Title: Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving
Venue: NeurIPS 2025
Accepted: yes
Keywords: Inverse Problems

Flow Matching

Posterior Sampling

ODE-based Inference

Training-Free Methods
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

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. The code will be publicly available at GitHub.

1 Introduction

Linear inverse problems are fundamental to a 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 x 0 ∈ R n , which is assumed to follow an unknown prior distribution q ( x 0 ) , from incomplete and noisy observations y ∈ R m , commonly modeled as:

where A ∈ R m × n represents a known linear operator and n denotes additive i.i.d. Gaussian noise. When the operator A is singular (e.g., if m<n ), the inverse problem becomes ill-posed [5], hindering the unique or stable recovery of x 0 from y . Consequently, accurate and plausible inferences demand strong priors that effectively integrate domain-specific knowledge to 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 [11-32]. Building on these advancements, diffusion-based 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-based guidance, which can lead to unstable sampling and reduced coverage of the posterior distribution [21, 30].

Recently, flow matching [41, 42] has gained prominence as a compelling alternative for generative modeling. By parameterizing transformation dynamics 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-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-probability 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 ( L atent Refinement via Flow s), 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 reduced-dimensional domains. Additionally, we introduce a well-founded, time-dependent variance for the latent identity posterior covariance, formulated using Tweedie's covariance formula and the optimal vector field under the assumption of a 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.

Our primary contributions are as follows:

Methodological: We propose a training-free framework based on latent flow matching and posterior-guided ODE sampling for solving linear inverse problems, significantly outperforming latent diffusion-based approaches in both efficiency and reconstruction quality.

Analytical: We derive a principled correction to the pretrained latent flow using the measurement likelihood gradient and introduce an analytically justified, time-dependent 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-art results without requiring substantial problem-specific 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 a parameterized surrogate posterior distribution, typically Gaussian, whose parameters are optimized using a variational objective [27, 52, 53]; (3) Asymptotically 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-based 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 to estimating the likelihood score ∇ x t log p ( y | x t ) during the reverse-time diffusion process governed by an SDE of the form:

where f ( · , · ) is the drift, g ( · ) the diffusion coefficient, w t standard Brownian motion, and s θ ( · , · ) a pretrained score network. Notable methods such as DPS [21], Π GDM[22], TMPD [23], OPC [29], and MMPS [30] define the likelihood score as ∇ x t log p ( y | x t ) = ∇ x t log ∫ p ( y | x 0 ) p ( x 0 | x t ) d x 0 . The main challenge lies in computing the expectation over all possible denoised states x 0 given x t , which requires sampling from p ( x 0 | x t ) at every reverse step-posing significant computational demands. A

common remedy is a local Gaussian approximation p ( x 0 | x t ) ≈ N ( E [ x 0 | x t ] , C ov[ x 0 | x t ] ) , reducing the problem to estimating posterior moments. These approaches differ mainly in how they specify C ov[ x 0 | x t ] . For instance, DPS sets it to zero, and Π GDM obtains a prior-agnostic , time-scaled identity matrix derived from the forward process under a Gaussian data assumption . OPC performs a post hoc constant variance optimization for each step. TMPD approximates the covariance by replacing the Jacobian in the Tweedie relation C ov[ x 0 | x t ] = σ ( t ) 2 α ( t ) ∇ ⊤ x t E [ x 0 | x t ] 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, with a diagonal rowremains computationally expensive in high dimensions, with prohibitive memory and runtime costs.

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 a supposedly Gaussian prior on the unconditional reverse sample with a Gaussian pseudo-likelihood centered at a forward-projected measurement-consistent posterior mean. Similarly, DAPS [37] enhances posterior sampling by decoupling consecutive diffusion steps through a two-step procedure: first drawing z 0 ∼ p ( z 0 | z t , y ) , then re-noising z t -∆ t ∼ N ( z 0 , σ 2 t -∆ t I ) , which provides global corrections-particularly 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.

Flow Matching in Inverse Problems has recently proven effective for achieving fast and highquality solutions across various tasks [45-50]. A prime example is OT-ODE [45], which adopts Π GDM[22] gradient correction within the flow regime and employs an ODE solver scheme based on the conditional OT path. CΠ GFM [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 prior-agnostic 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 { p t ( x t ) } t ∈ [0 , 1] that transports samples from a data distribution q ( x 0 ) , with x 0 ∈ R d , to a standard Gaussian N ( 0 , I ) at t = 1 . This evolution follows a time-varying vector field v : R d × [0 , 1] → R d , governed by the ODE d x t = v ( x t , t ) d t . In practice, v ( · , t ) is approximated by a learnable field v θ , trained via maximum likelihood, which requires expensive ODE simulations.

