Title: Precise Diffusion Inversion: Towards Novel Samples and Few-Step Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Diffusion model; Inversion; Diffusion based real image editing
Abstract: The diffusion inversion problem seeks to recover the latent generative trajectory of a diffusion model given a real image. Faithful inversion is critical for ensuring consistency in diffusion-based image editing. Prior works formulate this task as a fixed-point problem and solve it using numerical methods. However, achieving both accuracy and efficiency remains challenging, especially for few-step models and novel samples. In this paper, we propose ***PreciseInv***, a general-purpose test-time optimization framework that enables fast and faithful inversion in as few as two inference steps. Unlike root-finding methods, we reformulate inversion as a learning problem and introduce a dynamic programming-inspired strategy to recursively estimate a parameterized sequence of noise embeddings. This design leverages the smoothness of the diffusion latent space for accurate gradient-based optimization and ensures memory efficiency via recursive subproblem construction. We further provide a theoretical analysis of ***PreciseInv***'s convergence and derive a provable upper bound on its reconstruction error. Extensive experiments on COCO 2017, DarkFace, and a stylized cartoon dataset show that ***PreciseInv*** achieves state-of-the-art performance in both reconstruction quality and inference speed. Improvements are especially notable for few-step models and under distribution shifts. Moreover, precise inversion yields substantial gains in editing consistency for text-driven image manipulation tasks. Code is available at: https://github.com/panda7777777/PreciseInv

Precise Diffusion Inversion: Towards Novel Samples and Few-Step Models

Abstract

The diffusion inversion problem seeks to recover the latent generative trajectory of a diffusion model given a real image. Faithful inversion is critical for ensuring consistency in diffusion-based image editing. Prior works formulate this task as a fixed-point problem and solve it using numerical methods. However, achieving both accuracy and efficiency remains challenging, especially for few-step models and novel samples. In this paper, we propose PreciseInv , a general-purpose testtime optimization framework that enables fast and faithful inversion in as few as two inference steps. Unlike root-finding methods, we reformulate inversion as a learning problem and introduce a dynamic programming-inspired strategy to recursively estimate a parameterized sequence of noise embeddings. This design leverages the smoothness of the diffusion latent space for accurate gradient-based optimization and ensures memory efficiency via recursive subproblem construction. We further provide a theoretical analysis of PreciseInv 's convergence and derive a provable upper bound on its reconstruction error. Extensive experiments on COCO 2017, DarkFace, and a stylized cartoon dataset show that PreciseInv achieves state-of-the-art performance in both reconstruction quality and inference speed. Improvements are especially notable for few-step models and under distribution shifts. Moreover, precise inversion yields substantial gains in editing consistency for text-driven image manipulation tasks. Code is available at https://github. com/panda7777777/PreciseInv

1 Introduction

Large-scale pre-trained diffusion models [10, 17, 32, 33] have demonstrated strong generative capabilities in producing high-quality and diverse images. Built upon these models, many recent methods enable a wide range of image editing operations by guiding the generative process with user inputs, such as text prompts [4, 9, 16, 21, 22, 29], reference images [1, 6, 23, 25], and spatial masks [8, 28]. To ensure consistency and controllability in such edits, faithful inversion of real images into the diffusion process is essential [13].

Since the sampling process of diffusion models is inherently irreversible, mapping a real image back into the model domain remains a fundamental challenge. Existing methods on diffusion inversion can be divided into three major categories: forward optimization, backward optimization, and invertible samplers. Invertible samplers, such as EDICT [38], BDIA [43], and BELM [39], introduce explicit constraints into the sampling procedure to enable bidirectional generation. Backward optimization methods [21, 22, 29] fix the forward noising trajectory of a real image and optimize the reverse diffusion process to match this trajectory. Both types of methods improve reconstruction quality by modifying the denoising path, but often weaken the generative ability of the diffusion model. To address this issue, forward optimization methods estimate a noising trajectory that can be faithfully

InputImage

ReNoise

EasyInv

DDIMInversion

BELM

Ours

reversed by the diffusion model. For example, several methods [30, 34, 45] convert the problem into a fixed-point equation and solve it via Anderson acceleration [2], while ReNoise [13] refines an iterative renoising mechanism. However, these methods suffer from slow convergence and limited performance, especially under few-step inference or on novel samples, as illustrated in Fig. 1.

In this paper, we propose a general framework named PreciseInv that efficiently solves the diffusion inversion problem by progressively learning a parameterized noising trajectory. Specifically, we reformulate diffusion inversion as a learning problem and decompose it into T overlapping subproblems by leveraging the Markov property of the diffusion process. We then introduce a dynamic programminginspired strategy [5] to recursively solve these subproblems. In each subproblem, PreciseInv learns a local noise embedding that approximates the optimal intermediate state along the inversion trajectory. Benefiting from the smooth transitions in the latent space [44], this gradient-based optimization converges more accurately than numerical solvers. Moreover, the recursive formulation is highly memory-efficient: inversion on the SD v1.4 base model requires only 3.43 GB of GPU memory, regardless of the number of inference steps. Extensive experiments show that PreciseInv achieves state-of-the-art reconstruction performance in both quality and efficiency (Section 4.1). The improvements are especially pronounced under few-step inference (e.g., T =2 , 4 ) and on novel samples. Furthermore, precise inversion yields more semantically continuous latent representations (Fig. 5), which substantially improve consistency and controllability in text-driven editing tasks (Section 4.2).

Our main contributions are summarized as follows: (i) We propose PreciseInv , a general-purpose testtime optimization framework for diffusion inversion that enables fast and faithful image reconstruction under few-step inference and distribution shift. (ii) We reformalize diffusion inversion as a sequence of non-overlapping learning subproblems and introduce a dynamic programming strategy to solve them efficiently. (iii) We demonstrate that precise inversion yields smoother latent trajectories, which in turn improve consistency and controllability in text-driven image editing.

