cessuurement-iffomeeee likelihood centered on a consistent latent- the sample aligns with both the data manifold and observed measurements.
MPGD [38] accelerates inference by computing gradients only with respect to the clean latent estimate instead of the noisy input, thus avoiding heavy chain-rule expansions. Their gradient update in latent space is as follows:
Z DDIM(Z;) ry VE v. l1y A (Dip (E(zo/z.)))1
(38)
Note that in MPGD, we leveraged Stable Diffusion v1.5 as for PSLD. For more information, please refer the GitHub repository
DAPS [37] refines latent estimates via Langevin dynamics guided by a latent prior and a measure- ment likelihood. The initial estimate (0) is obtained by solving a probability flow ODE from ze using the latent score model. At each inner iteration j = O,...,. N.,,, the latent estimate zo is updated as follows:
Rdd
(5) (j+1) = 20 + n (V log ((z) 22) + 20
logply ()))) + 2m Ey, Ej G~N(0,
(39)
Here, Th denotes the Langevin step size. The first term reflects prior guidance via the latent score model, while the second enforces consistency with the measurement through decoding and evaluating the likelihood.
DAPS avoids the limitations of local Markovian updates in diffusion models by decoupling time steps and directly sampling each noisy state z, from the marginal posterior p(z: yy) It performs posterior sampling by alternating between posterior-guided denoising via MCMC and noise re-injection, enabling large global corrections and improved inference in nonlinear inverse problems.
SITCOM (Step-wise Priple-Consistent Sampling) [60] enforces three complementary consistency conditions conditions-mmasurreenn measurement, forward diffusion, and step-wise backward diffusion diffusion-alloving diffusion trajectories to remain of a pre-trained diffusion model at each step, SITCOM ensures triple consistency across the data manifold, measurement space, and diffusion process, leading to efficient inverse problem solving.
www.urremmen--oossstteee with fewer reverse steps. By optimizing the input
At each step f, SITCOM enforces three consistencies:
27

---

(S1) Meassurement-consiiseeet optimization:
V == arg min
A
VV=
v1 de co(().t)
> +1 + - vil1
(40)
vai
C
C₁
(S2) Denoising:
2/ -
[v. v1 de ())
(41)
vor
(S3) Sampling (Forward step):
2 v0- 20 + V 11-1. di-1c.
(42)
Together, these steps enforce C1: measurement, C2: backward trajectory, and C3: forward diffu- sion consistency. For implementation details and hyperparameters, we rely on the official GitHub repository 6
SITCOM nudges the input to the denoiser at each diffusion step so its denoised output matches the measurements while staying close to the current state. It then computes the clean estimate (Tweedie) and re-noises via the forward kernel to keep the next input in distribution.
DMPlug [51] views the entire reverse diffusion process R(-) as a deterministic function mapping seeds to objects, and solves the inverse problem by optimizing directly over the seed z:
z E arg min A(y.)((())) + ((((()),, x" = R()").
(43)
Most existing DM-based methods for inverse problems interleave reverse diffusion with measurement projections in step-wise manner, but this often breaks both manifold feasibility (staying on the data manifold M) and measurement feasibility (satisfying {x y =(x))). In contrast, DMPlug is not step-wise: it preserves manifold feasibility by keeping the pretrained reverse process intact while promoting s A(x) via global optimization.
IIGM [22] considers the following gradient update scheme
x = DDIM(X))))---)) - (1x- A xx))) (rAAAT + +
dE Xx
x,
A
(44)
Ix,
where 1) controls the step size, By represents the noise level of the measurement, and ra the time- dependent scale for identity posterior covariance. For this method, we utilize the official, reliable code provided by [29].
OT-ODE [45] extends the gradient guidance of TTGDM [22] to ODE sampling via an optimal a(t)²
transport path, resulting variance for the identity covariance (+) +(t)++(()) I. Moreover,
the conditional expectation E[[p xz] computed from the velocity field vo (x,, t), according to the
relation in A.3. For fair comparison, used the same solver LFlow, i.e., the adaptive Heun.
hhttps://github http://githuub..omuuuuuu wwm/sun-eeeoooooo..oo.......ww.m watiaaummmDMMMM0
28

---

D Additional Experiments and Ablations
D.1
CelebA-HQ (256 256 x 3).
Table 5: Quantitative results of linear inverse problem solving on CelebA-HQ samples of the validation dataset.
Bold and underline indicate the hest and second-best respectively. The method shaded in gray is in pixel space.
Deblurring (Gauss)
Deblurring (Motion)
SR (x4)
Inpainting (Box)
Method
PSNRT
SSIM
LPIPS
PSNR1
SSIM LPIPS.
PSNR1
SSIM LPIPS
PSNRT SSIM LPIPS.
LFlow (ours)
29.06
0.825
0.164
30.14
0.845
0.167
28.92
0.830
0.170
24.82
0.876
0.123
SITCOM [60]
28.74
0.792
0.275
28.05
0.776
0.324
28.45
0.812
0.208
21.45
0.733
0.216
DAPS [37]
26.66
0.773
0.314
27.22
0,766
0.251
28.29
0.798
0.227
21.15
0.807
0.202
Resample [36]
28.07
0.742
0.239
28.37
0.804
0.232
29.84
0.806
0.193
19:49
0.797
0.237
PSLD [33]
29.47
0.833
0.310
29.75
0.821
0.313
31.65
0.829
0.246
24.03
0.812
0.165
MPGD [38]
29.85
0.821
0.792
0.348
0.280
23.80
0.773
OT-ODE [45]
27.83
0.789
0.292
0.758
0.326
28.95
0.784
0.251
22.37
0.790
0.225
C-IIGFM [48]
28.26
0.801
0.280
27.18
0.743
0.335
29.52
0.805
0.226
22.84
0.798
0.219
Measurements
Resample
(ours)
IVIE
IVIE
IVIE
UVIS
Figure 7: Qualitative results on CelebA-HQ test set. Row 1: Deblur (gauss), Row 2: Deblur (motion), Row 3:
SRx4, Row 4: Inpainting.
CelebA-HQ LFlow attains the lowest LPIPS across all four tasks and the highest SSIM three of four tasks. It reports LPIPS values of 0.164 (Gaussian deblurring), 0.167 (motion deblurring, 0.170 (super-resolution), and 023 (inpainting), surpassing the second-best method by 0.042 in inpainting. PSNR remains competitive throughout, ranking first in two tasks and remaining close elsewhere. These quantitative gains are reflected in the visual results: for both deblurring tasks, LFlow restores sharper eye contours, facial edges, and skin textures while avoiding ringing oversharpening seen in MPGD and PSLDD. In super-resolution, it preserves fine details such as eyelashes and lips with smooth transitions, maintaining natural gradients without introducing artifacts. In the inpainting task, LFlow offers semantically consistent completions with coherent tone and geometry- -wheress other methods exhibit mismatched shading, seams, or patchy textures. These results highlight LFlow's ability to recover fine structures while maintaining perceptual realism across diverse facial reconstructions.
29

---

D.2
Ce (512 X 512 x 3).
We further tested and evaluated our method on the high nigh-resolution CelebA-HQ dataset (512 x 512 x 3), demonstrating its robust capabilities in handling complex image processing tasks. In Figure showcase the effectiveness and versatility of our approach in enhancing image quality across various tasks.
Measurements
References
LFlow (ours)
Figure 8: Additional results CelebA-HQ 512 (motion), Row 3: SRx4, Row Inpainting.
512 datasct. Row 1: Deblur (gaussian), Row 2: Deblur
30