Flow Matching (FM) [41] avoids inefficient likelihood-based training for CNFs by directly regressing a learnable vector field toward an analytically defined target field. In particular, Conditional Flow Matching (CFM) [41] introduces a time-dependent conditional vector field v ( x t | x 0 ) that governs the evolution of samples x t conditioned on an initial point x 0 . This field induces a conditional probability path p t ( x t | x 0 ) satisfying the boundary conditions p 0 = δ ( x 0 ) and p 1 = N ( 0 , I ) . The training objective then becomes:

Using Gaussian paths, we define p t ( x t | x 0 ) = N ( α ( t ) x 0 , σ ( t ) 2 I ) with interpolation x t = α ( t ) x 0 + σ ( t ) x 1 , where x 1 ∼ N ( 0 , I ) . Differentiating x t with respect to t gives ˙ α ( t ) x 0 +˙ σ ( t ) x 1 . Substituting the inversion x 0 = [ x t -σ ( t ) x 1 ] /α ( t ) yields the true vector field:

where ˙ α ( t ) and ˙ σ ( t ) denote time derivatives. In particular, choosing α ( t ) = 1 -t and σ ( t ) = t recovers the OT path, which induces straight trajectories and improves efficiency.

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4 Method

To address the ill-posedness of linear inverse problems, we adopt a Bayesian view and target the posterior p ( x 0 | y ) ∝ p ( y | x 0 ) p ( x 0 ) , where p ( y | x 0 ) is the likelihood and p ( x 0 ) 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 ( x 0 ) implicitly via a latent prior p ( z 0 ) and the decoder pushforward p ( x 0 ) = D φ # p ( z 0 ) . Specifically, let E ϕ : R d → R k and D φ : R k → R d be a pretrained autoencoder. The latent z 0 = E ϕ ( x 0 ) follows p ( z 0 ) , modeled by a flow-matching velocity v θ ( z t , t ) defining the ODE

This flow transports latent samples from z 0 ∼ p ( z 0 ) to a noise distribution z 1 ∼ N ( 0 , I ) . At inference, we integrate backward from z 1 to ˆ z 0 , adding measurement-driven guidance to follow the conditional trajectory, then decode ˆ x 0 = D φ (ˆ z 0 ) . Figure 1 illustrates the overall inference pipeline and provides a 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 ( z 0 | y ) by simulating a reverse-time ODE characterized by a conditional velocity field. The following proposition provides the theoretical foundation for this procedure, ensuring that if the conditional density evolves according to a continuity equation, then the reverse-time flow recovers samples from the posterior. A formal proof is deferred to Appendix A.1.

Proposition 4.1 (Posterior Sampling via Reverse-Time Conditional Flows) . Let p t ( z t | y ) denote the conditional distribution of latent variables z t ∈ R k at time t ∈ [0 , 1] , with terminal condition p 1 ( z 1 ) = N ( 0 , I ) . Suppose this distribution evolves over time according to the reverse-time continuity equation:

for some conditional velocity field v t ( z t | y ) . Then, the solution to the reverse-time ODE

yields samples from the posterior p ( z 0 | y ) as t → 0 .

Conditional Vector Field Estimation Having established that posterior samples can be obtained by integrating the reverse-time ODE in Eq. (7), it remains to estimate the corresponding conditional

vector field v t ( z t | y ) . Under Gaussian latent dynamics, this field takes the form:

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 a likelihood correction.

where the additional term ∇ z t log p t ( y | z t ) , 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

Here, the likelihood term can be expressed as p ( y | x 0 ) = N ( y ; AD φ ( z 0 ) , σ 2 y I ) , which is a Gaussian distribution with a nonlinear mean induced by the decoder D φ . Due to the nonlinearity of D φ , the marginal likelihood p ( y | z t ) deviates from Gaussianity. To facilitate the computation of its gradient, we approximate D φ ( z 0 ) via a first-order Taylor expansion around ¯ z 0 := E [ z 0 | z t ] as:

where J D := J D (¯ z 0 ) is the Jacobian of D φ at ¯ z 0 . Assuming the posterior p ( z 0 | z t ) ∼ N (¯ z 0 , Σ z ) , the distribution of the image x 0 = D φ ( z 0 ) becomes approximately Gaussian as well:

Using this approximation, we estimate the gradient of the log-likelihood in latent space based on the decoded mean and the propagated covariance (see Appendix A.4):

which involves a Jacobian-vector product ( J · v ), and can be efficiently computed using automatic differentiation (Appendix B). Furthermore, the Gaussian posterior assumption enables the application of Tweedie's formula [63], which connects the pretrained vector field v θ ( z t , t ) to the posterior moments E [ z 0 | z t ] and C ov[ z 0 | z t ] . Specifically, we have (refer to Appendix A.3):

Latent Posterior Covariance Computing the posterior covariance C ov[ z 0 | z t ] in low-dimensional latent spaces is more tractable and can be efficiently estimated via automatic differentiation. However, inverting the matrix ( σ 2 y I + A J D Σ z J ⊤ D A ⊤ ) in Eq. (13) remains a significant computational bottleneck. To avoid explicit inversion, one may solve the corresponding linear system using the Conjugate Gradient (CG) method. Yet, CG's convergence crucially depends on the symmetry and positive definiteness of the system matrix-conditions that may be violated in practice due to imperfections in the pretrained velocity field v θ , potentially leading to instability or divergence. To address this, we further analyze the structure of the posterior covariance and the Jacobian ∇ z t v θ ( z t , t ) , under certain regularity assumptions on the latent prior p ( z 0 ) .