2 Background

2.1 Diffusion Models

Given samples from the data distribution q ( x 0 ) , diffusion models [17] define a generative process by learning to reverse a fixed Markovian forward process that progressively corrupts data by adding Gaussian noise. The forward process is formulated as:

where x 0 ∼ q ( x 0 ) , { x t } T t =1 are latent variables, and { α t } T t =1 ∈ (0 , 1] T is a predefined noise schedule. The marginal distribution at an arbitrary timestep t is given by:

where ¯ α t := ∏ t s =1 α s . The generative process is modeled as a parameterized Markov chain:

where p ( x T ) = N ( x T ; 0 , I ) and θ denotes the model parameters. In practice, a denoising neural network ϵ θ is trained to predict µ θ ( x t , t ) , and Σ θ ( x t , t ) is set to a constant σ t according to α t . The training objective minimizes a simplified variational bound:

where

Starting from x T ∼ N ( x T ; 0 , I ) , the DDPM sampling procedure [17] iteratively applies the reverse step to generate x 0 as follows:

where z ∼ N ( z ; 0 , I ) . To reduce the number of inference steps, the DDIM sampler [36] converts Eq. (6) into a non-Markovian form:

To further enhance the inference efficiency, recent efforts have focused on training few-step diffusion models, such as Consistency Models (CM) [37], Latent Consistency Models (LCM) [27], Adversarial Diffusion Distillation (ADD) models [35], Shortcut models [12] and MeanFlow [14].

2.2 Image Inversion in Diffusion Models

Inverting real images into the latent space of a pretrained diffusion model is critical for diffusionbased image editing methods. Given a real image x 0 , our goal is to estimate a corresponding noisy latent x T , such that x 0 can be reconstructed by applying the reversed denoising trajectory to x T . Unfortunately, the neural network ϵ θ do not inherently predict noise direction ϵ from x t -1 to x t . DDIM Inversion [8] was among the earliest methods attempted to address this issue. Rewriting the Eq. (7), we obtain:

DDIM Inversion assumes ϵ θ ( x t -1 , t -1) ≈ ϵ θ ( x t , t ) and iteratively applies Eq. (8) from t = 1 to t = T . However, the denoising trajectory is nonlinear, resulting in discrepancies between predicted ˆ x t and actual x t .

Subsequently, there are three main lines of work on diffusion inversion: modifying the reversed denoising trajectory, designing invertible samplers or optimizing the initial noise latent x T . Acommon practice of the former is to optimize the denoising process to better align with the trajectory obtained by DDIM inversion. For example, Null-Text Inversion [29] optimizes unconditional text embeddings to implicitly adjust the denoising trajectory, while Eta Inversion [22] uses a hyperparameter η to control the trade-off between the stochastic denoising trajectory and the vanilla DDIM Inversion trajectory. Moreover, a DDPM inversion method [21] was proposed, which rectifies the DDPM denoising process using a forward trajectory constructed by independently applying Eq.(5) at each timestep t . More recently, several invertible samplers [38, 39, 43] have been proposed for bidirectional generation. While, in general, manually modifying the denoising trajectory enhancing reconstruction quality, it inevitably disrupts the original generative process of the diffusion model.

To preserve the original generative process, another line of methods has been proposed to estimate a more accurate value of ϵ θ ( x t , t ) in Eq. (8). For example, ReNoise [13] ensembles multiple DDIM Inversion results at each iteration. In addition, several methods reformulates the diffusion inversion

problem as a fixed-point problem and solves the problem by improving Anderson acceleration [2] methods. Let the estimated value of ϵ θ ( x t , t ) be denoted as ϵ ∗ t , then we have:

Substituting Eq. (9) into Eq. (8), we obtain:

By replacing x t in Eq. (9) with Eq. (10), we deduce that:

To effectively solve Fixed-Point Eq.(11), AIDI[30] uniformly mixes the outputs from two adjacent iterations in vanilla Anderson acceleration, while EasyInv [45] increases the influence of x 0 in estimating x ∗ T by directly predicting x ∗ t instead of ϵ ∗ t . Despite recent progress, achieving precise and efficient inversion remains challenging, especially under few-step inference or when the inputs are novel, as illustrated in Fig. 1.

3 Method

3.1 Reformulating Diffusion Inversion as a Learning Problem

Given a pretrained diffusion model that defines a mapping X : x T ↦→ x T -1 ↦→··· ↦→ x 0 from latent space to image space, we hypothesize that the inverse mapping X -1 : x 0 ↦→ x 1 ↦→··· ↦→ x T exhibits a smooth optimization landscape, owing to the smoothness of transitions in the latent space [44]. Based on this observation, we reformulate diffusion inversion as a learning problem. Let x 0 be an observed image, and let ϵ ∗ T denote a learnable noise embedding. The objective is defined as:

where

3.2 Progressive Denoising Trajectory Learning

Directly optimizing (12) requires backpropagation through all T timesteps, which incurs prohibitive memory and computational costs. To address this, we exploit the Markov property of the diffusion process to decompose problem (12) into T subproblems and solve them efficiently using dynamic programming principles.

Subproblem Decomposition Let { x t } T t =1 denote latent variables along the denoising trajectory from x 0 to x T , as indicated by the spherical markers along the blue arrow in Fig. 2. Since only x 0 is observed, we adopt a bottom-up dynamic programming strategy to sequentially estimate latent states from x 1 to x T . To this end, we define a collection of subproblems {P ( t ) } T t =1 , where each P ( t ) corresponds to the estimation of x t . Specifically,

with where ϵ ∗ t is the learnable noise embedding at timestep t ∈ { 1 , 2 , . . . , T } .

Optimal Substructure The inverse mapping X -1 : x 0 ↦→ x 1 ↦→··· ↦→ x T inherits the Markov structure of X , implying that the optimal solution to each subproblem P ( t ) depends only on the solution to the previous subproblem P ( t -1) for all t ≥ 2 . In particular, P (1) depends solely on the known observation x 0 . This recursive dependency establishes that the subproblems {P ( t ) } T t =1 exhibit the optimal substructure property.