---

D.3
FID Score Results
Table 6: FID + scores across four inverse problems on FFHQ. CelebA-HQ, and ImageNet.
Dehlurring (Gaussian)
Deblurring (Motion)
SR (x4)
Inpainting (Box)
Method
FFHQ
CelebA
ImageNet
FFHQ
ColebA
ImageNet
FFHQ
CelebA
ImngeNet
FFHQ
CelebA
ImageNet
LFlow (ours)
52.48
47.79
88.76
57.11
48.53
82.89
58.49
51.07
92.28
34.40
30.78
117.45
SITCOM [60]
74.80
76.72
72.95
70.18
70.26
68.64
67.30
62.58
90.55
45.25
50.05
123.62
DAPS [37]
72.45
65.58
75.51
76.23
67.23
89.17
65,78
48.47
83.42
51.28
45.59
126.36
DMplug [51]
78.50
75.35
80.86
60.36
Resample [36]
78.62
MPGD [38]
64,20
61.37
102.58
70.32
88.7
146.58
90.55
84.43
119.12
84.53
53.15
154,28
PSLD|33
62.49
57.92
87.39
68.94
75.21
124.73
66.22
70.66
80.58
43,89
40.18
119.12
OT-ODE [45]
56.72
62.23
53.55
58.12
60.71
47.83
40.31
37.92
C-TIGFM (48)
56.85
50.54
45.10
34.96
PSNR and SSIM are commonly used recovery metrics that quantify pixel-level fidelity, while LPIPS and FID are considered perceptual metrics that assess high-level semantic similarity or perceptual quality. In this paper, focus on image reconstruction tasks from noisy measurements. In such settings, perceptual metrics like FID- althouggghh effective for evaluating generative models that prioritize visual realism- primarily measure distribution-leva similarity and may overlook structural details, especially when fine- fine-graine grained information is critical. Moreover, FID can be misleading when reconstructed images appear perceptually plausible but deviate significantly from the ground truth [49]. In contrast, PSNR and SSIM offer objective evaluations of noise suppression and content preservation, which crucial in experiments. That said, report FID scores across all three datasets considered above for four different tasks. As shown in Table 6, our algorithm achieves balanced trade-off between recovery and perceptual metrics. In the task of noisy image reconstruction, it not only delivers the best recovery metrics but also achieves strong perceptual scores.
realism-primarily
D.4 Ablation on hyperparameter K
To evaluate the effect of hyperparameter K, we conducted an ablation study on the FFHQ dataset. As shown in Table 7, setting K=2 yields slightly better average performance across tasks compared to K=1. Although we also tested 2 3, the results showed negligible improvements while incurring higher computational costs. For this reason, we adopt K=2 it provides consistent gains while maintaining reasonable inference time.
Table 7: Ablation of parameter K on FFI Increasing K offers marginal gains with higher cost; K = achieves the best trade-off off between performance and efficiency.
GDB
MDB
SR
BIP
K
PSNRT
SSIMT
LPIPSL
PSNRT
SSIM
I.PISS].
PSNRT
SSIMT
1.PIPS.
PSNRT
SSIM1
1PPISS
28.72
0,82
0.172
29.71
0.42
0.173
28.80
0.834 0,841
0.183 0.176
25.59
0.859 0,867
0.136
29.10
0,877
0.166
30.04
0.549
0,168
29.12
23,85
0.132
D.5 Comparing LFlow with supervised methods
We incorporated supervised results obtained using a conditional diffusion model. Our method offers several key advantages over supervised inverse approaches such SR3 [81] and InvFussion [82]:
L.Flow a zero-shot method that generalizes across diverse tasks without retraining, whereas supervised methods require training separate model for each specific task.
LFlow is robust varying degradation types, while supervised methods often exhibit poor generalization when faced with distribution shifts.
LFlow achieves significantly better performance certain datasets and resolutions- for example, FFHQ at 256 x 256.
These advantages clearly demonstrated in the results presented in Table 8.
31

---

Table 8: Comparison of LFlow with supervised baselines on tasks: super-resolution and box inpainting. For super-resolution, we compare against SR3, trained on the ImageNet dataset. For box inpainting, we compare against InyFussion, trained on the FFHQ dataset.
SR
Inpainting (Box)
Method
PSNR
SSIM
LPIPS
PSNRT
SSIM
LPIPS
SR3 [81]
24.65
0.708
0.347
InvFussion [82]
20.12
0.827
0.215
LFlow (ours)
25.29
0.696
0.338
23.85
0.867
0.132
D.6 Pretrained Models and Algorithm Conversion
Conversion of Pretrained Models. Conversion from discrete-time diffusion model continuous- time flow model was first introduced in [45] by aligning the signal-to-noise ratio (SNR) of the two processes. In principle, this enables mapping discrete diffusion steps {T) to continuous flow times t and rescaling the noise accordingly. However, this mapping is practically valid under restrictive assumptions namely. that both trajectories follow the same distributional path (e.g.. Gaussian) and employ linear noise schedule. While such conversion is feasible for pretrained models with similar linear schedules, becomes non-trivial for discrete-time latent diffusion models (e.g.. IDMs, Stable Diffusion). Their custom nonlinear schedules break the clean SNR alignment, often requiring the solution of nonlinear or even cubic equations infer consistent flow times.
Algorithmic Conversion. Beyond converting the pretrained model itself, one must also consider the sampling algorithm. Several baselines are inherently discrete-time: their update rules depend on the availability of finite noise grid and stepwise re-noising kernels. Such designs do not always admit a continuous- time analogue, and therefore direct conversion is not universally possible. Nevertheless, we successfully extended two representative methods, PSLD and MPGD, to the continuous time flow setting. By reformulating their projection projection-correctioo steps as infinitesimal updates within an ODE sampler, we obtain continuous-time counterparts that closely follow the spirit of the original algorithms while operating with a pretrained flow prior. In contrast, algorithms that fundamentally rely on discrete re-noising (e.g., DAPS) cannot be faithfully mapped without substantial redesign.
Table 9: Quantitative comparison of LFlow results on FFHQ dataset against continuous- time versions of CT-MPGD and CT-PSLD across four inverse problems. Bold and underline indicate the best and second-best respectively.
Deblurring (Gaussian)
Deblurring (Motion)
SR (x4)
Inpainting (Box)
Method
PSNRT
SSIM+ LPIPS.
PSNR1 SSIM LPIPS
PSNR
SSIM LPIPS
PSNR1 SSIM LPIPS.
LFlow (ours)
29.10
0.837
0.166
30.04
0.849
0.168
29.12
0.841
0.176
23.85
0.867
0.132
PSLD [33]
30.28
0.836
0.281
29.21
0.812
0.303
29.07
0.834
0.270
24.21
0.847
0.169
CT-PSLD
27.11
0.783
0.332
25.98
0.770
0.356
26.53
0.772
0.331
20.60
0.790
0.252
MPGD [38]
29.34
0.815
0.308
27.98
0.803
0.324
27.49
0.788
0.295
20.58
0.806
0.324
CT-MPGD
24.96
0.765
0.343
24.30
0.751
0.375
24.69
0.740
0.338
20.01
0.780
0.353
The results in Table 9 clearly show that algorithmic conversion is non-trivial. Although successfully reformulated PSLD and MPGD into their continuous-ting counterparts (CT-PSLD and CT-MPGD), both suffer a significant performance drop compared to their original discrete-time versions. Importantly, we kept the comparison fair by employing the same ODE solver (adaptive Heun) as used in LFlow. This indicates that simply replacing the diffusion prior with flow prior, while retaining the algorithmic structure, does not guarantee competitive performance in continuous time. Instead, the gap highlights the necessity of designing sampling strategies that are intrinsically compatible with ODE-base flow formulations, as achieved in our proposed LFlow framework.
Discussion.
32

---

D.7 Additional Visual Results (Best Viewed When Zoomed in)
Measurements
Referonces
Resample
LFlow (ours)
a
Figure 9: Additional Gaussian deblurring results on the FFHQ dataset.
Measurements
References
MPGD
PSLD
Resample
DAPS
LFlow (ours)
Figure 10: Additional Gaussian deblurring results on the CelehA-HQ dataset.
33

---

Measurements
References
Resample
LFlow (ours)
Figure 11: Additional motion deblurring results on the CelebA-HQ dataset.
Measurements References
MPGD
Resample
DAPS
LFlow (ours)
Figure 12: Additional motion deblurring results on the ImageNet dataset.
34

---

Measurements
Resample
LFlow (ours)
Figure 13: Additional Super-resolution results on the FFHQ dataset.
Measurements
References
Resample
LFlow (ours)
Figure 14; Additional Super-resolution results on the CelebA-I1Q dataset.
35

---

Measurements
Rofaronces
MPGD
PSLD
Resample
DAPS
LFlow (ours)
Figure 15: Additional Inpainting results on FFHQ dataset.
36

---

Measurements
Referonces
MPGD
PSLD
Resample
DAPS
LFlow (ours)
Figure 16: Additional Inpainting results on CelebA-HQ dataset.
37

---

NeurIPS Paper Checklist
The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility. transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
You should answer [Yes], [No] [NA].
[NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.
Please provide short (1-2 sentence) justificatinn right after your answer (even for NA).
The checklist answers
integral part of your paper submission. They visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
The reviewers of your paper will be asked to use the checklist one of the factors in their evaluation.
While "[Yes] is generally preferable to "[No]" ", is perfectly acceptable to answer "[No] provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). ). In general, answering "[No] " or "[NA] is not grounds for rejection. While the questions are phrased in binary way, acknowledge that the true answer often more nuanced, so please just use your best judgment and write justification elaborate. All supporting evidence can appear either in the main paper the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
IMPORTANT, please:
Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist".
Keep the checklist subsection headings, questions/answers and guidelines below.
Do not modify the questions and only use the provided macros for your answers.
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer: [Yes]
Justification: We only claim to present a new method, which we describe in Section 4. We compare our method against previous ones in Sections 5.1 and 5.2.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long it is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes]
38

---

Justification:
We discuss the limitations in Section 2.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proof's
Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [NA]
Justification: For each theoretical result presented in the paper, we provide the full set of assumptions explicitly and include complete, step-by-step proofs. All derivations are checked for mathematical correctness, with no steps omitted. Where appropriate, we include formal statements of assumptions (e.g., regularity or convexity conditions), and provide full proofs either in the main text or in the appendix to ensure transparency and reproducibility.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
39

---

Justification:
The manuscript describes all methods and experiments. provided for the methods. The code will be available in a future update.
Algorithms are whether the code and data are provided or not.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurlPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm. (b) If the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [No]
Justification: We plan to publicly release the full codebase and instructions to reproduce all experimental results upon acceptance. At submission time, we are unable share the code due to anonymity requirements and ongoing clcanup of the repository. We confirm that all experiments are reproducible and will be documented with detailed scripts and setup instructions.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips.cm co/
conoolcc))) for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (c.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https: // httppp/ww//pggieeeee/eeeeeiiiata
for more details.
40