Assumption 4.2 (Strong Log-Concavity of the Latent Prior [64]) . Let p ( z 0 ) = exp( -Φ( z 0 )) be the latent prior over R d , where Φ is a twice continuously differentiable potential function. We assume that p ( z 0 ) is γ -strongly log-concave for some γ > 0 ; that is, ∇ 2 Φ( z 0 ) ⪰ γ I d , for all z 0 ∈ R d .

(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 bound on the Jacobian of the vector field.

Proposition 4.3 ( Bound on the Jacobian of the Vector Field ) . Let v θ ( z t , t ) denote the velocity field of the interpolant z t = α ( t ) z 0 + σ ( t ) z 1 between a standard Gaussian prior and a target distribution p ( z 0 ) , defined via coefficients α ( t ) , σ ( t ) ∈ R . Under Assumption 4.2, the Jacobian of the vector field satisfies the following bound:

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 a tractable special case where the latent prior is Gaussian, z 0 ∼ N (0 , σ 2 latr I ) , a choice that is both theoretically justified and widely used in practice-for instance, in variational autoencoders (V AEs), where the latent prior is regularized toward a standard Gaussian via KL divergence [65]. 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.6.

Proposition 4.4 ( Optimal Vector Field ) . Let z 0 ∼ N ( 0 , σ 2 latr I d z ) and z 1 ∼ N ( 0 , I d z ) be independent random variables. Define z t = (1 -t ) z 0 + t z 1 for t ∈ [0 , 1] . The optimal vector field v ⋆ ( z t , t ) that minimizes the expected squared error arg min v E [ ∥ v ( z t , t ) -( z 0 -z 1 ) ∥ 2 ] is given by

Remark 4.5 ( Tightness of Bound ) . When z 0 ∼ N (0 , σ 2 latr I ) , the optimal vector field v ⋆ ( z t , t ) achieves the Jacobian lower bound in Proposition 4.3. This follows from the potential function Φ( z 0 ) = 1 2 σ 2 latr ∥ z 0 ∥ 2 , yielding γ = σ -2 latr , and interpolation coefficients α ( t ) = 1 -t , σ ( t ) = t . Substituting into the bound confirms it matches the exact Jacobian of v ⋆ , making the bound tight.

Plugging the Jacobian of the optimal vector field into Eq. (15) yields:

where we assumed σ latr = 1 . As a result, the propagated covariance can also be simplified as

Computing the full Jacobian J D ∈ R d x × d z is often infeasible in practice. To simplify, we assume that the decoder acts approximately as a local isometry near ¯ z 0 [66], such that J D J ⊤ D ≈ · P , where P is the orthogonal projector onto the image of J D . For computational convenience, we approximate this behavior as isotropic in the full space, resulting in:

Comparison with Π GDM and OT-ODE The posterior covariance in Π GDM is derived solely from the forward process under the strong assumption of Gaussian data space , resulting in a variance for the identity covariance as r 2 ( t ) = σ ( t ) 2 α ( t ) 2 + σ ( t ) 2 I . Please see Appendix A.7 for details.

Additionally, Π GDMleverages the inverse of this posterior covariance as Fisher information for natural gradient [67] updates on samples during the guidance step, thereby enhancing performance. OTODE can be regarded as a flow-based extension of Π GDM, where it adopts the same covariance expression but replaces the diffusion forward process with a flow-specific noise scheduler, specifically setting α ( t ) = 1 -t and σ ( t ) = t . Therefore, both methods rely on 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 sampling, ultimately improving reconstruction quality. Figure 2 visually illustrates how our time-dependent variance differs in magnitude and effect from the simpler identity-based covariance used in OT-ODE.

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Initiating the Flow Sampling Process Inspired by prior works in pixel spaces [68, 45], we propose initializing the reverse ODE-across all tasks-from a partially corrupted version of the encoding of y in the latent space. Specifically, rather than starting from pure noise z 1 ∼ N (0 , I ) at t = 1 , we initialize at t s < 1 :

This initialization ensures that z t s is closer to the posterior mode z 0 | y , making the subsequent backward integration more stable and likely to remain on a plausible manifold.