ET-1

Et+1

X

x

mineillk(x) -αol12

minelμ(x-1) -α-2||²2

mineillμ(x)-∞-12

mine|l(x+1)-∞|12

X1

minel(x-1)-a-l2

mineyill(xt)-a-12

I-x

XT

XT1

Xt+1

Overlapping Subproblems Each subproblem P ( t ) seeks a latent variable x t such that the denoising trajectory from x t accurately reconstructs x 0 . As adjacent subproblems P ( t ) and P ( t -1) share this reconstruction objective and lie on successive steps of the trajectory, their optimization processes overlap. Specifically, the solution to P ( t -1) influences and partially informs the optimization of P ( t ) , resulting in substantial overlap across subproblems.

Bottom-Up Computation Without loss of generality, we adopt the DDIM sampling strategy. For convenience in the subsequent analysis, we reframe Eq. (7) as:

Substituting into Eq. (14), the first subproblem becomes:

Exploiting the optimal substructure of the subproblems {P ( t ) } T t =1 , we derive a recursive formulation f that maps P ( t -1) to P ( t ) :

where x ∗ t -1 denotes the optimized solution obtained from P ( t -1) . Leveraging the recursive formulation f , we solve the subproblems {P ( t ) } T t =1 iteratively from t = 1 to t = T , as indicated by the red dotted arrows in Fig. 2. We define a convergence threshold η for P ( t ) , and stop the optimization when ∥ µ ( x ∗ t ) -x ∗ t -1 ∥ 2 < η . In practice, we set η in the range of 10 -2 to 10 -5 . η and the inference step number T jointly control the trade-off between reconstruction accuracy and inference speed, as detailed in Table 5.

Optimization Procedure. The overall optimization algorithm of PreciseInv is presented in Algorithm 1. The procedure is memory-efficient since backpropagation is performed only over a single timestep. Moreover, PreciseInv is general and model-agnostic. We show its adaptability to different diffusion backbones (e.g., rectified flow [26]) and samplers (e.g., DDPM [17]) in Appendix A, and provide additional quantitative results in Appendix D.1.

1: Input: Real image x , diffusion model ϵ , convergence threshold η , number of inference steps T

3.3 Theoretical Analysis of Convergence

While the progressive optimization strategy above is conceptually appealing, its convergence has yet to be formally established. Here, we provide a theoretical proof to establish its convergence. We begin by stating two standard assumptions that are widely used in diffusion models [37].

Assumption 1. The diffusion model ϵ θ ( x t , t ) is Lipschitz continuous in x t with constant L m , i.e., ∥ ϵ θ ( x 1 , t ) -ϵ θ ( x 2 , t ) ∥ ≤ L m ∥ x 1 -x 2 ∥ .

Assumption 2. Let L t := ∥ µ ( x t ) -x t -1 ∥ 2 . The function L t has L -Lipschitz continuous gradients with respect to x t , i.e., ∥∇L t ( x t ) -∇L t ( x ′ t ) ∥ ≤ L ∥ x t -x ′ t ∥ .

Under these assumptions, standard non-convex optimization results [15] apply. With gradient descent using a fixed step size γ < 2 /L for each subproblem P ( t ) , the local loss L t decreases monotonically. After K iterations,

indicating convergence to a stationary point δ t . Since each step minimizes a local reconstruction loss and the optimization proceeds recursively, the global loss L is non-increasing. Given that µ is Lipschitz in x t , the accumulated reconstruction error satisfies:

If L ( j ) m ≤ L m < 1 (satisfies due to Assumption 1) and let δ = max { δ 1 , · · · , δ T } , a geometric upper bound follows:

Theorem 1. Under Assumptions 1 and 2, gradient descent on P ( t ) with step size γ < 2 /L converges monotonically to a stationary point of L t .

Theorem 2. Let Γ( t ) := ∥X ( x t ) -x 0 ∥ 2 . Under Assumption 1 and Theorem 1, Γ( T ) admits a geometric upper bound, i.e., Γ( T ) ≤ δ/ (1 -L m ) .

Detailed proofs of Theorem 1 and 2 are provided in Appendix B.

3.4 Prompt-driven Image Editing

We apply PreciseInv to enable prompt-driven editing by replacing the inversion stage without modifying the editing pipeline. Let x 0 denote the input image, and let c s and c t denote the source and target prompts, respectively. We perform inversion conditioned on c s , following the progressive optimization strategy in Eq. (14). At each step t , we solve

where µ c ( · ) generalizes Eq. (16) to conditional generation. This yields a latent x ∗ T that faithfully reconstructs x 0 under the conditioning of the source prompt.

To generate the edited image, we reuse x ∗ T and apply DDIM sampling under classifier-free guidance (CFG) [18], where the target and source prompts act as positive and negative conditioning signals, respectively. The conditional mean of the guided denoising process is:

where X c pos , c neg denotes the deterministic denoising trajectory governed by Eq. (23). This pipeline procedure enables prompt-driven editing without task-specific designs and improves fidelity and localization by aligning the initial latent more precisely with the structure of the input image.

4 Experiments

In this section, we present experimental results with the aim of (i) demonstrating the state-of-the-art performance of PreciseInv for image reconstruction in terms of both quality and efficiency; (ii) verifying that faithful inversion enhances consistency in prompt-guided image editing; and (iii) providing insights into its local behaviors and interaction mechanisms.

