Free Lunch for Stabilizing Rectified Flow Inversion
Authors: Chenru Wang
Beier Zhu
Chi Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Flow Inversion
Rectified Flow
Image Editing
Primary area: generative models

Abstract: Rectified-Flow (RF)-based generative models have recently emerged as strong alternatives to traditional diffusion models, demonstrating state-of-the-art performance across various tasks. By learning a continuous velocity field that transforms simple noise into complex data, RF-based models not only enable high-quality generation, but also support training-free inversion, which facilitates downstream tasks such as reconstruction and editing. However, existing inversion methods, such as vanilla RF-based inversion, suffer from approximation errors that accumulate across timesteps, leading to unstable velocity fields and degraded reconstruction and editing quality. To address this challenge, we propose Proximal-Mean Inversion (PMI), a training-free gradient correction method that stabilizes the velocity field by guiding it toward a running average of past velocities, constrained within a theoretically derived spherical Gaussian. Furthermore, we introduce mimic-CFG, a lightweight velocity correction scheme for editing tasks, which interpolates between the current velocity and its projection onto the historical average, balancing editing effectiveness and structural consistency. Extensive experiments on PIE-Bench demonstrate that our methods significantly improve inversion stability, image reconstruction quality, and editing fidelity, while reducing the required number of neural function evaluations. Our approach achieves state-of-the-art performance on the PIE-Bench with enhanced efficiency and theoretical soundness.

Published as a conference paper at ICLR 2026
FREE LUNCH FOR STABILIZING RECTIFIED FLOW IN- VERSION
Chenru Wang
Beier Zhu
AGI Lab, Westlake University wangchenru@west ake. edu.cn
University of Science and Technology of China beier. thuuuste. edu cn
Chi Zhang*
AGI Lab, Westlake University chi chizhang@westlake ed
ABSTRACT
Rectified-Flow (RF)-based generative models have recently emerged as strong al- ternatives to traditional diffusion models, demonstrating state-of-the-art perfor- mance across various tasks. By learning continuous velocity field that transforms simple noise into complex data, RF-based models only enable high-quality generation, but also support training-free inversion, which facilitates downstream tasks such as reconstruction and editing. However, existing inversion methods, such as vanilla RF-based inversion, suffer from approximation errors that accu- mulate across timesteps, leading to unstable velocity fields and degraded recon- struction and editing quality. To address this challenge, propose Proximal- Mean Inversion (PMI), a training-free free gradient correction method that stabilizes the velocity field by guiding it toward rumning average of past velocities, con- strained within a theoretically derived spherical Gaussian. Furthermore, we in- troduce m mi c-CFG, lightweight velocity correction scheme for editing tasks, which interpolates between the current velocity and its projection onto the histor- ical average, balancing editing effectiveness and structural consistency. Extensive experiments on PIE-Bench demonstrate that our methods significantly improve in- version stability, image reconstruction quality, and editing fidelity, while reducing the required number of neural function evaluations. Our approach achieves state- of-the-art performance on the PIE-Bench with enhanced efficiency and theoretical soundness.
INTRODUCTION
Recently, Rectified -lowf (RF)-based generative models have gained increasing popularity over tra- ditional diffusion models, achieving remarkable progress in wide range of applications such as image synthesis (Labs et al. 20255] Esser et al. 2024), video generation (Wang et al. 2024; Li al.) Zhao et al. 2025| Zhou et al.) 2025), 3D content creation (Li et al. 2025c Go et al. 2025 Li a. 2024| Song et al. 2025). and other areas(Jin et al. 2025; Ma et al., 2026). These models operate by learning continuous velocity field that transforms samples from simple noise distri- bution (typically standard Gaussian) into complex data distributions. Beyond generation, RF-based models also offer the potential for inversion task, the process of mapping observed data samples back into their corresponding latent representations using a pre-trained RF-based model. This in- version capability opens up a range of downstream applications without requiring further training or fine-tuning of the model. For instance, high-quality inversion latent can support tasks such as image
2025a
reconstruction and image editing (Deng al. 2024 Rout al. 2025
Wang al. 2025a Xu al.
2025
Jiao et al. 2025 Ma et al. 2025).