We summarize the complete sampling algorithm via latent ODE flows in Algorithm 1 (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 a resolution of 256 × 256 × 3 pixels. We use 200 randomly selected validation samples per dataset. All images are normalized to the range [ -1 , 1] . We present our findings on 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 a Gaussian blurring kernel of size 61 × 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 × 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 a margin of [16 , 16] pixels, following the methodology of [21]. All measurements in these experiments are corrupted by Gaussian noise with a standard deviation of σ y = 0 . 01 , ensuring that the reconstruction methods are evaluated under realistic noisy conditions.

Baselines and Metrics Since the primary goal of this study is to improve solving inverse problems in the latent space, we focus on comparing our proposed method against state-of-the-art 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 Π GDM [22], OT-ODE [45], and CΠ GFM [48], when reported, for a more comprehensive comparison. Following the evaluation protocols of prior works, we report quantitative metrics including Learned Perceptual Image Patch Similarity (LPIPS) [72], Peak Signalto-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], we 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 a lower-dimensional latent space of 32 × 32 . After evaluating various numerical ODE integration methods, we selected adaptive_heun as the default solver, utilizing its reliable implementation from the open-source torchdiffeq 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 hyper-parameters are provided in Appendix C.

5.1 Results

For clarity, the quantitative results for the four tasks on both FFHQ and ImageNet are presented in Tables 1 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-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, LFlow similarly excels in perceptual quality, restoring structures like the dog's snout, fur, and facial wrinkles more faithfully. The reconstructions

Measurements

References

MPGD

PSLD

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LFlow(ours)

remain sharp and natural, free from the spurious high-frequency artifacts or texture distortions that affect other approaches.

Motion Deblurring LFlow reconstructs motion-blurred scenes with strong structural consistency and minimal artifacts, effectively preserving sharp transitions along motion boundaries. Fine details-such as hair contours, facial edges, and accessories-are recovered with smooth gradients and natural appearances. Without introducing ringing or over-enhancement, LFlow balances fidelity and realism across both FFHQ and ImageNet, achieving state-of-the-art performance on all metrics. 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-qualities that are visually evident in the restored hair and facial structures.

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 a perceptual gain of 0.021 LPIPS over the next-best method, while also maintaining competitive PSNR. On ImageNet, it establishes a clear lead across all metrics, with an 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, or PSLD and Resample, which occasionally yield hazy regions, our approach preserves the natural appearance of 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, it 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, it continues to lead in LPIPS and SSIM, recovering fine textures and edges with minimal boundary artifacts.

Measurements

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LFlow(ours)

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-resulting in reconstructions that appear complete and visually seamless.

Perceptual-Fidelity Trade-off While LFlow occasionally reports slightly lower PSNR than pixelfidelity-oriented baselines such as 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

Posterior Covariance To examine how the time-dependent posterior covariance influences overall performance, we conducted a series of ablation studies. As shown in Table 3, adapting the posterior covariance (labeled Cov_LFlow ) provides systematic gains over the baseline ' Cov_ Π GDM '. Across both FFHQ and ImageNet, Cov_LFlow yields higher PSNR and lower LPIPS for deblurring and × 4 SR, indicating that LFlow helps reduce perceptual artifacts while preserving fine details. Figure 6 presents qualitative comparisons between Cov_LFlow and Cov_ Π GDM on FFHQ. Across both motion 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.

Starting Time t s We also investigated the impact of the starting time t s on the results. Figure 5 illustrates the impact of varying the start time t s for the flow process (SR task). We see that overly large t s values tend to slightly degrade perceptual quality, whereas overly small t s values can cause excessive smoothing or artifacts. The plot indicates that an intermediate choice of t s (around 0.7-0.8) strikes a favorable balance, leading to consistently lower LPIPS on both FFHQ and ImageNet.

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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 Cov_ Π GDM , which employs a less accurate covariance, converges more slowly-demonstrating 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 a justified latent Gaussian representation assumption , our approach introduces a theoretically sound, time-dependent latent posterior covariance that enhances gradient-based inference. Experimental results across such tasks as deblurring, super-resolution, and inpainting demonstrate that LFlow outperforms current latent diffusion models in reconstruction quality.

Limitation. One limitation of LFlow lies in its runtime: the current implementation can require approximately 3:30 -10 minutes to 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 is 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 to real-world scenarios and downstream tasks.

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A Proofs

Lemma A.1 ( Tweedie's Mean Formula ) . Suppose the joint distribution of z 0 and z t factors as

Proof. Starting from the definition of the score,

Rearranging completes the proof.