4.1 Image Reconstruction

Datasets. We evaluate PreciseInv on three domains: LAION-aligned, low-light, and stylized. (i) The COCO 2017 [24] validation set contains 5,000 natural images with diverse everyday scenes. It serves as a close proxy to the LAION distribution used in training most text-to-image diffusion models. (ii) The DarkFace [41] validation set includes 6,089 nighttime images captured in real-world low-light conditions, exhibiting extreme visibility degradation. (iii) The Cartoon dataset consists of 722 stylized images collected from the internet. These samples exhibit abstract shapes, exaggerated structures, and vivid palettes, reflecting rare visual styles far from the training distribution.

Baselines. We compare our method with forward optimization methods, including DDIM Inversion [8], ReNoise [13], and EasyInv [45]; and invertible samplers, including EDICT [38], BDIA [43], and BELM [39]. Following prior work [45, 39], we don't compare with backward optimization methods, as they are not designed for recovering the original generative trajectory of diffusion models. We carefully tune hyperparameters for all methods.

Metrics. We evaluate reconstruction quality using LPIPS, SSIM, and RSNR, which respectively measure perceptual similarity, structural alignment, and signal fidelity. To assess efficiency, we also report the average inference time per image.

Results and Analysis. Table 1 reports quantitative results on COCO, DarkFace, and Cartoon datasets using the SD v1.4 base model. With a tight convergence threshold ( η =10 -5 ), PreciseInv achieves the lowest LPIPS, highest SSIM, and best PSNR across all datasets. Despite the high quality, its inference time remains competitive. With a relaxed threshold ( η =10 -2 ), PreciseInv still surpasses most methods in LPIPS, SSIM, and PSNR, while reducing inference time to 5.33 s. Among forward optimization methods, EasyInv and ReNoise underperform the 1000-step DDIM Inversion , which suggests that numerical or iterative methods suffer from a performance bottleneck. In contrast, our method breaks through this bottleneck by a large margin. In particular, on the stylized Cartoon dataset,

the performance gains are more pronounced, yielding lower LPIPS ( -0 . 127 ), higher SSIM ( +0 . 081 ), and better PSNR ( +7 . 92 dB). Moreover, different invertible samplers exhibit similar reconstruction quality. Compared to EDICT and BDIA , BELM significantly decreases inference time from 45.5 s to 9.00 s. Although they achieve remarkable success on the Cartoon dataset, their performance remains limited on COCO and DarkFace. By comparison, our method shows robustness across data types and further improves overall efficiency.

For generality, we also present the reconstruction results on the COCO dataset using the few-step model LCM-SD v1.5 and the high-resolution model SDXL in Table 2. PreciseInv consistently achieves the best reconstruction quality across the two base models. With η =10 -2 , it already significantly outperforms all baselines while maintaining fast inference (8.83 s for LCM-SD v1.5 and 7.43 s for SDXL). When using a tighter threshold ( η =10 -5 ), the performance of PreciseInv is further improved. These results demonstrate that our method is general across different model architectures.

As shown in Table 3, we report the reconstruction performance of different inversion methods under the challenging few-step setting ( T = 2 , 4 ). ReNoise exhibits stronger few-step inference capability compared to other baselines. Nevertheless, our method achieves substantially better reconstruction quality across all metrics.

4.2 Prompt-driven Image Editing

Here, we present the results of PreciseInv on prompt-driven image editing, as shown in Fig. 3, and compare its editing performance with existing methods in Fig. 4. We demonstrate that PreciseInv enables the preservation of fine-grained details that are irrelevant to the editing prompt across a wide range of editing scenarios. For example, when editing a down jacket into a leather jacket, the original fabric folds are well preserved (left example of the first row in Fig. 3). Furthermore, compared to

11

12

"a down leather jacket"

"a dark green clock"

"a fox wolf"

"a sketch style dress"

Original Images

DDIM

ReNoise

BDIA

BELM

Ours

"a sandy desert grass surface"

"a man with pink hair"

"a tiger lion"

" pixel kirby"

existing methods, PreciseInv enables more faithful and controllable prompt-based editing. As shown in the first row of Fig. 4, our method better preserves both the tree structure and sky consistency during the texture transformation from desert to grassland, while baseline methods introduce sky artifacts (e.g., BELM) or distort tree details (e.g., ReNoise).

Notably, the purpose of these experiments is not to position PreciseInv as a state-of-the-art image editing method. As a generic inversion method without editing-specific design, it is inherently not comparable to advanced pipelines that involve domain-specific training [20, 40], attention manipulation [11, 16, 31, 42], or test-time optimization [7, 19]. Rather, PreciseInv serves as a foundation that can be integrated with such techniques. Incorporating precise inversion into editingoriented frameworks is a promising direction for future work.

Ours

BELM

EasyInv

ReNoise

DDIM

To further explore why PreciseInv enables controllable and faithful editing, we perform an interpolation-based diagnostic experiment, as shown in Fig. 5. Specifically, we first select two similar real images and apply different inversion methods to recover their corresponding latents. Then, we perform spherical linear interpolation (slerp) between the two inverted latents and reconstruct all intermediate states using the same diffusion model. As shown in Fig. 5, PreciseInv produces a smooth and semantically meaningful transition, maintaining structural coherence and identity consistency throughout the interpolation path. In contrast, DDIM and EasyInv often yield abrupt domain shifts or introduce foreign objects, indicating a drift away from the correct semantic manifold. BELM fails to preserve visual coherence due to accumulated noise artifacts, while ReNoise partially retains structure but suffers from geometric degradation. These findings suggest that accurate inversion leads to latents that better respect the geometry of the diffusion space, thereby enabling more localized, stable, and prompt-aligned edits.

5 Conclusion

We propose PreciseInv , a general-purpose test-time optimization framework that enables fast and faithful inversion for pre-trained diffusion models. Unlike prior fixed-point methods, we reformulate diffusion inversion as a progressive learning problem and introduce a dynamic programming-inspired strategy to solve it effectively. Theoretical analysis shows that PreciseInv converges under mild assumptions. Extensive experiments across diverse data domains and model architectures demonstrate that the proposed method achieves state-of-the-art performance in image reconstruction. Notably, the performance gains are especially pronounced under few-step settings and on novel samples. Moreover, we verify the effectiveness of PreciseInv in prompt-driven image editing tasks and further analyze its underlying mechanism. In future, we attempt to integrate PreciseInv with diverse editing techniques to explore a unified, controllable, and consistency-aware framework for image editing.