---

The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data, The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments should state which ones are omitted from the script and why.
reproducible, they paper) is recommended, but including URLs to data and code is permitted.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible supplemental material (appended to
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Experiment details provided in Appendix C. Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [No]
Justification: Computing eπror bars for Table 1, 2, and 7 would require retraining every model several times for different datasets.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that out of range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
Question: For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
41

---

Answer: [Yes]
Justification: All experiments were conducted on a local machine equipped with a single NVIDIA RTX 3090 GPU. Experiment details provided in Appendix C.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics https http:///eeuppppeeeee https//ccpppiiicccccceccuiieeeee
Answer: [Yes]
Justification:
We have reviewed and agree with the NeurlPS Code of Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. there special consid- eration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification: There is no societal impact of the work performed.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deploymeet of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied partieular applications, let alone deployments. However, if there is direct path any negative applications, the authors should point it out. For example, it is legitimate to point that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and funetioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of MI..
11. Safeguards
42

---

Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification:
The paper poses no such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: The FFHQ, ImageNet and CelebA datasets are used under their respective non-commercial research licenses and cited appropriately.
Any third-party code or pretrained models (e.g.. for flow matching) are used under compati- ble open-source licenses (e.g., MIT) and are cited in the main paper.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website). the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. www.coo////////o datasets has curated licenses for some datasets. Their licensing guide can help determine the license dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
The paper does not release new assets.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/moodel as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
43

---

The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
The paper does not involve crowdsoureing nor research with human subjects
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsoureing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Answer: [NA]
Justification: This work does not involve large language models (LLMs)) in any part of the core methodology. LLMs were not used for designing algorithms, generating model outputs, conducting any experiments. They were only used for minor writing support and clarity improvement, which does not impact the scientific originality or rigor of the research.
Guidelines:
The answer NA means that the core method development in this research does not involve L.LMs as any important, original, or non-standard components.
Please refer to our LLM policy (https : http://s.....ooeeeeeeeee//////// for what should or should not be described.
44