Despite these potentials, achieving accurate and robust inversion remains significant challenge, and the quality of inversion directly impacts the fidelity, consistency, and controllability of downstream
"Coreesooningg author.

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tasks. A representative example is the DDIM inversion (Song al., 2022). which has been widely adopted for inverting diffusion models (Pan et al. 2023: Samuel et al. [025] Mokady et al., 2023] Han et al. 2024a Kadosh et al. 2025). It uses deterministic sampling equation that includes neu- ral network term to predict the noise at each timestep. Since the exact latent variable is inaccessible during inversion, the network input is typically approximated using the observed data the current timestep. This approximation introduces estimation errors at each step, and these errors compound as the inversion progresses through multiple steps. As result, the inversion latent representation deviates from the original noise prior (e.g,, standard Gaussian) (Lukasz Staniszewski et al., 2025), which ultimately degrades the performance of tasks such as image reconstruction and editing. Simi- larly, the inversion of these RF-based models also introduces approximation errors, which affect the performance of downstream tasks.
To address the instability introduced by approximation errors in velocity fields, we aim to improve the reliability of inversion and editing by guiding the perturbed velocity field toward a more stable evolution. Existing research (Zhang et al. 2025 has found that such instability from the inherent sparsity of the generative distribution, which makes the approximation errors during inference dif- ficult to eliminate entirely. These errors accumulate across timesteps, leading to disturbed velocity fields and ultimately degrading the stability of the inversion and reconstruction processes. To mit- igate this issue, we propose Proximal-Mean Inversion (PMI), which performs gradient correction on the velocity field. At each timestep, PMT computes a running average of the predicted velocities from the initial step to the current one and applies a proximal update that guides the current veloc- ity prediction toward this averaged direction. Furthermore, to prevent the corrected trajectory from deviating into low-density regions of the data distribution, we constrain each correction step within spherical Gaussian, whose radius is theoretically derived in our work based on the instability anal- ysis. This correction method stabilizes the inversion process and reduces the probability of falling into low-density regions, without additional training or neural function evaluations (NFESs, and can be seamlessly integrated into any RF-based model. PMI yields improved reconstruction quality and editing performance while maintaining sampling efficiency.
Beyond reconstruction, we identify that velocity perturbations also adversely affect editing tasks (Lukasz Staniszewski al. 2025 Jiao et al. 2025), one of the most practical applications of inversion. Excessive perturbations to the velocity can degrade structural consistency, while ve- locities with insufficient deviation can compromise editing capabilities. To strike balance between structural consistency and editing capabilities, we introduce mimic-CFG, a lightweight velocity correction strategy inspired by classifier-free guidance. Specifically, each step, we compute the running average of the past velocities and project the current predicted velocity onto this average di- rection. The final corrected veloeity is then obtained by interpolating between the predicted velocity and its projection. Experimental results on the PIE-Bench (Ju et al. (024) show that our meth- ods effectively correct velocity, balancing the two aforementioned factors, significantly reducing the number of sampling steps while enhancing editing quality and structural consistency.
We conduct extensive experiments across multiple benchmarks and tasks to evaluate the effective- ness of our methods. Results show that our methods consistently enhance baselines across multiple key metrics. Notably, by integrating our methods, many baselines can reach or exceed state-of- the-art (SOTA) quality with fewer NFEs, while preserving theoretical soundness and computational efficiency. Our contributions are summarized as follows:
We propose PMI, a training-fres method that corrects the perturbed velocity field via gradient correction toward running average, effectively reducing inversion errors in RF-based models.
We propose mimi C-CFG, a lightweight velocity correction strategy that mirrors the structure of Classifier-Free Guidance (CFG) (Ho & Salimans] (002)) by interpolating between the current velocity and its projection onto the running average direction.
Our approach establishes new SOTA on the PIE-Bench, achieving superior editing and recon- struction performance with fewer sampling steps and no additional NFE.
2 RELATED WORK
Inversion. DDIM inversion aims to map real data to noise that approximates standard Gaussian. However, diffusion model's approximations at each timestep introduce errors, causing deviations
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from the ideal noise
Zhu al. (2025a): Wang et al. (2025c b): Zhu et al. (2025b). To address
this, various methods have been proposed (Mokady et al. [2023 Dong et al. 2023a Cho et al.