Acknowledgment

This work was supported by the Hainan Provincial Joint Project of Li'an International Education Innovation Pilot Zone (Grant No.624LALH008), BUPT Kunpeng & Ascend Center of Cultivation, NSFC (Grant No.61601042), the Program for Youth Innovative Research Team of BUPT (Grant No. 2023QNTD02), and the Super Computing Platform of BUPT.

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Fangyikang Wang, Hubery Yin, Yue-Jiang Dong, Huminhao Zhu, Hanbin Zhao, Hui Qian, Chen Li, et al. Belm: Bidirectional explicit linear multi-step sampler for exact inversion in diffusion models. Advances in Neural Information Processing Systems , 37:46118-46159, 2024.

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Appendix & Supplementary Materials

Contents

A Formulations

In this section, we extend the formulation of PreciseInv to different combinations of base models and samplers, highlighting the generality and adaptability of our approach.

A.1 Formulation of Stochastic Sampler: DDPM

We begin with the stochastic DDPM sampler [17]. For clarity, we first restate its sampling step:

The added noise term σ t z t in Eq. (24) complicates the inversion task, as deterministic reconstruction from x t is no longer guaranteed. To overcome this, we treat z t as an additional learnable variable. Specifically, we modify each subproblem P ( t ) (defined in Eq. (14)) to jointly optimize the noise embedding ϵ ∗ t and the stochastic component z ∗ t :

where

This formulation preserves the dynamic programming structure of our method. In particular, the optimal substructure property remains intact, with each subproblem P ( t ) dependent only on the solution to the previous subproblem P ( t -1) :

By learning both the latent noise and the stochastic perturbations at each step, our method generalizes to stochastic samplers and enables accurate inversion under the DDPM setting.

A.2 Formulation of Rectified Flow Models: SD3

Preliminary. Rectified Flow [26] defines score-based diffusion modeling by learning a velocity field over a linear interpolation path between the data distribution q ( x 0 ) and the Gaussian prior N ( 0 , I ) . The forward process is formulated as:

A neural network learns a time-dependent velocity field v θ ( x t , t ) by minimizing the following training objective:

The backward (generative) process recovers x 0 from a sample x 1 ∼ N ( 0 , I ) by solving the following ordinary differential equation (ODE) in reverse time:

Unlike discrete-time diffusion models such as DDPM and DDIM, Rectified Flow does not define an explicit update rule. Instead, the reverse trajectory is obtained by numerically integrating Eq. (30), typically via Euler discretization:

SD3 [10] refines Rectified Flow to enable large-scale pretraining for text-to-image generation. First, instead of adopting a uniform timestep schedule, it samples timesteps from a logit-normal [3] distribution defined as:

This design allows SD3 to flexibly control the bias and width of the timestep distribution. It further introduces a multimodal diffusion transformer that unifies cross-attention and self-attention mechanisms. This results in improved alignment between text and image modalities.

PreciseInv for the SD3 Model We further extend PreciseInv to the continuous-time rectified flow model, SD3. To construct a tractable denoising trajectory, we first sample a sequence of timesteps { t 1 , t 2 , . . . , t T } from the logit-normal distribution π ln ( i ; m,s ) defined in Eq. (32). These sampled timesteps discretize the continuous-time trajectory into a sequence of latent variables { x t 0 , x t 1 , . . . , x t T } , where each x t i is recursively approximated by Euler integration:

with x t T ∼ N ( 0 , I ) and i ∈ { T, T -1 , . . . , 2 } .

Following Section 3, we define T subproblems {P ( t i ) } T i =1 , each targeting a local reconstruction step along the denoising path x t T ↦→··· ↦→ x t 1 . For each t i , we parameterize the latent variable x ∗ t i with a learnable noise embedding ϵ ∗ t i as:

Each subproblem P ( t i ) seeks to minimize the reconstruction error between the estimated and true signal:

where X denotes the Euler-discretized backward trajectory from x t T to x t 1 . For clarity, we rewrite the Euler step in Eq. (31) as:

With this notation, each subproblem can be reformulated recursively as:

By progressively solving these subproblems from P ( t 1 ) to P ( t T ) , PreciseInv achieves a precise inversion of the image x 0 into the SD3 latent space x 1 .

B Proofs

We provide formal proofs for the theoretical results presented in Section 3.

B.1 Proof of Theorem 1

Proof. For the function L t ( x t ) with L -Lipschitz continuous gradients (Assumption 2), we define the gradient-based update as:

where k denotes the iteration index and γ is the learning rate. By the smoothness lemma [15], for any x and x ′ ,

where f denotes any function with L -Lipschitz continuous gradients. Then, when γ < 2 /L , we obtain

After K iterations, we have

which implies that L t converges to a stationary point δ t during the optimization of P ( t ) .