Reviews:
[{"review_id": "VVjBEAb4TJ", "reviewer": "Reviewer_R8py", "summary": "In this work, the authors propose LFlow that exploits a pretrained latent flow matching prior for solving linear inverse problems. It mainly exploits flow matching and ODE sampling in a latent space. The work develops a time-varying posterior covariance for latent representation for gradient-based estimation. Results are shown for a few image restoration tasks with comparison to diffusion models.", "questions": "1.\tThe term “training-free inference” is a confusing one for diffusion models and the like. Given many methods do use pre-trained models, so there is typically expensive training on datasets. Some recent diffusion methods even do test-time adaptation of the pre-trained model when training and test data are significantly mismatched. There are other classes of image restoration methods like deep image prior, which do not involve any pre-training but still involve test-time adaptation. They could be better thought of as training-free but test-time adaptive.\n\n2.\tCan the authors include a derivation for (3)?\n\n3.\tBelow (14), Cov[z0|zt]  = (t^2/(1-t))E[z0|zt] is missing a gradient w.r.t. zt per (13)?\n\n4.\tEquation (20) uses E_phi(y). Is this an error since y (measurement) and x (image) have different dimensions? The following sentence says “ensures z_ts is closer to the posterior mode z0|y”, which is unclear. Is the y assumed to have noise?\n\n5.\tThe Algorithm pseudocode could be better stated in the paper rather than the appendix.\n\n6.\tThe Related Work section is misplaced before the final discussion in my opinion.\n\n7.\tThere need to be more insights provided on how the flow is happening in the proposed model. Could authors add more results and insights?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n1.\tThe work proposes latent space based flow matching for image restoration.\n\n2.\tThe algorithms are theoretically motivated, with proofs included in the Appendix, which is a strength.\n\nWeaknesses\n\n1.\tDespite rigor, there are numerous over-simplifying assumptions and approximations made throughout the paper that appear to make the work less complete and reduce interest, such as ones noted below.\n\n   a.\tFor example, Proposition 3.1 assumes the reverse-time Fokker-Planck model.\n\n   b.\tBelow (8), the dimensionality of zt and y are assumed matched, which is ad hoc. The authors say this can later be relaxed via an appropriate mapping. Please discuss more what you mean?\n\n   c.\tThe linear approximation to the decoder in (10) seemed overly simplistic.\n\n   d.\tThe Gaussian assumption on z0|zt wasn’t motivated.\n\n   e.\tIn (11), it seems incorrect that the distribution of images x0 depends on zt for variable t. Approximations are also made in (11) and (12).\n\n   f.\tAssumption 3.2 could hold for cases like Gaussian priors with well-trained VAEs. But more needs to be said for other architectures or more complex priors.\n\n   g.\tBelow (18),  the decoder Jacobian is assumed to be orthonormal, which is a very strong assumption.\n\n2.\tThe comparisons to diffusion models in the experiments use only a few sample methods. There have been other diffusion works significantly outperforming DAPS etc., across tasks recently.\n\n3.\tComparison to state of the art supervised image restoration methods is lacking.\n\n4.\tFrom Table 2, reconstruction accuracy metrics like PSNR are typically worse for LFlow although LPIPS seems slightly better. This makes me concerned the method may recover spurious realistic features. More discussion is needed.\n\n5.\tFor results in Figure 2, the quantitative metrics could be indicated. For example, the dog on the bottom right for LFlow shows spurious features that DAPS doesn’t.", "was_revised": true, "final_justification": "The authors responded to my comments in detail. While I do have some remaining concerns such as comparisons/baselines and what authors claimed they included in revised manuscript, I find value in the work. Hence, I increased my score to borderline accept.", "rebuttal": "We thank the reviewer for their time and constructive comments. \n### **1. On the Necessity and Scope of Our Assumptions**  \n\n> **a. Proposition 3.1.** This proposition is based on the **reverse-time continuity equation** (**Liouville** equation) for deterministic ODE flows, rather than the reverse-time **Fokker-Planck** equation, which describes stochastic diffusion processes (SDEs). Thus, our theoretical results apply directly to deterministic ODEs, not SDEs. This follows a standard setup used in prior works (e.g., Lipman et al., 2023) to justify sampling in continuous-time generative models. We revised the text to reflect this correction and avoid confusion.\n\n> **b. Dimension relaxation.** We thank the reviewer for this question. While Eq. (8) assumes matched dimensions for notational simplicity, our method is fully general and does not require $\\mathbf{z}_t \\in \\mathbb{R}^k$ and $\\mathbf{y} \\in \\mathbb{R}^m$ to have the same dimension. The likelihood is defined in image space as $p(\\mathbf{y} \\mid \\mathbf{x}_0)$, with $\\mathbf{x}_0 = \\mathcal{D} _\\varphi(\\bar{\\mathbf{z}}_0)$ and $\\bar{\\mathbf{z}}_0 = \\mathbb{E}[\\mathbf{z}_0 \\mid \\mathbf{z}_t]$. The guidance gradient is computed using the chain rule: $\\nabla _{\\mathbf{z}_t} \\log p(\\mathbf{y} \\mid \\mathbf{z}_t) =(\\nabla _{\\mathbf{z}_t} \\bar{\\mathbf{z}}_0)^\\top J _\\mathcal{D}^\\top \\mathcal{A}^\\top \\nabla _{\\mathbf{x}_0} \\log p(\\mathbf{y} \\mid \\mathbf{x}_0)$. For a Gaussian likelihood, this yields the explicit formula given in Eq. (12) of the paper. We have clarified this mapping and removed any ambiguity in the revised manuscript.\n\n> **c. Decoder linearity.** We thank the reviewer for this point. Most latent-based inverse solvers ignore decoder nonlinearity during inference and effectively assume local linearity, but often *without formal justification*—see, e.g., MPGD, Resample, LatentDMplug, and LatentDAPS. While PSLD [31] attempts to explicitly address the decoder’s nonlinearity, recent work **[1]** has shown this approach to be ineffective and prone to artifacts. In our method, we use a first-order Taylor expansion around $\\bar{\\mathbf{z}}_0$, which provides a tractable and theoretically grounded way to propagate uncertainty. This linearization is empirically justified by the observed smoothness of pretrained decoders and the fact that posterior samples typically remain in high-density regions of the latent prior, where the decoder behaves approximately linearly.\n\n> **d. Motivation for the Gaussian assumption on $p(\\mathbf{z}_0 | \\mathbf{z}_t)$.** We thank the reviewer for this question. The Gaussian assumption for $p(\\mathbf{x}_0 | \\mathbf{x}_t)$ is widely used in training-free (zero-shot) guidance methods for pixel-space inverse solvers (see, e.g., **[2]**). This approximation is motivated by tractability: computing $\\nabla{\\mathbf{x}_t} \\log \\mathbb{E} _{\\mathbf{x}_0 \\sim p(\\mathbf{x}_0 \\mid \\mathbf{x}_t)} [p(\\mathbf{y} | \\mathbf{x}_0)]$ is otherwise intractable, as it requires sampling from the full conditional at each step. In our case, the Gaussian assumption is further justified by the Gaussian latent prior and the smoother, more linear dynamics in latent space, which make the conditional distribution $p(\\mathbf{z}_0 | \\mathbf{z}_t)$ closely approximate Gaussianity—especially near high-density regions where the posterior concentrates. This is also supported by standard results for linear Gaussian systems in diffusion and flow models. We have clarified this motivation in the revised manuscript.\n\n> **f. Latent Prior $p(\\mathbf{z}_0)$.**  Our assumption is grounded in a bound on Jacobian of vector field $\\nabla_{\\mathbf{z}_t} \\mathbf{v} _\\theta(\\mathbf{z}_t, t)$ (Revised Proposition 3.3, which holds for any strongly log-concave prior, including  Smoothed Laplace,  Logistic Distribution, Generalized Gaussian with hight $\\beta$,  and Sub-Gaussian priors with strong tail curvature, ensuring theoretical robustness even when the prior deviates mildly from exact Gaussianity. In Remark 3.4 (Please see our response to Reviewer xFYk), we show that the Gaussian prior is a tight instance of this bound, justifying the closed-form posterior covariance in Eq. 17. \n\n> **g. Local isometry.** While assuming $J_{\\mathcal{D}}^\\top J_{\\mathcal{D}} \\approx \\mathbf{I}$ is indeed a simplification, we emphasize that: first, many decoder networks used in generative models (e.g., VAEs, diffusion decoders) are designed to approximately preserve local structure in the latent space, especially when trained with reconstruction loss and smooth latent priors. This implies that the decoder acts locally as a near-isometry around well-behaved regions of the latent distribution **[3]**, **[4]**; second, our assumption is only invoked locally around the posterior mean $\\bar{\\mathbf{z}}_0$, not globally across the entire latent space. In this regime, the decoder Jacobian typically varies smoothly, and near-orthonormal behavior has been observed empirically in prior work (e.g., VAEs, latent diffusion); third, importantly, the assumption allows us to arrive at a tractable and interpretable approximation of the likelihood gradient, which facilitates training-free inference. \n\n---\n### **2. Baselines** \n\n> We initially compared against five strong baselines. In response to all reviewer feedback, we have added additional baselines. Results are now presented in two categories: **latent-based methods** (*MPGD*, *Resample*, *PSLD*, *LatentDAPS* on three datasets, and *LatentDMPlug* on FFHQ) and **pixel-based methods** ($\\Pi$GDM, *OT-ODE*, and *C-$\\Pi$GFM* on CelebA-HQ). We kindly refer the reveiwer to **Table 2** (Reviewer **xFYk**) and **Table 3** (Reviewer **H6V9**).\n\n---\n### **3. Supervised baseline**\n\n>  Our focus was on training-free (zero-shot) methods, and none of the baselines we compare against—including recent state-of-the-art approaches—assume access to paired training data or require supervised fine-tuning. This aligns with the standard protocol adopted in many zero-shot studies. Supervised methods typically require task-specific training and do not generalize across different degradation operator settings.\n\n---\n### **4. Table 2: PSNR-LPIPS tradeoff and spurious features** \n\n> We thank the reviewer for this observation. As shown in Table 2, LFlow achieves consistently superior perceptual similarity (lower LPIPS) and SSIM across all tasks, while maintaining PSNR values comparable to the strongest baselines. This is in line with widely observed trends in generative models: methods optimized for perceptual quality may slightly sacrifice pixel-wise metrics like PSNR to better capture semantic structure and fine details. Occasionally, this can result in enhanced textures or subtle features that, while visually plausible, do not perfectly align with the ground truth—referred to as “spurious features.” Importantly, our qualitative results (Fig. 2) show that LFlow faithfully reconstructs key structures without introducing unnatural or fabricated artifacts. We have added further discussion of this well-known tradeoff and its implications for reconstruction quality in the revised manuscript.