Lemma A.2 ( Tweedie's Covariance Formula ) . For any distribution p ( z 0 ) and

the posterior p t ( z 0 | z t ) is also Gaussian with mean E [ z 0 | z t ] and covariance C ov[ z 0 | z t ] . These are connected to the score function ∇ z t log p t ( z t ) via

Proof. We start with the Hessian of log p t ( z t ) :

Solving for C ov[ z 0 | z t ] yields the stated formula.

with

Then

Lemma A.3 (Connection between Posterior Mean and Vector Field) . Let z t = α ( t ) z 0 + σ ( t ) z 1 be a one-sided interpolant, where z 0 ∼ p data and z 1 ∼ N ( 0 , I ) . Then the vector field v ( z t , t ) satisfies

Proof. Starting from the definition,

where ˙ σ ( t ) , ˙ α ( t ) represent the first-order time derivatives of σ ( t ) and α ( t ) , respectively. Noting that

we obtain

Lemma A.4 (Cramér-Rao inequality) . Let µ (d x ) = exp( -Φ( x ))d x be a probability measure on R d , where Φ : R d → R is twice continuously differentiable 1 . Then for any f ∈ C 1 ( R d ) ,

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-Rao) . Let z t = α ( t ) z 0 + σ ( t ) z 1 , where z 0 ∼ p ( z 0 ) ∝ exp( -Φ( z 0 )) with Φ ∈ C 2 ( R d ) , and z 1 ∼ N (0 , I d ) are independent. Suppose:

(i) The likelihood is Gaussian: p ( z t | z 0 ) = N ( α ( t ) z 0 , σ ( t ) 2 I d ) .

(ii) Φ is γ -strongly convex, i.e., ∇ 2 Φ( z 0 ) ⪰ γ I d ,

Then the posterior satisfies:

Proof. Consider assumption (i). By applying Bayes' rule, the negative log-posterior is:

Taking second derivatives yields:

Using assumption (ii), ∇ 2 Φ( z 0 ) ⪰ γ I d , we obtain:

Finally, applying Lemma A.4, the Cramér-Rao bound gives:

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 t ∈ R d is generated by a known forward process of the form:

where z 0 is the initial (clean) latent sample and α ( t ) , σ ( t ) are scalar functions of time.

This process induces a time-indexed family of conditional distributions ¯ p t ( z | y ) , where the expectation is taken over the joint distribution of z 0 | y and the noise ε .

We define the characteristic function of the marginal at time t as:

where ˜ E denotes expectation over the conditional joint distribution of z 0 | y and ε .

We now differentiate this characteristic function with respect to time:

Since z t is a deterministic function of z 0 and ε , we may condition on z t and write:

On the other hand, by the definition of the characteristic function:

Equating the two expressions for ∂ t ˆ p t ( k ) , we obtain:

We now use the identity ∇ z e i k ⊤ z = i k · e i k ⊤ z and integration by parts to move the gradient from e i k ⊤ z to the density term:

Finally, since both expressions match for all k , their inverse Fourier transforms must be equal. Hence, we conclude:

This is the continuity (Fokker-Planck) equation with drift field ˜ v t . Therefore, integrating the ODE:

backward in time from z 1 ∼ N ( 0 , I ) , yields samples z 0 ∼ ¯ p 0 ( z | y ) , as desired.

A.2 Conditional Vector Field

Proof. From Lemma A.3, the unconditional vector field is

Substituting this in,

Conditioning on y , the score becomes

Hence, the conditional vector field is

Finally, choosing the linear schedule α ( t ) = 1 -t and σ ( t ) = t simplifies this to

Approximating the unconditional velocity v ( z t , t ) by a parametric estimator v θ ( z t , t ) gives

A.3 Derivation of Posterior Mean (Eq. (14)) and Posterior Covariance (Eq. (15))

Proof. From Lemma A.3, we know

With α ( t ) = 1 -t and σ ( t ) = t , one obtains

Meanwhile, from Lemmas A.1 and A.2, one can show

A.4 Proof of Noise Conditional Score Approximation in Latent Space

Proof. We assume the likelihood and posterior over x 0 are Gaussian:

with:

Thus,

Define:

Then:

The log-likelihood becomes:

We compute its gradient w.r.t. x t ∈ R n in two parts:

(1) Log-determinant term:

So,

(2) Quadratic term:

Putting both together:

Assuming C ov[ x 0 | x t ] is slowly varying w.r.t. x t , we neglect its gradient, yielding the approximation:

Now, we aim to derive a principled extension of this formula to the latent space . Let z 0 and z t be latent variables, and D φ be a (generally nonlinear) decoder such that x 0 = D φ ( z 0 ) . Then the likelihood can be written as:

which is a Gaussian with nonlinear mean . We further assume:

Since D φ is nonlinear, p ( y | z t ) is not Gaussian. However, we approximate D φ ( z 0 ) via a first-order Taylor expansion around ¯ z 0 :

where J D (¯ z 0 ) is the Jacobian of D φ at ¯ z 0 .

Using this approximation, the distribution over x 0 becomes approximately Gaussian:

We now apply the same posterior score approximation used in pixel space, but using the decoded mean and propagated covariance:

This expression is now a well-defined and justified approximation to the posterior score in latent space, based on first-order decoder linearization and Gaussian propagation.