B.2 Proof of Theorem 2

Proof. Let the accumulated reconstruction error of P ( t ) be Γ( t ) := ∥X ( x t ) -x 0 ∥ 2 , where x t denotes the estimated latent representation and x 0 is the ground-truth image. We define X ( x 0 ) = 0 . From Theorem 1, for all t ∈ { 1 , . . . , T } , we have:

Considering Eq. (18) and Assumption 1, we obtain:

Under Assumption 1, there exists a constant L m < 1 such that L ( j ) m ≤ L m . Hence,

Let δ = max { δ 1 , . . . , δ T } . By recursively applying Eq. (44) from t = 1 to T , we derive:

Thus, Γ( T ) is bounded by δ/ (1 -L )

C Implementation Details

C.1 Training Details

We implement our method using PyTorch and the Hugging Face diffusers library. All diffusion model weights are frozen during training, and only the per-timestep noise embeddings ϵ ∗ t are optimized via gradient descent. Following EasyInv , we initialize the value of ϵ ∗ t by performing a single step of DDIM Inversion . We use the AdamW optimizer with β 1 = 0 . 9 , β 2 = 0 . 999 , and a weight decay of 0 . 01 . The learning rate is set to 0.1 for SD v1.4, 0.05 for both LCM-SD v1.5 and SDXL, and 0.025 for SD3. We use the DDIM sampler unless otherwise specified; for SD3, the Euler discrete sampler is adopted due to architectural compatibility. We keep the number of inversion steps equal to the number of sampling steps, denoted as T . All experiments are conducted under mixed precision: float16 is used for SD v1.4, LCM-SD v1.5, and SDXL, while bfloat16 is used for SD3 to ensure numerical stability. We run all experiments on a single RTX 4090 24GB GPU, except for SD3, which requires a single A100 40GB GPU due to its higher memory and compute demands.

C.2 Hyperparameters for Image Reconstruction

As described in Section 4, we tune all methods for optimal reconstruction quality. We set the number of inference steps T as follows: T =1000 for DDIM Inversion , T =50 for ReNoise and BDIA , T =100 for EasyInv and BELM , and T =2 for EDICT and our PreciseInv . For ReNoise , we disable the editing loss by setting λ =0 and sweep the renoising steps n ∈ { 1 , 2 , . . . , 9 } . EasyInv , EDICT , BDIA , and BELM are used with default configurations from their respective papers. For PreciseInv , we set the convergence threshold η to 10 -2 , 10 -3 and 10 -5 , respectively. To eliminate the effect of text conditioning, all methods are evaluated with empty prompts and guidance scale set to zero.

C.3 Pseudocode of PreciseInv

We present the pseudocode of the proposed PreciseInv . Specifically, three cases are covered: diffusion models using the DDIM sampler, diffusion models using the DDPM sampler, and rectified flow models using the Euler discrete sampler, as shown in Algorithm 2, 3, and 4, respectively.

Algorithm 2 PreciseInv for Diffusion Models with DDIM Sampler

Algorithm 3 PreciseInv for Diffusion Models with DDPM Sampler

D Additional Results

D.1 Quantitative Results.

We provide additional quantitative results of our method on the image reconstruction task. Table 4 presents the performance of PreciseInv with different combinations of base models and samplers, as described in Section A. The experiments run on the COCO 2017 validation set [24]. We set the number of inference steps to T = 3 for SD3 and T = 2 for the other models. The convergence threshold η = 10 -3 is used for all models.

Input: Real image x 0 , rectified flow model ϵ θ , convergence threshold η , number of inference

Table 4 shows that all models achieve comparable reconstruction quality. SD3 attains higher SSIM and PSNR scores, while other models perform better on LPIPS. Compared to the results reported in Tables 1 and 2, PreciseInv maintains state-of-the-art reconstruction performance, further demonstrating the generality and robustness of our method. However, SD3 exhibits lower inference efficiency relative to other models, which can be attributed to its larger model size and the absence of an effective initialization manner.

To address this, future work will explore a progressively decreasing timestep optimization strategy for large models such as SD3. We believe that this strategy will result in a smoother optimization landscape. Additionally, exploring better initialization techniques, such as using initial values derived from existing flow-based inversion methods, would also be a promising direction.

We also report reconstruction performance under varying convergence thresholds η and inference step counts T in Table 5. We randomly sample 20 images from the COCO 2017 validation set and evaluate using the Stable Diffusion v1.4 model with a DDIM sampler. The number of inference steps is varied as T ∈ 2 , 5 , 10 , 20 , 50 , 100 and the convergence threshold as η ∈ 10 -2 , 10 -3 , 10 -5 , 10 -6 , 10 -7 . We report metrics including LPIPS, SSIM, and PSNR, as well as inference time in seconds.

From Table 5, we draw the following observations. First, for a fixed number of inference steps T , decreasing the convergence threshold η consistently improves reconstruction quality across all metrics-LPIPS decreases, while SSIM and PSNR increase. However, this comes at the cost of

significantly longer inference time, especially as η drops below 10 -5 . Second, we observe diminishing returns in reconstruction quality when η < 10 -5 , particularly for small-step regimes (e.g., T ≤ 10 ). This suggests that extremely strict convergence criteria may be unnecessary when the number of inference steps is limited. Third, when η is fixed, increasing the number of inference steps T initially degrades the reconstruction quality before improving it. This phenomenon results from a tradeoff: while smaller per-step initialization errors benefit from higher T , the total accumulated error at convergence (approximately T · η ) also increases. Notably, due to the smooth structure of the diffusion latent space, error accumulation is sublinear and does not dominate overall performance. Fourth, inference time exhibits a non-monotonic trend with respect to T . When η ≥ 10 -5 , increasing T initially reduces inference time-likely due to faster convergence from smaller per-step errors-but then increases it again as the step count becomes the dominant factor. In contrast, for stricter thresholds ( η ≤ 10 -5 ), inference time generally decreases with larger T , implying that the time saved through easier optimization outweighs the overhead introduced by additional steps. Finally, when T ≤ 10 , tightening the threshold from η = 10 -5 to η = 10 -6 yields negligible gains in reconstruction quality. Moreover, for large-step configurations ( T > 10 ), using η = 10 -7 achieves reconstruction performance on par with small-step setups at η = 10 -5 . Taken together, these findings suggest that setting η = 10 -5 and T = 2 strikes a favorable balance between efficiency and quality.

D.2 Qualitative Results.

Here, we present additional qualitative results on image reconstruction, prompt-based image editing, and image interpolation.