\n\n---\n### **5. Figure 2: dog case** \n\n> We thank the reviewer for the observation. Upon closer inspection, we note that LFlow’s reconstruction of the dog preserves the pose and structure consistent with the ground truth, while enhancing texture and fur detail. We believe the perceived \"spurious features\" may stem from higher-frequency detail restoration, not from hallucinated or incorrect content. In the revised manuscript, we have now annotated the visual examples in Figure 2 with their corresponding PSNR and LPIPS values for each method, including the dog example in the last row. This allows a direct quantitative comparison between perceptual and distortion-based metrics for the highlighted regions. We hope this clarifies how each method balances detail restoration and faithfulness to the reference image.\n\n---\n### **6. Questions** \n\n> **1.** We would like to clarify that the phrase “training-free inference” appears only once in our paper (Line 28), and we are happy to rephrase it if the wording is unclear. First, we acknowledge that diffusion models have proved to be effective for both **training-free inference** and **test-time adaptation**, the latter of which can involve updating the model during inference. Second, we recognize that the term **test-time adaptation** is broad and applies to various tasks—including classification, object detection, and inverse problems—whenever one has access to observations from a target domain and a pre-trained model from a source domain. In our case, however, we do not adapt the model at inference time; thus, no additional training is involved.\n\n> **2.** We have now included a derivation for Equation 3.\n\n> **3.** It is fixed now. Thank you.\n\n> **4.** In practice, $\\mathbf{y}$ is a *degraded version* of $\\mathbf{x}_0$, and we resize or zero-fill it to match the input size of the encoder $\\mathcal{E} _\\phi$, ensuring dimensional compatibility.  Also, as clarified in the experimental section, $\\mathbf{y}$ indeed contains noise. Yes — we directly pass the measurement to the encoder rather than a reconstructed image ($\\mathcal{A}^\\dagger \\mathbf{y}$).\n\n> The phrase *closer to the posterior mode* $\\mathbf{z}_0 | \\mathbf{y}$'' refers to initializing in a *plausible latent region* guided by $\\mathbf{y}$, which improves the stability of reverse-time sampling.\n\n> **5.** We did this now. \n\n> **6.** We have now revised the Related Work section and moved it to Section 2.\n\n> **7.** We added a trajectory of image flow to demonstrate the generation process.\n---\n**[1]. Prompt-tuning latent diffusion models for inverse problems (chung2023prompt)**\n\n**[2]. A Survey on Diffusion Models for Inverse Problem (daras2024survey)**\n\n**[3]. When is Unsupervised Disentanglement Possible? (horan2021unsupervised)**\n\n**[4]. Variational Autoencoders Pursue PCA Directions (by Accident) (2019Rolınek)**"}, {"review_id": "GM0WmRZEwj", "reviewer": "Reviewer_kCk5", "summary": "This work proposes a framework to solve linear inverse problems via latent flow matching models. Prior works based on diffusion models assume posterior covariance that is independent of the learned score or vector field which results in suboptimal solutions of the inverse problem. In contrast, this work derives an expression for posterior covariance  in latent space based on Tweedie’s formula.  In addition, the paper provides an expression for sampling in latent space from posterior by simulating a reverse time ODE. The formal proof shows that the probability density of the marginal distribution evolves according to the continuity equation. In practice, the expression for computing the conditional vector field is a sum of the unconditional vector field and a likelihood correction term. The latter can be evaluated with a Jacobian vector product. The method has been evaluated on different linear inverse problems like Gaussian deblurring, super-resolution, box inpainting, motion deblurring etc. under the set up of noisy linear inverse problem on $256 \\times 256$ resolution images.", "questions": "__Questions__\n1. Why is K-step update of vector field needed in Algorithm 1 (Line 7)? What quantity is re-used between the two iterations in this case (this is unclear from the current Algorithm 1)? As K is a hyper parameter, how does the performance of the algorithm vary with different values of K?\n2. Line 118-119 correctly state that the derivation assumes that the dimensionality of $z_t$ matches that of $y$. However, it is unclear to me how the paper correctly relaxes this assumption.\n3. How many NFEs on average are needed by adaptive heun sampler to converge? Are the NFEs in Table 3 for $\\Pi$GDM and LFlow comparable?\n4. How does LFlow perform in completely noiseless setting (i.e. $\\sigma_y = 0.01$? How does its performance compare against $\\Pi$GDM in this case? In my understanding, $\\Pi$GDM performs very well under the noiseless setting. \n\n__Suggestions__\n1. Table 2 and Table 4 could be augmented with NFEs for each method too for a more comprehensive  overview of the methods.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "__Strengths__\n1. Writing; The paper explains the main steps of the method well. Even though the method is quite involved, all the steps and assumptions involved in the algorithm have been listed well. \n2. The setting considered in this work of latent space flow matching model is more realistic and opens avenues to solve inverse problems on high resolution images. \n3. Qualitative and quantitative results seem good on different datasets and metrics as shown in Table 1 and Table 2. \n\n__Weaknesses__\n1. The method is slow and can take 4-10 mins to solve inverse problems even though it is faster than prior methods as shown in Table  4.\n2. The PSNR metric seems consistently weak and it is unclear how it is reflected in qualitative results. Theoretically, it should result in slightly more burrier images.\n3. (Minor) The paper only considers the noisy setting for linear inverse problem and not the noiseless setting which is also widely used.", "was_revised": true, "final_justification": "The reviewers provided additional empirical results that addressed my concerns about baselines. My other questions and concerns have been satisfactorily addressed. Therefore I'm increasing my score.", "rebuttal": "We are grateful for the reviewer’s insights and suggestions. We have carefully considered each point and provide our responses below.\n\n\n### **1. Inference time** \n\n> We appreciate the reviewer’s observation. While our inference time is already faster than that of prior methods (as shown in Table 4), we acknowledge that the current implementation can still require approximately **3 min 40 s** to **10 min** to solve inverse problems. Specifically, for **deblurring** and **super-resolution**, the average time ranges from about **2 min 30 s** to **5 min 40s**, whereas for **inpainting** it is around **3 min 40 s** to **10 min**. This is why we generally report a range of **4 to 10 minutes**. We believe this runtime is primarily due to two factors:\n\n\n\n\n> - **Use of adaptive solvers**: Our inference relies on an adaptive ODE solver, which results in variable computation time across samples. While some samples complete in ~3 minutes, others—depending on task complexity—can take longer.  \n\n> - **Repeated decoding for data consistency**: Similar to other latent-based inverse solvers, our approach enforces measurement consistency by repeatedly decoding intermediate latent states during inference. This decoding step, while necessary for accurate reconstruction, is computationally expensive unless the measurement operator is learned during training.  \n\n> For context, D-Flow **[1]** reports sampling times ranging from **5 to 15** minutes on average.  We believe our method offers a reasonable trade-off, and in future work we aim to further reduce inference time through **solver optimization or by exploring alternative numerical methods like accelerated integration techniques** **[2]**.\n\n---\n### **2. PSNR values** \n\n> We appreciate the reviewer’s observation. While our PSNR is competitive but not always the highest among all baselines, our method consistently outperforms all baselines in SSIM and LPIPS across all tasks, as shown in Table 1. These two metrics are more aligned with perceptual and structural fidelity, which is reflected in the qualitative results in Figure 1—our reconstructions preserve fine textures (e.g., skin, hair) and edges more faithfully.  The slightly lower PSNR may be due to small misalignments or high-frequency texture recovery, which benefit perceptual quality but increase pixel-wise error. This trade-off has also been observed in prior perceptual-focused methods such as Resample. Please note that on **CelebA-HQ**, our method achieves the best PSNR in three tasks.\n\n---\n### **3. Noiseless setting**\n \n> We thank the reviewer for pointing this out. All our baselines implemented in the latent space are evaluated only under the noisy setting, so we followed this setting for a fair and consistent comparison. We now included results for the completely noiseless setting on the CelebA-HQ dataset for super resolution and inpainting task. \n\n---\n### **4. Questions**\n\n> **1.** For computational efficiency, we experimented with  $K=1$ and $K=2$. As shown in the ablation results provided below (**Table 1**), using $K=2$ yields slightly better average performance across tasks while maintaining reasonable inference time. We also tested $K=>3$, but observed little noticeable improvement—but increased computational cost. In what follows, we present our ablation study of hyperparameter $K$ on FFHQ dataset.\n\n---\n| LFLow        |        | **GDB** |           |        | **MDB**     |           |        | **SR** |           |        | **BIP**     |           |\n|---------------|--------|-------------|-----------|--------|-------------|-----------|--------|------------------|-----------|--------|-------------|-----------|\n|               | PSNR↑  | SSIM↑       | LPIPS↓    | PSNR↑  | SSIM↑       | LPIPS↓    | PSNR↑  | SSIM↑            | LPIPS↓    | PSNR↑  | SSIM↑       | LPIPS↓    |\n| *K* = 1       | 28.72  | 0.829       | 0.172     |29.71  | 0.842       | 0.173    | 28.80  | 0.834            | 0.183     | 23.59  | 0.859       | 0.136     |\n| **K** = 2  | **29.10**  | **0.837**       | **0.166**     | **30.04**  | **0.849**       | **0.168**     | **29.12**  | **0.841**            | **0.176**     | **23.85**  | **0.867**       | **0.132**     |\n> **Table 1.