A.5 Proof of Proposition 4.3

Proof. We aim to compute the Jacobian ∇ z t v ( z t , t ) . From Lemma A.3, we have:

Taking the gradient with respect to z t , we obtain:

From Lemma A.1, the posterior mean is given by:

and therefore its gradient is:

Substituting this into the expression for the Jacobian, we get:

Applying Lemma A.2, the Hessian of the log-density can be expressed as:

Therefore, the Jacobian becomes:

By Lemma A.4, if p ( z 0 ) is γ -semi-log-convex, then the Cramér-Rao inequality yields:

Substituting this bound into the Jacobian expression gives:

which implies:

This expression simplifies to:

Finally, observe that the right-hand side can be written as:

Now, assume the posterior covariance is positive semidefinite:

Define the scalar coefficient

Then the Jacobian simplifies to

Since C ov[ z 0 | z t ] ⪰ 0 , the sign of c ( t ) determines the direction of the inequality:

If c ( t ) ≥ 0 , then c ( t ) · C ov[ z 0 | z t ] ⪰ 0 , and

If c ( t ) < 0 , then c ( t ) · C ov[ z 0 | z t ] ⪯ 0 , and

In particular, for the common case where α ( t ) = 1 -t and σ ( t ) = t , we have c ( t ) < 0 for all t ∈ (0 , 1] , 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 z 0 ∼ N ( 0 , σ 2 latr I ) and z 1 ∼ N ( 0 , I ) be independent. Define z t = (1 -t ) z 0 + t z 1 for t ∈ [0 , 1] . The optimal vector field v ⋆ ( z t , t ) that minimizes

is

Proof. Since the loss function is quadratic and v ⋆ ( z t , t ) depends only on z t and t , the optimal vector field is the conditional expectation:

Given that z 0 and z 1 are independent Gaussian random variables, and z t is a linear combination of z 0 and z 1 , the joint distribution of z 0 , z 1 , and z t is multivariate Gaussian. We will compute E [ z 0 | z t ] and E [ z 1 | z t ] using the properties of multivariate normal distributions.

First, we identify the covariance matrices:

since z 0 and z 1 are independent.

Next, compute the covariance between z 0 and z t :

Similarly, the covariance between z 1 and z t :

The variance of z t is:

The joint covariance matrix of z 0 , z 1 , and z t is:

Using the properties of conditional expectations for multivariate normals, we compute the conditional expectations:

Therefore, the optimal vector field is:

A.7 Proof of Covariance in Π GDM

( Π GDM [22]) . Let x 0 ∼ N ( 0 , σ 2 data I ) , and consider the forward process

Then the conditional covariance C ov[ x 0 | x t ] is

Proof. Again x 0 and x 1 are independent Gaussians, and x t is a linear combination. Thus x 0 , x t are jointly Gaussian with and

Using the standard formula for conditional covariances in a Gaussian,

Since C ov[ x 0 ] = σ 2 data I , one obtains

which simplifies to

B Closed-form Solutions for computing vector V 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. C ov[ z 0 | z t ] = r ( t ) 2 I . We begin by introducing essential notation.

Notations.

Let m ∈ { 0 , 1 } d × 1 represent the sampling positions in an image or signal.

The downsampling operator associated with m is D m ∈ { 0 , 1 } ∥ m ∥ 0 × d . It selects only those rows (i.e. entries) of a vector or matrix corresponding to the non-zero entries of m . For example, s -fold downsampling with evenly spaced ones is denoted D ↓ s .

D ⇓ s represents a distinct block downsampler, which averages s blocks (each of size d/s ) from a vector.

F is the (unitary) Fourier transform matrix of dimension d × d , and F ↓ s is the analogous transform matrix for signals of dimension d/s .

ˆ v denotes the Fourier transform of the vector v , and ¯ v denotes its complex conjugate.

The notation ⊙ refers to element-wise (Hadamard) multiplication. Divisions such as ' / ' 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,

Proof. Please see [79, 29] for details.

B.1 Image Inpainting

The observation model for image inpainting can be written as

where n is noise. A convenient zero-filling version of y can be defined as

The closed-form solution for v in image inpainting is then

Proof. Starting with the more general form,

Since D m D ⊤ m = I on the support of y and ( σ 2 y I + r ( t ) 2 I ) -1 = 1 / ( σ 2 y + r ( t ) 2 ) , it simplifies to

Recalling that ˜ y = D ⊤ m y = m ⊙ ( x 0 + ˜ n ) , we arrive at the stated closed-form solution.

B.2 Image Deblurring

For image deblurring, the observation model is

where k is the blurring kernel and ∗ is the circular convolution operator. Using the Fourier transform, this can be written as

where ˆ k is the DFT of k . Under isotropic-covariance assumption, the closed-form solution for v is

Proof. Because A is a real linear operator of convolution type, we have A T = A H . Thus,

Substituting A = F -1 diag( ˆ k ) F and simplifying in the Fourier domain (using the diagonal structure in frequency space), one obtains

B.3 Super-Resolution

Following [79], the super-resolution observation model is approximately

which, in 'canonical form,' can be written as

Hence, A = D ↓ s F -1 diag( ˆ k ) F . The closed-form solution under the isotropic assumption is

where ⊙ s denotes block-wise Hadamard multiplication.