2024
Han al.,
2024b; Miyake et al. 2025 Wallace al. 2022] Huberman Huberman-Spiegeeeee Spiegelglas et al.
2024 including Fixed-Point Iteration (FPI), which iteratively refines predictions to reduce local
2023 Bao al. 2025 Samuel et al. 2025 Garibi al. 2025 Meiri al. 2023) errors (Pan al.
yielding more accurate noise estimates.
With the rise of flow-based models, inversion has shifted from solving stochastic differential equa- tions (SDEs) to ordinary differential equations (ODEs). Building on this evolution, several methods have been proposed: RF-Inversion (Rout et al., 2025) uses dynamic optimal control to reduce in- version enrors; RF-Solver (Wang et al., 2025a) employs high-order Taylor expansions to minimize integration errors; FireFlow (Deng et al.) 2024 achieves second-order accuracy with only one NFE per step; and Two-stage Inversion (Xu et al. 2025) applies multiple fixed-pone iterations per step to further refine predictions. While these approaches improve accuracy, especially those using FPI, they typically increase sampling steps and NFEs, leading to higher computational cost. To overcome this, we propose PMI, which reduces inversion error while requiring fewer sampling steps.
Editing. Image editing requires conditionally modifying semantic content while preserving origi- nal structure. Diffusion-basee methods typically follow two steps: (1) invert the input image to its latent noise; (2) generate the edited image via guided sampling under editing constraints. However, inversion errors often degrade control and final image quality. To improve editing accuracy, some efforts introduce additional guidance, such as masks or image features. For instance, Prompt-to- Prompt (Hertz et al., (202)) manipulates cross-attention maps for precise control; MasaCtrl (Cao) ensure consistency; and Direct Inversion (Ju al., 20224 sepa- rates source and target diffusion to better preserve content. However, these approaches often require extra inference steps or feature extraction, increasing computational cost.
al. 2023) adjusts self-attentio
Beyond inversion-basee methods, recent works explore the inversion-free editing, avoiding explicit reconstruction of the latent noise. InfEdit (Xu et al. 2024) introduces an inversion-free editing framework that reconstructs consistent denoising path from noise, enabling fast and semantically coherent edits. FlowChef (Patel et al. 2024) exploits the straightness of RF models trajectories to propose gradient-free, inversion-free steering strategy, significantly reducing computational costs for editing. FlowEdit (Kulikov et al.) 2025) introduces a direct ODE mapping between source and target distributions, enabling efficient editing. These approaches substantially improve efficiency, but their reliance on approximate trajectory manipulation can lead to reduced fine-grained consis- tency. In contrast, mi mi C-CFG preserves the controllability of inversion-basee pipelines while quiring no auxiliary models or structured inputs. By reducing inversion error through PMT, it enables accurate semantic edits with fewer sampling steps, striking practical balance between efficiency and fine-grained consistency.
3 PRELIMINARIES
Before introducing our method, we present preliminaries on RF model training and sampling, the inversion process, and the inherent instability of existing inversion methods.
3.1 RECTIFIED FLOw
Rectified Flow (RF (Liu et al. over the trajectory, enabling faster and more stable sampling. maps real data Zo ~ Pdata Gaussian noise 2 ~ '0 I) via the following ODE:
2022 is built on ODEs and learns an almost constant velocity field
dz. dt
()) E 10.
Here, Ug is a time-dependent velocity field parameterized by neural network. The model is trained to match the ideal direction z Zn throughout the trajectory by minimizing the following loss:
CR = Exo o1 [1lvo (2,, () (z) =20)
where z, = (1 - t)zn + tz) denotes a linear interpolation between 2n and z During inference, generation involves solving the learned ODE in feverse-from 2 back to 2g. Using Fuler's method,
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the reverse update from to f is given by:
-t,-1 Ze, (tz-1 ( (1)00
(3)
where {t, E [0, 1]} is decreasing sequence.
3.2
DDIM INVERSION
DDIM inversion (Euler scheduler) performs the reverse the sampling process, mapping a real image into its latent noise. Based on the DDIM update rule, the inversion step of Euler scheduler using approximation is given by:
Ze. + (ax, or Doo(zz, (.1)
(4)
where ot, and o are scheduling parameters.