** *Ablation of parameter* $K$ *on FFHQ. Increasing* $K$ *offers marginal gains with higher cost;* $K{=}2$ *achieves the best trade-off between performance and efficiency.*\n\n> **2.**  While Eq. (8) assumes matched dimensions for notational simplicity, our method is fully general and does not require $\\mathbf{z}_t \\in \\mathbb{R}^k$ and $\\mathbf{y} \\in \\mathbb{R}^m$ to have the same dimension. The likelihood is defined in image space as $p(\\mathbf{y} \\mid \\mathbf{x}_0)$, with $\\mathbf{x}_0 = \\mathcal{D} _\\varphi(\\bar{\\mathbf{z}}_0)$ and $\\bar{\\mathbf{z}}_0 = \\mathbb{E}[\\mathbf{z}_0 \\mid \\mathbf{z}_t]$. The guidance gradient is computed using the chain rule: $\\nabla _{\\mathbf{z}_t} \\log p(\\mathbf{y} \\mid \\mathbf{z}_t) =(\\nabla _{\\mathbf{z}_t} \\bar{\\mathbf{z}}_0)^\\top J _\\mathcal{D}^\\top \\mathcal{A}^\\top \\nabla _{\\mathbf{x}_0} \\log p(\\mathbf{y} \\mid \\mathbf{x}_0)$. For a Gaussian likelihood, this yields the explicit formula given in Eq. (12) of the paper. We have clarified this mapping and removed any ambiguity in the revised manuscript.\n\n> Later, posterior mean of $\\mathbf{z}_t$ given $\\mathbf{z}_t$ is passed through the decoder, and consistency is enforced in pixel space as following\n\n\n>- $\\nabla_{\\mathbf{z}_t} \\log p(\\mathbf{y} \\mid \\mathbf{z}_t) \\approx \\underbrace{(\\nabla _{\\mathbf{z}_t} \\bar{\\mathbf{z}}_0)^\\top J _{\\mathcal{D}}^\\top} \\underbrace{.... (\\mathbf{y} - \\mathcal{A} \\mathcal{D} _\\varphi(\\bar{\\mathbf{z}}_0))} _{\\mathbf{v}}$\n\n\n>- $ \\underbrace{(\\nabla _{\\mathbf{z}_t} \\bar{\\mathbf{z}}_0)^\\top J _{\\mathcal{D}}^\\top} = (J _{\\mathcal{D}} \\cdot \\nabla _{\\mathbf{z}_t} \\bar{\\mathbf{z}}_0)^\\top = (\\nabla _{\\mathbf{z}_t} \\mathcal{D}(\\bar{\\mathbf{z}}_0))^\\top $\n\n> **3.** Please note that we use an adaptive solver, where the number of function evaluations (NFEs) naturally varies across samples. For this reason, instead of reporting a fixed NFE, we focus on inference time, which provides a more practical and consistent measure of efficiency across tasks and methods.\n\n> **4.** We thank the reviewer for the question. While our main focus is the noisy setting—in line with all baselines implemented in latent space—we included the results for the completely noiseless setting on the CelebA dataset for two tasks. We agree that $\\Pi$GDM performs well in the noiseless setting; however, its performance drops significantly under noisy conditions.\n\n---\n\n**[1]. D-flow: Differentiating through flows for controlled generation (ben2024d)**\n\n**[2]. FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing (deng2024fireflow)**\n\n\n**==============================****==============================****==============================**\n**==============================****==============================****==============================**\n\n\n\n\n**Note**. Due to space constraints in the individual response sections, we provide a brief summary of our main revisions and clarifications here. Reviewer-specific details are included where space allowed.\n\n\n**Summary of Revisions:**\n\n We would like to provide all reviewers with a concise summary of the steps we have taken to address the comments and suggestions. For each major point, we have incorporated changes as recommended by the specific reviewer(s), as detailed below:\n\n- 1. **As recommended by Reviewer R8py**, we revised and modified the Related Work section and moved it before the Methodology section to clearly highlight prior approaches and better position our own contributions.\n\n- 2. **As suggested by Reviewer H6V9**, we improved the Methodology section—especially our contributions on latent likelihood approximation and posterior covariance estimation—and restructured it to better highlight our key contributions.\n\n- 3. **Following Reviewer R8py**’s suggestion, we added a conceptual figure illustrating the flow of our proposed method. We also moved the original figure (previously on page 25) to appear alongside this new figure, clearly presenting our methodology and highlighting the differences from prior work.\n\n- 4. **In response to Reviewer xFYk and Reviewer H6V9**, we added additional baselines. Results are now presented in two categories: **latent-based methods** (*MPGD*, *Resample*, *PSLD*, *LatentDAPS* on three datasets, and *LatentDMPlug* on FFHQ) and **pixel-based methods** ($\\Pi$GDM, *OT-ODE*, and *C-$\\Pi$GFM* on CelebA-HQ). Accordingly, we have revised and updated the Implementation Details section for improved readability and completeness.\n\n- 5. **Based on feedback from Reviewer kCk5**, we included an ablation study on the hyperparameter $K$, which controls the number of gradient update steps. This analysis clarifies the impact of $K$ on performance.\n\nWe hope these targeted revisions directly address each reviewer's feedback and strengthen the overall clarity and quality of the paper."}, {"review_id": "XQwD8ALXqF", "reviewer": "Reviewer_xFYk", "summary": "This paper presents an inference-time inverse problem solver with pretrained flow models. The proposed algorithm (LFlow) extends posterior sampling methods from diffusion models to flow models, especially latent flow models. Similar to previous works on diffusion posterior sampling, LFlow approximates the noise-conditional likelihood $\\nabla_{z_t} \\log p(y|z_t)$ using Gaussian approximation on $p(z_0|z_t)$, and derive a close-form expression assuming the linearity of the forward model. The authors further approximate the covariance matrix $Cov(z_0|z_t)$ assuming the prior is Gaussian. Experimental results on image restoration tasks show advantage of  LFlow against several existing baselines.", "questions": "I have some more questions regarding the experiments:\n\n- What is the detailed setup in posterior covariance ablation study that results in Table 3 and Figure 4?\n- Why is DAPS included in Table 2 but not in Table 4?", "limitations": "Yes. The authors address the limitation of this paper, e.g., relatively slow generation.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength: \n\n- This paper targets an interesting task of solving blind inverse problems with pretrained generative priors, and adapts diffusion posterior sampling methods to flow models.\n- The methodology, from estabilishment to practical implementation with approximation, is supported by rigorous theory.\n- The experimental results show promising advantages of the proposed method against previous methods.\n\nWeakness:\n\n- Flow models and diffusion models are known to be two sides of a coin, as they can be transferred from one to another by altering parameterization. The posterior sampling of conditional flow is not a result as exciting as it seems in the paper, given that diffusion posterior sampling is well-established.\n- Missing conceptual comparison to previous methods. There is no comparison to existing methods in likelihood approximation and posterior covariance approximation in Section 3, which obfuscates existing techniques with the original contribution of this paper. The likelihood approximation looks essentially the same as $\\Pi$GDM[1] and TMPD[2], and approximating the covariance matrix (e.g., assuming Gaussian prior) is also discussed in these previous works. It is not clear what is unique in LFlow at least from how it is presented in the current form.\n\n- The experiments are missing several recent baselines, such as DMPlug [3] and SITCOM[4]. The selected pixel-space method  $\\Pi$GDM is not the most competitive baseline as well. Also, LFlow is only applicable to linear inverse problems, while most of the compared baselines are applicable to general inverse problems.\n\n\n\n[1] Pseudoinverse-Guided Diffusion Models for Inverse Problems. Song et al. ICLR 2023.\n\n[2] Boys et al. Tweedie Moment Projected Diffusions for Inverse Problems\n\n[3] Wang et al. DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Models. NeurIPS 2024.\n\n[4] Alkhouri et al. SITCOM: Step-wise Triple-Consistent Diffusion Sampling for Inverse Problems. ICML 2025.", "was_revised": true, "final_justification": "The authors have addressed most of my original concerns on the manuscript, and I therefore decide to increase my score to 4.", "rebuttal": "We appreciate the reviewer’s comments, which helped us improve the clarity of the paper.\n\n### **1. Flow vs. Diffusion: Two Sides of the Same Coin?** \n \n> - Diffusion trajectories can be converted into flow paths by matching the signal-to-noise ratio, which allows one to derive the corresponding **time** and **scale** mappings. However, this holds only under certain assumptions—namely, that both processes share the same distributional path (e.g., Gaussian) and the diffusion model is trained in continuous time. For discrete-time models, which operate at fixed steps, this mapping becomes nontrivial. Although heuristic rounding methods exist [Karras 2022], they are often impractical—*particularly* for **latent diffusion models** or **Stable Diffusion** with non-linear noise schedulers—because they require solving a **cubic equation** to infer the flow time, making conversion extremely difficult, if not infeasible in practice.\n\n> - Flow-based models offer significant advantages in both **training** and inference efficiency, thanks to their straighter sampling trajectories. This led the researchers to apply it for inverse problems in *pixel space*, as demonstrated by methods such as OT-ODE [43] and D-Flow [44].\n\n> - **Latent** flow models, such as Stable Diffusion3 (Esser et al., 2024) and OpenSora-2, have shown great success in synthesis but remain underexplored for inverse problems. Our work takes a step in this direction by enabling efficient and accurate posterior sampling in the latent flow framework.\n\n---\n### **2. Missing Conceptual Comparison**\n\n> - We have a conceptual figure in **Appendix (page 25)** that illustrates the differences between our method and prior methods such as OT-ODE and $\\Pi$GDM (we moved this now to the method section). We have also clarified the distinctions in the revised Related Work section (now Section 2). Additionally, we have now included a conceptual figure that illustrates the overall flow of our proposed method. \n\n---\n### **3. Likelihood and Posterior Covariance Estimation**\n \n> We should have included our related work section earlier, before the method section (this has now been revised), which would have made the differences more apparent. We approximate the likelihood as $\\nabla _{\\mathbf{x}_t} \\log \\mathbb{E} _{\\mathbf{x}_0 \\sim p(\\mathbf{x}_0 | \\mathbf{x}_t)}[p(\\mathbf{y} | \\mathbf{x}_0)]$, whereas previous methods approximate it as $\\nabla _{\\mathbf{z}_t} \\log \\mathbb{E} _{\\mathbf{x}_0 \\sim p(\\mathbf{x}_0 | \\mathbf{z}_t)}[p(\\mathbf{y} | \\mathbf{x}_0)]$. We believe that, roughly, all methods assume the reverse process $p(\\mathbf{x}_0|\\mathbf{x}_t)$ is Gaussian, characterized by a posterior mean and posterior covariance. The key difference lies in modeling posterior covariance: Most of methods, whether in pixel or latent spaces, such as DPS, DAPS, Resample, and so on, consider it zero. $\\Pi$GDM assumes **Gaussian data distribution** and derives scaling from a **forward process**, which is prior-agnostic (proved in page 22 and 25), TMD applies a second-order Tweedie’s formula with a row-sum approximation. In contrast, we achieved as follows (revised version). \n\n**Latent Posterior Covariance.