Proof. Since A T = A H and

we get A ⊤ = F ⊤ diag ( ¯ ˆ k ) ( F -1 ) ⊤ D ⊤ ↓ s . Applying Lemma B.1, namely D ↓ s F -1 = F -1 ↓ s D ⇓ s , and its conjugate-transpose version, reduces the inverse ( σ 2 y I + r ( t ) 2 AA ⊤ ) -1 to diagonal form in the 'downsampled' Fourier domain. One obtains

C Implementation details

Algorithm 1 LFlow Sampling: Posterior-Guided Latent ODE Inference for Linear Inverse Problems 1: Input: measurements y , encoder E ϕ , decoder D φ , forward operator A , pre-trained vector field v θ ( · , t ) for t ∈ [ t s , 0] , t s = 0 . 8 , and K = 2 . 2: Initialize. z t s ← (1 -t s ) E ϕ ( y ) + t s z 1 , z 1 ∼ N (0 , I ) 3: for t = t s down to 0 do 4: ¯ z 0 ← z t -t v θ ( z t , t ) ▷ Posterior mean prediction Eq. (14) 5: r 2 ( t ) ← t 2 1 -t (1 -t ∇ z t v ⋆ ( z t , t )) ▷ Posterior covariance estimation Eq. (18) 6: ∇ z t log p ( y | z t ) ← ( ∇ z t D φ (¯ z 0 )) ⊤ A ⊤ ( y -AD φ (¯ z 0 )) ( σ 2 y I + r 2 ( t ) AA ⊤ ) ▷ Eq. (13) 7: for k = 1 to K do 8: v θ ( z t , y , t ) ← v θ ( z t , t ) -t 1 -t ∇ z t log p ( y | z t ) ▷ K -step update of vector field Eq. (9) 9: end for 10: z t -∆ t ← ODESolverStep( z t , v θ ( z t , y , t )) 11: end for 12: Return: x 0 ←D φ ( z 0 )

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 K = 2 and t s = 0 . 8 for all tasks. These values were selected via ablation on validation performance and were found to balance guidance strength and reconstruction fidelity across tasks.

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 as in [29] across both our method and the relevant baselines.

For inpainting tasks, we 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 as x 0 = A T A x 0 +( I -A T A ) D φ ( z 0 ) . 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 to 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., x 0 = D φ ( z 0 ) .

C.2 Comparison methods

PSLD [33] applies an orthogonal projection onto the subspace of A between decoding and encoding to enforce fidelity:

We use a fixed step size of ρ and select γ as recommended in [36]. For our experiments, we rely on the official PSLD implementation 2 with its default configurations. Specifically, we conduct ImageNet experiments using Stable Diffusion v1.5, which is generally considered more robust compared to the LDM-VQ4 models.

PSLD aims to ensure that latent variables remain close to 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].

Resample [36] estimates first a clean latent prediction z est 0 ( z t +1 ) from the previous sample z t +1 using Tweedie's formula, as described in Eq. (22). This prediction is then used to update the latent state via DDIM:

The updated sample z ′ t is then projected back to a measurement-consistent latent variable z proj t via:

where z cond 0 is a latent vector that satisfies the measurement constraint, obtained by solving the following optimization problem:

Here, σ 2 t is a tunable hyperparameter controlling the trade-off between the prior and the data fidelity, and ¯ α t is a predefined DDIM noise schedule parameter. For our experiments, we adopt the publicly available implementation provided by the authors 3 , using the pre-trained LDM-VQ4 models on FFHQ and ImageNet [59], along with their default hyperparameters and a 500-step DDIM sampler.

Resample refines latent diffusion sampling by balancing the reverse-time prior from the unconditional model with a measurement-informed likelihood centered on a consistent latent-ensuring 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:

Note that in MPGD, we leveraged Stable Diffusion v1.5 as for PSLD. For more information, please refer to the GitHub repository 4 .

DAPS [37] 5 refines latent estimates via Langevin dynamics guided by a latent prior and a measurement likelihood. The initial estimate z (0) 0 is obtained by solving a probability flow ODE from z t using the latent score model. At each inner iteration j = 0 , . . . , N -1 , the latent estimate z ( j ) 0 ∈ R d is updated as follows:

Here, η t 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 t from the marginal posterior p ( z t | y ) . 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 Triple-Consistent Sampling) [60] enforces three complementary consistency conditions-measurement, forward diffusion, and step-wise backward diffusion-allowing diffusion trajectories to remain measurement-consistent with fewer reverse steps. By optimizing the input 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.