3.3 INSTABILITY IN INVERSION
Zhang et al. 2025) show that using ODEs map noise images is inherently unstable in high dimensions: small perturbations in the latent space can cause large reconstruction errors. To quantify this effect, they introduce the geometric mean instability coefficient:
Jr JF(Z))
EF(z) = II
(5)
where (z) is the Jacobian of z and u, orthonormal vectors, and when (r 1, it indicates that the mapping amplifies any infinitesimal perturbation. They prove that for any threshold M > 1, instability occurs with probability:
PM Treal
(fz: & (z)) M)) e-
where
(f 2: Pron(z) (2 22
1
2n43v22
Traal
(fz: (z::11=((2)11
real
2n 3 2m
Here, Treal is the real data distribution, and as dimension n increases, both and 8 vanish, implying that instability is almost guaranteed in high-dimensional ODE mappings.
3.4 INVERSION EVALUATION WITHOUT PROMPT BIAS
The inversion quality is ultimately assessed through downstream tasks such reconstruction or editing. These downstream processes typically rely on text prompts or other conditional inputs. As result, the observed reconstruction editing performance may be inffuenced not only by the inversion accuracy but also by the quality and relevance of the conditioning prompt (Dong et al. 2023b Li et al. (2225b). This makes conditional evaluation insufficient for isolating inversion errors, since prompt mismatch or weak textual alignment can obscure the true stability of the inversion trajectory.
To avoid the confounding effect introduced by prompts and to obtain a more faithful measure of inversion quality, additionally report reconstruction performance under unconditional setting. This prompt-free evaluation isolates the inversion process and reveals trajectory drift or accumulated emrrr that may be masked in conditional reconstruction. This diagnostic perspective offers clearer assessment of inversion stability and complements the conditional metrics used in downstream edit- ing tasks.
METHODOLOGY
4.1 PROXIMAL-MEAN INVERSTON
As detailed in Sec. 3.3 RF in inversion sampling is susceptible to errors arising from intrinsic insta- bility, often leading suboptimal results. We posit that these errors stem from the disruption of the
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originally near-constant velocity field during inversion, which compromises the stability essential for both inversion and reconstruction.
Algorithm Proximal- Inversion (PMI): Euler Case
Algorithm 2 mimi CFG Editing: Euler Case
Input: velocity function (e("), image 2g. time schedule T { 0, ........ IN 1}. interpolation weight w.
Input: velocity function 0i (.), image Ze. time
schedule T = (40 -,.... IN
Output: 21
Output: 2n
1: Xdist = 0
1: 21 = PMI (Ug Zo: t)
2: for i = to
do
2: Xdist 0
3: Vt; Up uo(2e, (i)
3: for i = N to do
4: Xdist + (t,+1 tx) ve
4: Vt uo v0(2 (i)
5: Xdist + = (ti-1 -14) Ve:
5:
Vt: 44-1 Xdist (F(v
6: ti
Xdist
6: Ve: = =V4, - Th |V 112
t--1 tn
Vi, vodie
yproj
Vedit
7: 26.11 Lc, + (t++1 4x) Vt.
| 112
8: end for
Ve
w) w)v Va.
9: return 21
9:
2t.-1
= 21. + (ti-1 - t,
10: end for
11:
return 20
To address this issue, we gradually correct the disturbed velocity field towards a more stable and near-constant one. Following the principle of global consistency, we guide velocity correction using an average velocity as a reference. Let Vi, = uo (Z,, tú) denote the predicted velocity at time 4,, the average velocity from the initial time to the current time t, is defined
Σ (1++1- tt)v
(8)
t++1 to
for & E {O..... N 1}. To stabilize the perturbed velocity field, we employ proximal objective using the average velocity:
1 F(v) ||v vu- 2 ll vo2
The corrected velocity V., at timestep fy is then
Vea prox, (Vr.) arg min F(v),
(10)
equation 10 balances two objectives with a trade-off factor (results see Appendix B.I and B.2
V| Via | encourages local consistency by keeping the corrected velocity close to the previ- ous step. An analysis experiment on the norm choice in ||v - Vit-ill - Ilp p 1 yields the best performance.
(see Sec. 5.4) shows that
||v Vea. encourages the velocity toward the global average direction Ves-
However, solving equation I0 exactly at each timestep is computationally expensive. To improve efficiency, we approximate the objective with a first-order Taylor expansion around the current ve- locity Vt.:
F(v) R F(v..) (V(...)) TV v..).