** \n\nTo address this, we further analyze the structure of the posterior covariance and the Jacobian $\\nabla _{\\mathbf{z}_t}\\mathbf{v} _{\\boldsymbol{\\theta}}(\\mathbf{z}_t, t)$, under certain regularity assumptions on the latent prior $p(\\mathbf{z}_0)$.\n\n **Assumption (Strong Log-Concavity of the Latent Prior \\cite{cattiaux2014semi})**\nLet $p(\\mathbf{z}_0) = \\exp(-\\Phi(\\mathbf{z}_0))$ be the latent prior over $\\mathbb{R}^d$, where $\\Phi$ is a twice continuously differentiable potential function. We assume that $p(\\mathbf{z}_0)$ is $\\gamma $-strongly log-concave for some $\\gamma > 0$; that is, $\\nabla^2 \\Phi(\\mathbf{z}_0) \\succeq \\gamma \\, \\boldsymbol{I}_d, \\text{for all } \\mathbf{z}_0 \\in \\mathbb{R}^d.$\n\n**(1) Log-concave Prior:** This assumption imposes a uniform lower bound on the Hessian of the prior's potential function, which translates to a lower and upper bound on the Jacobian of the vector field, as formalized below.\n\n\n**Proposition (Bound on the Jacobian of the Vector Field).** Let $\\mathbf{v}(\\mathbf{z}_t, t)$ denote the velocity field of the interpolant $\\mathbf{z}_t = \\alpha(t)\\mathbf{z}_0 + \\sigma(t)\\mathbf{z}_1$ between a standard Gaussian prior and a target distribution $p(\\mathbf{z}_0)$,  defined via coefficients $\\alpha(t), \\sigma(t) \\in \\mathbb{R}$. Under Assumption 1, the Jacobian of the vector field satisfies the following bound:\n\n\n$\\frac{\\mathrm{d}}{\\mathrm{d}t}( \\tfrac{1}{2} \\log(\\alpha(t)^2 + \\gamma \\sigma(t)^2)) \\cdot \\boldsymbol{I}_{d_z} \\preceq \\nabla _{\\mathbf{z}_t} \\mathbf{v} _{\\boldsymbol{\\theta}}(\\mathbf{z}_t, t) \\prec \\frac{\\mathrm{d}}{\\mathrm{d}t} \\log \\sigma(t) \\cdot \\boldsymbol{I} _{d_z} \\quad \\text{for all } t \\in (0,1],\\; \\mathbf{z} \\in \\mathbb{R}^d$\n\nThe proof is provided in Appendix A. The resulting sandwich bound ensures the Jacobian remains well-behaved—the upper bound guarantees valid and stable posterior covariance, while the lower bound prevents overestimated uncertainty during posterior-guided inference.\n\n**(2) Gaussian Prior:** We now consider a tractable special case where the latent prior is Gaussian, i.e., $\\mathbf{z}_0 \\sim \\mathcal{N}(0, \\sigma _{\\textit{latr}}^2 \\boldsymbol{I})$,  a choice that is both theoretically justified and widely used in practice—for instance, in variational autoencoders (VAEs), where the latent prior is regularized toward a standard Gaussian via KL divergence~\\cite{vahdat2021scorebased}. This enables us to derive the optimal velocity field and its Jacobian in closed form, which is characterized in the next proposition, whose proof can be found in Appendix A. \n\n**Proposition (Optimal Vector Field).** Let  $\\mathbf{z}_0 \\sim \\mathcal{N}(\\mathbf{0}, \\sigma _{\\text{latr}}^2 \\boldsymbol{I} _{d_z})$  and $\\mathbf{z}_1 \\sim \\mathcal{N}(\\mathbf{0}, \\boldsymbol{I} _{d_z})$ be independent random variables.  Define $\\mathbf{z}_t = (1 - t) \\mathbf{z}_0 + t \\mathbf{z}_1$ for $t \\in [0, 1]$. The optimal vector field $\\mathbf{v}^{\\star}(\\mathbf{z}_t, t)$ that minimizes the expected squared error \n\n\n$\\arg \\min_{\\mathbf{v}} \\, \\mathbb{E}\\left[ \\left\\| \\mathbf{v}(\\mathbf{z}_t, t) - (\\mathbf{z}_0 - \\mathbf{z}_1) \\right\\|^2 \\right]$ is given by  $\\mathbf{v}^{\\star}(\\mathbf{z}_t, t) = \\frac{(1 - t) \\sigma _{\\textit{latr}}^2 - t}{(1 - t)^2 \\sigma _{\\textit{latr}}^2 + t^2} \\mathbf{z}_t.$\n\n\n **Remark (Tightness of Bound).** When $\\mathbf{z}_0 \\sim \\mathcal{N}(0, \\sigma _{\\textit{latr}}^2 \\boldsymbol{I})$,  the optimal vector field $\\mathbf{v}^\\star(\\mathbf{z}_t, t)$ achieves the Jacobian lower bound in Proposition 3. This follows from the potential function $\\Phi(\\mathbf{z}_0)=\\frac{1}{2\\sigma _{\\textit{latr}}^2} \\|\\mathbf{z}_0\\|^2$,  yielding $\\gamma = \\sigma _{\\textit{latr}}^{-2}$, and interpolation coefficients $\\alpha(t) = 1 - t$, $\\sigma(t) = t$. Substituting into the bound confirms it matches the exact Jacobian of $\\mathbf{v}^\\star$, making the bound tight.\n\nPlugging the Jacobian of the optimal vector field into \\eqref{} yields:\n\n$\\mathbb{C}\\mathrm{ov}[\\mathbf{z}_0 \\mid \\mathbf{z}_t] = r^2(t) \\cdot \\boldsymbol{I} _{d_z}, \\quad \\text{with} \\quad \nr^2(t)  = \\frac{t^2 \\left[(1 - t)(1 - 2t) + 2t^2 \\right]}{(1 - t)\\left[(1 - t)^2 + t^2\\right]},$\n\nwhere we assumed $\\sigma_{\\text{latr}} = 1$. As a result, the propagated covariance can also be simplified as \n\n$\\mathbb{C}\\mathrm{ov}[\\mathbf{x}_0 \\mid \\mathbf{z}_t] \\approx J _{\\mathcal{D}} \\cdot \\mathbb{C}\\mathrm{ov}[\\mathbf{z}_0 \\mid \\mathbf{z}_t] \\cdot J _{\\mathcal{D}}^\\top \\approx r^2(t) \\cdot J _{\\mathcal{D}} J _{\\mathcal{D}}^\\top.$\n\nComputing the full Jacobian $J_{\\mathcal{D}} \\in \\mathbb{R}^{d_x \\times d_z}$ is often infeasible in practice. To simplify, we assume that the decoder acts approximately as a local isometry near $\\bar{\\mathbf{z}}_0$ \\cite{dai2020usual}, such that $J _{\\mathcal{D}} J _{\\mathcal{D}}^\\top \\approx \\cdot P$, where $P$ is the orthogonal projector onto the image of $J _{\\mathcal{D}}$.  For computational convenience, we approximate this behavior as isotropic in the full space, resulting in:\n\n$\\mathbb{C}\\mathrm{ov}[\\mathbf{x}_0 \\mid \\mathbf{z}_t] \\approx r^2(t) \\cdot \\boldsymbol{I} _{d_x}.$\n\n---\n### **4. Baselines**\n\n> We considered five baselines and provided both quantitative and qualitative results for four tasks of each. We notice that **SITCOM** was published at ICML 2025, with its latest arXiv version appearing after our NeurIPS submission. Nonetheless, we now cite it appropriately. For **DMPlug**, we have included additional results **Table 2**. \n\n| Methods      |        | **GDB** |        |        | **MDB** |        |        | **SR**  |        |        | **BIP** |        |\n|-------------|--------|---------|--------|--------|---------|--------|--------|---------|--------|--------|---------|--------|\n|             | PSNR↑  | SSIM↑   | LPIPS↓ | PSNR↑  | SSIM↑   | LPIPS↓ | PSNR↑  | SSIM↑   | LPIPS↓ | PSNR↑  | SSIM↑   | LPIPS↓ |\n| LatentDMplug       | 27.43  | 0.784   | 0.240  | 27.95  | 0.828   | 0.243  | **29.45**  | 0.838   | 0.183  | 22.50  | 0.819   | 0.220  |\n| LFlow | **29.10** | **0.837** | **0.166** | **30.04** | **0.849** | **0.168** | 29.12 | **0.841** | **0.176** | **23.85** | **0.867** | **0.132** |\n\n**Table 2:** *Quantitative comparison of LFlow and LatentDMplug across four inverse problems (BIP: Box InPainting, GDB: Gaussian DeBlurring, MDB: Motion DeBlurring, SR: Super-Resolution) on the FFHQ dataset*.\n\n\n> We respectfully note that several recent top-conference works have also focused exclusively on linear inverse problems, including but not limited to **OT-ODE** (TMLR 2024), **PSLD** (NeurIPS 2024), **PnP-Flow** (ICLR 2025), **FPS-SMC** (ICLR 2025), and **C-$\\Pi$GFM** (NeurIPS 2024).\n---\n### **5. Questions**\n\n> 1. Details of the posterior covariance ablation are on page 25, but now explained in the right place.\n> 2. We added DAPS result in Table 4."}, {"review_id": "Qz1riVhDT1", "reviewer": "Reviewer_H6V9", "summary": "This paper introduces LFlow, a novel training-free framework for solving linear inverse problems in imaging. The core idea is to leverage a pretrained latent flow-matching model to define a prior distribution. The method operates in a compressed latent space for computational efficiency. The key contributions are twofold: (1) performing ODE-based sampling in the latent space of a pretrained autoencoder, which is more scalable than pixel-space methods , and (2) introducing a theoretically-grounded, time-dependent posterior covariance for the guidance term, which is derived under a Gaussian latent representation assumption to improve sampling accuracy and convergence.", "questions": "no", "limitations": "no", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "pros:\n1. A major practical advantage of LFlow is its computational efficiency. The ablation study on inference time (Table 4) shows that LFlow is substantially faster than its latent diffusion-based counterparts.  This is a crucial benefit, as the computational cost of iterative refinement models is often a significant barrier to their practical application. The paper effectively demonstrates that the proposed posterior covariance model leads to faster convergence.\n2. The paper is well-written, clearly structured, and easy to follow. The authors provide a detailed appendix containing full proofs, algorithm specifics, and extensive additional results, which bolsters the credibility of the work and provides a clear path for reproducibility.\n\ncons:\n1. The derivation of the optimal time-dependent posterior covariance (Proposition 3.5 and Eq. 17) hinges on the assumption that the latent prior is a standard Gaussian $(z_0 \\sim \\mathcal{N}(0,\\sigma_{latr}^{2}I))$.  While this is a common regularizer for VAEs, the true latent distribution of a powerful autoencoder is unlikely to be perfectly Gaussian. In fact, the VAEs used in diffusion models—such as Stable Diffusion or Flux—are only weakly regularized by a Gaussian prior. The paper does not analyze the sensitivity of the method to deviations from this assumption, which could affect its performance in practice.\n    - This also raises concerns about whether the Optimal Vector Field can be effectively applied to pixel-space inverse problem solving.\n\n2. The authors state they use an LFM-VAE framework based on a DiT transformer architecture, while the baseline methods primarily use LDM/Stable Diffusion models built on U-Nets. Although the authors claim their prior's quality does not exceed that of the baselines, this architectural difference is a potential confounding variable. It makes it harder to isolate the performance gains as being solely due to the proposed LFlow methodology versus the underlying generative architecture. Note that for solving inverse problem with pretrained diffusion models, to the best of my knowledge, all the related papers conducted benchmarks on the same backbone model.