At each step t , SITCOM enforces three consistencies:

(S1) Measurement-consistent optimization:

(S2) Denoising:

(S3) Sampling (Forward step):

Together, these steps enforce C1: measurement, C2: backward trajectory, and C3: forward diffusion 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] 7 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 :

Most existing DM-based methods for inverse problems interleave reverse diffusion with measurement projections in a step-wise manner, but this often breaks both manifold feasibility (staying on the data manifold M ) and measurement feasibility (satisfying { x | y = A ( x ) } ). In contrast, DMPlug is not step-wise: it preserves manifold feasibility by keeping the pretrained reverse process intact while promoting y ≈ A ( x ) via global optimization.

Π GDM[22] considers the following gradient update scheme

where η controls the step size, σ y represents the noise level of the measurement, and r t is the timedependent 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 Π GDM [22] to ODE sampling via an optimal transport path, resulting in a variance for the identity covariance as r 2 ( t ) = σ ( t ) 2 α ( t ) 2 + σ ( t ) 2 I . Moreover, the conditional expectation E [ x 0 | x t ] is computed from the velocity field v θ ( x t , t ) , according to the relation in A.3. For a fair comparison, we used the same solver as LFlow, i.e., the adaptive Heun .

D Additional Experiments and Ablations

D.1 CelebA-HQ (256 × 256 × 3).

Measurements

References

MPGD

PSLD

Resample

DAPS

SITCOM

LFlow(ours)

CelebA-HQ LFlow attains the lowest LPIPS across all four tasks and the highest SSIM in three of four tasks. It reports LPIPS values of 0.164 (Gaussian deblurring), 0.167 (motion deblurring), 0.170 (super-resolution), and 0.123 (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 or oversharpening seen in MPGDand PSLD. 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-whereas 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.

D.2 CelebA-HQ (512 × 512 × 3).

We further tested and evaluated our method on the high-resolution CelebA-HQ dataset (512 × 512 × 3), demonstrating its robust capabilities in handling complex image processing tasks. In Figure 8, we showcase the effectiveness and versatility of our approach in enhancing image quality across various tasks.

Measurements

References

LFlow(ours)

D.3 FID Score Results

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, we focus on image reconstruction tasks from noisy measurements. In such settings, perceptual metrics like FID-although effective for evaluating generative models that prioritize visual realism-primarily measure distribution-level similarity and may overlook structural details, especially when fine-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 are crucial in our experiments. That said, we report FID scores across all three datasets considered above for four different tasks. As shown in Table 6, our algorithm achieves a 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.

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 K ≥ 3, the results showed negligible improvements while incurring higher computational costs. For this reason, we adopt K=2 as it provides consistent gains while maintaining reasonable inference time.

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 as SR3 [81] and InvFussion [82]:

LFlow is a zero-shot method that generalizes across diverse tasks without retraining, whereas supervised methods require training a separate model for each specific task.

LFlow is robust to varying degradation types, while supervised methods often exhibit poor generalization when faced with distribution shifts.

LFlow achieves significantly better performance on certain datasets and resolutions- for example, FFHQ at 256 × 256 .

These advantages are clearly demonstrated in the results presented in Table 8.

D.6 Pretrained Models and Algorithm Conversion

Conversion of Pretrained Models. Conversion from discrete-time diffusion model to continuoustime 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 { τ } 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 a linear noise schedule. While such conversion is feasible for pretrained models with similar linear schedules, it becomes non-trivial for discrete-time latent diffusion models (e.g., LDMs, Stable Diffusion). Their custom nonlinear schedules break the clean SNR alignment, often requiring the solution of nonlinear or even cubic equations to 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 a 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-correction 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.

Discussion. The results in Table 9 clearly show that algorithmic conversion is non-trivial. Although we successfully reformulated PSLD and MPGD into their continuous-time 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 a 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-based flow formulations, as achieved in our proposed LFlow framework.

D.7 Additional Visual Results (Best Viewed When Zoomed in)

Measurements

References

MPGD

PSLD

Resample

DAPS

Measurements

References

MPGD

PSLD

Resample

DAPS

Measurements

References

MPGD

PSLD

Resample

DAPS

Measurements

References

MPGD

PSLD

Resample

DAPS

LFlow(ours)

Measurements

References

MPGD

PSLD

Resample

DAPS

Measurements

References

MPGD

PSLD

Resample

DAPS

Measurements

References

MPGD

PSLD

Resample

DAPS

Measurements

References

MPGD

PSLD

Resample

DAPS

LFlow(ours)

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\npros:\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.\n\nQUESTIONS:\nno\n\nLIMITATIONS:\nno", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: \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.\n\nQUESTIONS:\nI 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?\n\nLIMITATIONS:\nYes. The authors address the limitation of this paper, e.g., relatively slow generation.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n__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.\n\nQUESTIONS:\n__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.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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.\n\nQUESTIONS:\n1.\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?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-09"}