DDIM Inversion

ReNoise

EasyInv

BELM

Ours

DDIM Inversion

ReNoise

EasyInv

BELM

Ours

DDIM Inversion

ReNoise

EasyInv

BELM

Ours

STOP

STOP

STOP

STOP

MAMERTINE

MAMMETIME

MAVMETIME

ANMRTIME

DDIM Inversion

ReNoise

BELM

Ours

DDIM Inversion

ReNoise

BELM

Ours

DDIM Inversion

ReNoise

BELM

Ours

DDIMInversion

ReNoise

EasyInv

Ours

InputImage

DDIM Inversion

ReNoise

EasyInv

Ours

DDIM Inversion

ReNoise

EasyInv

Ours

16

16

16

16

Original Images

DDIM

ReNoise

BDIA

BELM

Ours

"a red light moon"

"a blue green bag"

"quartzwoodenbeadedbracelet"

"a sheep deer"

"a colorful styleboot"

"a man stands alone gets on a vehicle at the crossroads"

"a fabrie leather sofa"

"a taeo hotdog"

OriginalImages

DDIM

BELM

ReNoise

Ours

" a dog tiger"

" a straw leather hat"

" a red pink natural handphones"

" an range red car"

Original Images

DDIM

ReNoise

Ours

an orange red car"

an abstract painting style man"