(11)
As result, the constrained optimization problem becomes: Ves))
arg MinVF((v..)))
(12)
s.t. ||v - v = r
Solving equation 12 yields a closed-fome fored-foommmm update (proof in Appendix A.I
VF(ve)
Vic Vis
(13)
/VF(v
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Source
RF-Solver
-PMI (Ours)
FireFlow
PML (Ours)
Conditional
Unconditional
Qualitative comparison results for the inversion and reconstruction task on PIE-Bench. Baselines enhanced with our method consistently outperform their vanilla counterparts, particularly under the unconditional setting.
Figure 1:
Here, the gradient direction is normalized, so the update magnitude depends only on the radius Tte- Moreover, we derive Proposition [ which provides condition Tt, ensure stability during inversion, based on probability bounds for low-density regions in instability theory.
Proposition (Stability Condition) Suppose the inverted latent vector 21 E RR" follows a Gaussian distribution
21 ~ o²1),
and define == =nno o, where n is the latent dimension and a > O is the scaling factor. The radius that make the sampling points fall within the high-density region satisfies
At. T
v 2n + 3/2n
+
(14)
where T. is the total time, Al; = 14 - li-1, and r; denotes the step-wise correction radius at the i-th timestep.
The full procedure is summarized in Algorithm| andd referred to as PMI. Notably, PMI can be seam- lessly integrated into existing inversion methods (e.g., RF-Solver and FireFlow); here, we illustrate it using Fuler's method. It requires no additional training and effectively stabilize inversion.
4.2 MIMIC-CFG EDITING
Although our instability theory is primarily developed for the Inversion and Reconstruction phases, observe that similar instability propagate into the Editing stage due to their shared reliance

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RF-Inversion
RF-Solver Ours
FireFlow
Cleat > Hore
Classes
Cocomut
Ohi Paint
Figure 2: Qualitative comparison results for the editing task on PIE-Bench. The lefumost column shows the source images. Each of the three columns on the right contains two sets of results: the left side shows outputs from the vanilla baseline, while the right side shows outputs enhanced with our PMI and mimic- methods. The results show that our methods can enhance the background preservation and editing quality in different editing categories.
on the velocity field. Therefore, we apply our correction strategy during Editing as well, to mitigate residual errors.
Similarly, for the editing stage (traversing from iN 1 to to 0), we define the average velocity up to timestep 4 as:
Vedie l
(1:-1 - 1:)00 (Zt,, 4.).
(15)
1--1
IN
==N
At each timestep tg, we shift the predicted velocity Vt, toward the average velocity vedit to enhance stability. We aim to achieve this through interpolation. Prior study (Fan et al. 2025) has shown that interpolating between Vr, and its projection onto vadir performs better than direct interpolation between the two vectors. We also empirically verify this finding in our setting (see Appendix [B.8).
Therefore, we apply the shortest path update along this direction: we first compute the projection of Via onto the direction of vedit follows:
Vedit v
proj
adit
(16)
This projected vector proo represents the shortest correction path from Ut, towards vedit, Subse- quently, we perform linear interpolation between voro and Vt, to obtain the corrected velocity:
Ve (1 w) + w- V
(17)
Here, w E [0, 1] is an interpolation parameter used control the degree of correction. An analysis experiment of w (see Sec. B.I) shows that w 0.94 yields the best performance. It is worth noting that if we treat y the "unconditional" velocity and Vt. the "conditional" velocity, the above interpolation form closely resembles Classifier-Free Guidance (CFG) (Ho & Salimans) 2022).
Therefore, refer to this method as mimic-CFG.
We present the detailed algorithmic procedure in Algorithm 2 state obtained via PMI, perform reverse Euler process over the editing timesteps. The method introduces no additional training and improves stability during editing by aligning the velocity field with the average direction.
Starting from a corrected initial
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Source
RI-nnversion
RF-Solver
Laugh - Angzy
Figure 3: Qualitative comparison of editing results PlE-Bench (first three) and real-world (last) images. Our method achieves superior adaptation in both synthetic and real-world editing scenarios.