\n\n3. Baseline comparisons and fairness of evaluation:\nThe paper would benefit from a more comprehensive comparison with existing flow matching baselines. While the authors mention OT-ODE, they do not include it in the experimental comparison. Although OT-ODE was originally applied to pixel-space models, it can be readily adapted to the latent space—just as with \\$\\Pi\\$GDM. Similarly, other recent flow-matching solvers, such as [C-\\$\\Pi\\$GFM](https://proceedings.neurips.cc/paper_files/paper/2024/file/2f46ef5725a8eca24f7f24a17955ad1a-Paper-Conference.pdf), should also be considered. Moreover, some empirical studies (e.g., Stable Diffusion 3) suggest that the conventional noise schedules used in diffusion models are less efficient than flow matching. This raises a concern about the fairness of comparisons between the proposed method and diffusion-based baselines. To strengthen the claim that \"we designed a better sampler,\" it would be helpful to control for the training objective (e.g., flow matching vs. diffusion) and isolate the benefit from the proposed sampler design itself.\n\n4. Section 3 presents a mixture of existing methods and the proposed contributions, which makes it difficult to clearly identify the novelty of the paper. For example, techniques such as posterior sampling, conditional vector fields, and the initialization of the flow sampling process have already been introduced in prior works. I would recommend restructuring this section to more explicitly distinguish between what is novel in this paper and what is borrowed from previous literature.", "was_revised": true, "final_justification": "While there are still issues remained, the authors have resolved most of my concerns.", "rebuttal": "We sincerely thank the reviewer for their feedback. We address all comments individually below.\n### **1. What are Assumptions on Latent Prior $p(\\mathbf{z}_0)$ ?**\n\n> - **(1). Log-concave Prior.** Our assumption is first grounded in a bound on Jacobian of vector field $\\nabla_{\\mathbf{z}_t} \\mathbf{v} _\\theta(\\mathbf{z}_t, t)$ (Revised Proposition 3.3, which holds for any strongly log-concave prior, including **Smoothed Laplace**,  **Logistic Distribution**, **Generalized Gaussian with hight $\\beta$**,  and **Sub-Gaussian priors with strong tail curvature**, ensuring theoretical robustness even when the prior deviates mildly from exact Gaussianity. We further refer the reviewer to our complementary response to the third question of Reviewer xFYk regarding this point. \n\n\n> - **(2). Gaussian prior.** Using a Gaussian latent prior is standard practice in models such as Stable Diffusion and LDMs, enabling closed-form sampling and efficient inference. In VAEs, the latent codes are explicitly regularized towards a standard normal prior ($\\mathcal{N}(0, \\mathbf{I})$) during training, so the aggregate posterior $q(\\mathbf{z})$ is encouraged to match this prior—which also serves as the base distribution for score-based generative modeling (e.g., Vahdat et al., 2021). Although the true posterior may be complex, this procedure results in an aggregate latent distribution that is approximately Gaussian, as established in prior work. Importantly, if the latent prior were highly non-Gaussian, reliable sampling from the autoencoder (and thus from any latent generative model) would become fundamentally challenging. Since standard practice in VAEs and latent diffusion models is to sample $\\mathbf{z}_0 \\sim \\mathcal{N}(0, \\mathbf{I})$ for generation, a significant mismatch would lead to unrealistic or out-of-distribution decodings—impacting all approaches relying on pretrained autoencoders, not just ours. Thus, our assumption is both practical and broadly consistent with the requirements of latent generative modeling. Furthermore, our assumption is significantly less restrictive than the Gaussian assumption made by $\\Pi$GDM on the **data** itself (i.e., $p(\\mathbf{x}_0) \\sim \\mathcal{N}(0, \\boldsymbol{\\Sigma}))$. This is shown in proof A.7.\n\n>- **(3). Remark.** In this remark (please see response **3** to **xFYk**), we show that the Gaussian prior is a tight instance of this bound (The Gaussian prior makes the inequality in the bound turn into an equality), justifying the closed-form posterior covariance in Eq. 17.\n\n> - **Posterior Covariance in pixel space.** \n> Under *Gaussian data assumption*, an analogous posterior covariance can be derived in **pixel space**, making the Optimal Vector Field applicable there as well. However, the latent space formulation is much more natural and theoretically justified: the latent Gaussian prior is better supported (e.g., by VAE training), and the reduced dimensionality enables more stable and tractable posterior estimation.\n\n---\n### **2. Architectural Differences**\n\n> We respectfully offer a different perspective and provide two complementary reasons:\n\n> - **First**, the generative process is evaluated based on **image quality**, not the architecture itself. What ultimately matters in evaluating a generative prior is the quality of the samples it produces, as measured by such metrics as LPIPS and FID—not the specific architecture used. Therefore, if the prior in our method does not outperform the baselines in **unconditional generation** (as we explicitly state), any observed improvement in inverse problem performance can be reasonably attributed to the **inference strategy** rather than architectural differences. The comparison remains valid and well-justified.\n\n> - **Second**, solving inverse problems inherently requires **balancing** *prior fidelity* and *measurement consistency*. While strong priors (e.g., Stable Diffusion 1.5) have advantages, they may compromise consistency and overall performance, as shown in baseline experiments. This reflects a general principle: even if a stronger prior is used, it does not necessarily yield better inverse problem results unless it aligns well with the measurement model.\n\n---\n### **3. Baselines and Fair Comparisons**\n\n> **Baseline Comparisons:** \n>- All existing flow matching baselines (e.g., OT-ODE, C-$\\Pi$GFM) operate in **pixel space**, whereas our method is designed for **latent space**. In this work, we primarily focused on latent-space baselines to enable a fair and meaningful comparison within the same modelling paradigm.  That said, we acknowledge that OT-ODE can, in principle, be extended to the latent space. In fact, we already explored such a variant in our **ablation study**, using the OT-ODE posterior covariance (denoted **Cov-$\\Pi$GDM**), as described in the Appendix (p. 25).\n\n\n>-  We have now included additional **pixel-space** comparison on the CelebA-HQ dataset. For completeness, we also provide results for the **C-$\\Pi$GFM**. The additional experimental results are shown in the following table.  Corresponding qualitative results are available but cannot be included here due to policy restrictions.\n\n>- Results are now presented in two categories: **latent-based methods** (*MPGD*, *Resample*, *PSLD*, *LatentDAPS* on three datasets, and *LatentDMPlug* on FFHQ) and **pixel-based methods** ($\\Pi$GDM, *OT-ODE*, and *C-$\\Pi$GFM* on CelebA-HQ).\n  \n\n---\n| Methods        |        | **GDB**     |        |        | **MDB**     |        |        | **SR**     |        |        | **BIP**      |        |\n|---------------|--------|-------------|--------|--------|-------------|--------|--------|-------------|--------|--------|-------------|--------|\n|               | PSNR↑  | SSIM↑       | LPIPS↓ | PSNR↑  | SSIM↑       | LPIPS↓ | PSNR↑  | SSIM↑       | LPIPS↓ | PSNR↑  | SSIM↑       | LPIPS↓ |\n| $\\Pi$GDM        | 27.93  | 0.795       | 0.187  | 27.42  | 0.754   |  0.231 | 27.16   | 0.801   | 0.198  | 25.77  | 0.774  | 0.208  |\n| OT-ODE    | 28.62  |0.804 |  0.175 | 27.27 | 0.767  | 0.224 | 27.47 | 0.812 |  0.192 | 26.81 | 0.798 | 0.184 |\n| C-$\\Pi$GFM        |  28.85  |  0.813 |  0.172   |  26.55  | 0.775   | 0.219   | 26.85  |  0.808  |  0.204 |  26.79 |  0.789 | 0.186 |\n| LFlow (**ours**)     | **29.06**     |   **0.825**         | **0.164**      | **30.14**      | **0.845**   | **0.167**      | 28.92   | **0.830**   | **0.170**  | 24.82     | **0.876**   | **0.123**   |\n\n**Table 3:** *Quantitative comparison of LFlow against other Pixel-based flow solvers across four inverse problems (BIP: Box InPainting, GDB: Gaussian DeBlurring, MDB: Motion DeBlurring, SR: Super-Resolution) on CelebA-HQ datasets*. \n\n\n---\n\n> **Fair comparison:** \n>- As noted in our paper, flow matching models offer greater **efficiency** during both training and inference by producing straighter sampling paths under modified noise schedules. This efficiency can be further improved when applied in the latent space. However, this does not necessarily mean that ODE-based sampling yields **higher sample quality** compared to SDE-based sampling. This is precisely why other flow-based methods still compare their performance against **diffusion-based counterparts**.\n\n\n\n### **4. Restructuring the Method Section**\n\n> Thank you for this helpful suggestion. We have revised the methodology section to improve clarity and better distinguish our novel contributions from prior work. We also updated and relocated the Related Work section to Section 2. We added a conceptual figure showing the flow of our method and its difference with OT-ODE in posterior covariance estimation. Regarding the specific examples you mentioned:\n\n\n> - **Posterior sampling:**  While posterior sampling using $\\underline{\\textit{unconditional} \\text{ vector fields}}$ has been explored in **pixel space** for inverse problems, it has not been *theoretically justified*. In this work, we provide a formal **proposition 3.1** establishing when and why posterior sampling is valid under our setting. Building on this foundation, we further extend the framework to handle **conditional vector fields**, enabling guided inference consistent with measurements. We further improved our methodology section (for example, for posterior covariance estimation, we refer the reviewer to our response to the third question of Reviewer xFYk).\n\n\n> - **Initialization strategy:** We have now moved this explanation to the Implementation section. While we acknowledge being \"inspired by\" prior works, our initialization strategy is notably distinct. Specifically, **across all tasks**, we encode the measurement directly using the encoder $\\mathcal{E}$ to obtain the latent code—without applying the forward operator $\\mathcal{A}$ or its pseudoinverse $\\mathcal{A}^\\dagger$, as commonly done in other latent diffusion or pixel-space flow-based methods (e.g., C-$\\Pi$GFM)."}]

Author rebuttal:

Meta-review: Three reviewers recommend acceptance (two borderline), and one is a borderline reject. The paper is well-written and theoretically rigorous. The main concerns raised by the reviewers are the number of simplifying assumptions made and the limited comparison to state-of-the-art methods. The authors' detailed rebuttal, which included additional experiments, successfully addressed many of these concerns and led one reviewer to raise their score.
Decision: Accept (poster)