Ours

BELM

EasyInv

ReNoise

DDIM

Ours

BELM

EasyInv

ReNoise

DDIM

000

000

000

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies *inverse* diffusion, the goal is to find the noisy $x_T$, which has generated the provided image $x_0$. Where $x_T$ is the noise in the *latent* space of a Latent Diffusion Model. This is not a trivial task, given that the diffusion model itself is asymmetric, and simply inverting the sampler does not yield satisfactory results. In this paper inverse diffusion is seen as an optimization problem, where iteratively the optimal noise is sought, based on the optimal noise of the previous time: $x_{t-1} \\rightarrow x_t$. This makes use of the Markov property of diffusion, and is solved iteratively starting from $t=1$ to $t=T$. Experimentally the method is validated on 3 different datasets and in various settings (different values for T, different LDMs), the results show that compared to related works the proposed method is advantageous (especially in LPIPS). Moreover for prompt rewriting some quantitative results are shown and also an interpolation between two images.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n* To my opinion the current manuscript is excellent in describing the setting of inverse diffusion, the difficulties and the proposed solution.\n\n### Weakness\n* The manuscript has some confusing elements, it puts emphasize on a 'learning framework' (*e.g.* in the abstract and introduction), and hence I was first convinced that $\\epsilon_\\theta$ would be **learned** to predict the inverse diffusion problem. But, after a second and third read, it seems that is not the case, this is a 'test-time algorithm', where the dynamic programming optimization is run for each example at test/evaluation time, without learning any parameters. In that sense nothing is learned. Please clarify that (eg in the introduction, below Eq 18, and in the experiments) and add pseudo code for the final algorithm.\n* A part of the experimental setup is not clear to me, how is $T$ used. And why does smaller T values lead to better results? From the results it seems that using T=1 gives better results than using T=2 or T=4, while in general for Diffusion holds that a larger T value will provide better results.\n* In the experiments there is a convergence threshold ($\\eta=10^{-5}$), which is not discussed before. Please describe how the threshold is used during optimization.\n* In **Bottom up computation** it is stated that *without loss of generality, we adopt the DDIM sampling strategy*. Would it be possible to use the DDPM strategy (Eq 6), and would that be easier or more difficult due to the additional noise term $z$?\n\n\n### Remaining remarks\n* I would like to get more insights in the optimization, how many steps (on average) are needed to reach the threshold? Is the resulting noisy image indeed a sample from a N(0,1) Gaussian? Is there a clear tradeoff between the used threshold, the number of steps T, and the image reconstruction quality.\n* Table 2: How many iterations are used, ie what is the value for T?\n* Table 2 & 3: It seems that using T=2 performs better than T=4 (on LPIPS, SSIM, PSNR). Is that expected? Is there a trade-off of quality versus number of steps T? (And is it better to have a few steps with a stricter convergence threshold or more steps with a tighter threshold?)\n* In Figure 2, there are two $\\epsilon^{*}_{t}$ depicted.\n  * Fix: one of these should be $\\epsilon^*_{t+1}$\n\nQUESTIONS:\nPlease provide more details and insight on the optimization implementation: how is the threshold used, what is the impact of the value of $T$, how do the obtained latents look like, and is it really easy to use a different sampler?\n\nLIMITATIONS:\nLimitations are not discussed in the main paper.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes to reframe the diffusion inversion problem as a series of step-wise optimization sub-problems, rather than treating it as a traditional fixed-point problem. The method specifically addresses challenges related to accuracy and efficiency in inverting few-step inference diffusion models. Experimental results on both reconstruction and editing tasks demonstrate the effectiveness of the proposed method.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- S1. The proposed inversion method appears reasonable, and the experimental results are convincing.\n- S2. The proposed method performs well on few-step inference diffusion models such as LCM, showing improvements over comparison methods.\n\n**Weaknesses**\n- W1. The most critical issue is that, despite the paper framing the method as a **learning-based** inversion approach, it does not actually involve **learning** in the conventional sense. Throughout the main text, it is unclear what model or parameters are being trained. Only in Appendix B is it clarified that the proposed method **directly optimizes** the inverted noise, indicating that it is a direct optimization procedure rather than a learning-based model. This mischaracterization may mislead readers. Conventionally, learning-based inference using parametric or non-parametric models is expected to perform worse than direct optimization. Therefore, presenting the method as \"learning-based\" could result in an overestimation of its generality and performance.\n- W2. The analysis and experimental results on image editing is insufficient. In the context of image generation models, the primary motivation for inversion is to enable controlled editing through manipulation of the inverted input or generation trajectory. However, the editing examples provided are limited to minor modifications. Color editing reflects only simple intensity changes, and style transfer is demonstrated only through sketching or pixelation, lacking complexity. Moreover, no quantitative evaluation is provided. The authors should include a broader range of edit prompts and define a standardized prompt set for benchmarking, to ensure reproducibility and diversity. Additionally, quantitative metrics and user studies are necessary to objectively validate editing performance.\n- W3. The set of baseline algorithms is limited and includes outdated methods. To better establish the advantages of the proposed method, comparisons with more diverse and recent works are needed.\n\nQUESTIONS:\n- Q1. *(This question arose before I read Appendix B, but I'm still uncertain whether my interpretation is correct, so I'm leaving it here for clarification.)* What is the architecture of the PreciseInv model? Given that it appears to be comparable in size to the denoising backbone $\\epsilon_\\theta$, wouldn't such a large model introduce inefficiencies for inversion? Moreover, according to Eq. 20, $\\epsilon_\\theta$ must also reside in memory during PreciseInv training. Has the computational burden imposed by this requirement been analyzed, particularly in terms of limiting model size?\n  - Training denoisers for T2I diffusion models is computationally expensive. Wouldn't training an inversion model that generalizes across diverse prompts and image (=$x_0$) also require substantial data and training time? How much training data is required?\n  - Sec. 4.1 mentions only a validation set. What dataset was used for training?\n\n- Q2. According to l29-32, backward optimization methods improve reconstruction quality by following the reverse diffusion trajectory. However, l179-180 claim these methods are not designed for accurate reconstruction and are excluded from comparison. Isn't this contradictory?\n- Q3. What does the threshold $\\eta$ in l186 represent?\n- Q4. In Table 1 and Table 2, are the number of inversion steps for the proposed method and the baseline algorithms the same? The proposed method appears to use gradient descent to optimizer $\\epsilon_t^*$, which involves both forward and backward passes through the diffusion model at each iteration. Since one gradient descent step is unlikely to suffice per time step, it seems that the proposed method would require more iterations to reach a final output if the number of time steps is fixed. Then, why is it claimed to be faster?\n- Q5. In Sec. 3.3 on prompt-driven image editing, what is the rationale for denoising from the inverted latent $x_T^*$ using Eq. 21? Is there any theoretical justification for Eq. 21? Why is denoising not performing using $x_{c_t, \\emptyset}$?\n- Q6. In Figure 5, what prompt was used for denoising the interpolated latent $x_T^*$? Were interpolation-aware prompts also used in the baseline methods for a fair comparison?\n\nLIMITATIONS:\nThe authors do not discuss the limitations of their proposed method", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a diffusion inversion method that allows few-step inversion and generalization to novel samples by reformulating diffusion inversion as a progressive learning problem and using DP like strategy to solve this. The proposed method outperforms baseline methods in most settings and the difference is the most pronounced in the few-step inference setting.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- By solving subproblems separately, the given method is more memory-efficient compared to baseline methods, which might be helpful for applying to larger models. \n- The proposed method consistently shows better reconstruction accuracy compared to baseline methods with a much faster speed (Table 1).\n- The proposed method works better than baseline methods especially under a few-step setting.\n- Qualitative examples are still impressive, including the editing applications.\n- Performing slerp between inverted latents results in a better qualitative result across all interpolated positions.\n- The method can be extended to rectified flow models such as SD3 as well (Appendix A.2), and still shows a good reconstruction performance..\n\n**Weaknesses**\n- When adjusting the convergence threshold, the computation time seems to increase quite a lot, especially in the the case of SDXL. The speed-quality tradeoff needs to be discussed.\n- In fact, the method seems to be quite slower for SD3 (Appendix C.1, Table 4), which may suggest that the computation time saving might not scale to larger models such as SD3 or FLUX, which can limit the applicability. A comparison against baseline methods for SD3 or FLUX might also be necessary.\n\nQUESTIONS:\n- Can the method be used with shortcut models as well?\n- Do you have a particular explanation as to why the proposed method results in a smoother interpolation between latents? It would be interesting to know a more detailed reasoning behind this.\n- It is unclear as to whether the method is viable for SD3 just looking at Table 3 as there is no baseline comparison. How does this inference speed compare to other baselines in SD3?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes PreciseInv, which is a novel learning-based method for diffusion model inversion that achieves accurate and efficient inversion in a few inference steps. The previous methods are usually based on fixed-point or root-finding. Different from them, PreciseInv reformulates inversion as a progressive learning task. In detail, it leverages the Markov property of diffusion models. It decomposes the problem into several overlapping subproblems and solves them using a dynamic programming-inspired strategy. The authors conduct experiments on COCO 2017, DarkFace, and a cartoon dataset. The results show that PreciseInv achieves better reconstruction quality and efficiency compared with existing methods. Besides, it also improves semantic consistency in image editing.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper addresses an important problem. Diffusion inversion is important for image editing. The current fixed-point and root-finding method cannot find an accurate noise embedding. The paper aims to propose an accurate and efficient inversion method.\n2. The method is novel. The paper reformulates diffusion inversion as a learning problem. It also decomposes the problem into recursive subproblems and proposes a dynamic programming-inspired strategy to solve it.\n3. The results are good. The proposed method achieves strong trade-offs between reconstruction accuracy and efficiency under a step inference. The method also improves semantic consistency and controllability in image editing. The results demonstrate good performance in various datasets and diffusion architecture.\n\n\nWeaknesses:\n1. The theoretical analysis is missing. Although the proposed method is straightforward, the analysis of convergence or approximation error in the optimization should be provided.\n2. Comparison in image editing is limited. The paper only provides a qualitative comparison of image editing. However, quantitative metrics are missing.\n3. Some ablation studies are missing. The paper uses different thresholds. The studies about the choice should be provided.\n\nQUESTIONS:\n1. Could the author provide the analysis of convergence or approximation error about the optimization?\n2. Could the author provide a quantitative comparison in image editing? BIE-Benchmark is a good choice.\n3. How sensitive is the method to the choice of threshold?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}