Table 1: Quantitative results for inversion and reconstruction task. Our method consistently outper- forms baseline models, achieving higher PSNR and SSIM, and lower MSE and LPIPS. Best results are in bold.
Conditional
Unconditional
Method
PSNR
LPIPS
MSE
SSIM 20102
PSNR
LPIPS X 100
MSE
104
SSIM 102
Fuler
22.10
168.48
96.58
79.36
21.96
189.26
101.57
77.45
(ours)
166.71
83.59
Heun
29.16
63.62
30.76
89.93
27.76
78.84
40.42
88.51
+ PMI (ours)
30.38
53.25
21.51
90.34
29.86
62.67
32.44
90.26
Ro-Solver
28.18
39.68
+ PMI (ours)
30.72
53.69
22.42
91.23
29.87
62.66
29.88
90.29
FireFloo
29.72
55.87
23.05
90.86
28.87
66.04
27.03
89.76
PMI (ours)
30.42
53.48
18.44
91.22
29.73
62.51
23.72
90.34
EXPERIMENTS
5.1
SETUP
Baseline. To demonstrate the effectiveness of our method, we selected several fundamental and commonly used RF-based baselines. For the inversion and reconstruction tasks, we incorporate our method into four solvers, Euler, Heun, RF-Solver (Wang et al.) 2025a). and FireFlow (Deng et al., 2024]. and compare their performance against their respective vanilla counterparts. For the editing task, we apply our method to the same four solvers as well as RF-Inversion (Rout et al.) 2025 resulting five edited variants, and evaluate them against their vanilla counterparts.
Implementation details. In all experiments, we use Flux. (Labs et al. 2025) as the underlying reconstruction experiments, we
model for both reconstruction and editing tasks. For inversion ensure that each baseline method, both with and without proposed velocity correction strategy (PMI), shares the same NFE within each group to ensure fair comparison. For the editing task, adopt the best settings reported in FireFlow for In-Inverii,,, RF-Solver, and FireFlow, while maintaining the NFE around 50 for both the Euler and Heun samplers. More details see Appendix(C
Evaluate metrics. All experiments conducted the PIE-Bench (Ju al. 2024). which sists of 700 images spanning different editing categories. For the inversion and reconstruc- tion task, evaluate reconstruction quality using four widely adopted metries: Peak Signal-to-
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Table 2: Quantitative results for editing task. Our methods consistently improve CLIP similarity while maintain and slightly improve the background preservation metrics PSNR and SSIM. Best results are bold and identical results are underlined.
Mothod
Structure
Background Preservation MSE
CLIP Similarily
NFE J
Distance
LPIPS
SSIM
Whole f
Edited
50.08
20.57
187.47
135,03
25.73
23.00
+ Ours
46.06
21.41
185.55
121.78
76.04
25.87
23.01
50
Heen
44.51
20.51
176.09
137.53
77.45
25.79
23.08
48
+ Ours
42.24
20.56
175.27
124.57
77.57
25.91
23,09
48
RF-Inversion
63.41
18.03
252.89
21.44
62.65
25.02
22.56
56
21.44
RF-Solver
33.71
21.29
156.51
96.24
79.63
25.29
22.61
60
+ Ours
32.69
22.18
146.85
85.64
80.39
25.49
22.72
FireFlow
29.39
22.40
136.49
79.35
80.78
25.35
22.52
+ Ours
28.01
22.79
128.55
78.02
81.76
25.50
22.68
(Huynh-Thu Ghanbari 2008), Learned Perceptual Image Patch Similarity 2018), Mean Squared Error (MSE), and Structural Similarity Index Mcasure (SSIM) (Wang ct al. 2004). For the editing task, the same four metrics are used to assess background introduce the CLIP similarity (Rad-
Noise Ratio (PSNR)
(LPIPS) (Zhang et al.
preservation unedited areas. To evaluate editing fidelity, ford et al. 2021) for both the whole image and the edited regions. Furthermore, we employ Structure Distance (Ju et al. 2024) evaluate edit-irreaative context preservation.
5.2 INVERSION AND RECONSTRUCTION
Quantitative comparison.
To prove the effectiveness of our proposed correc- tion method, we compare it with various baseline models on inversion and reconstruction tasks, shown in Tab. I The results indicate consistent improvements with our correction method (PMT) across all baseline samplers. Compared to their vanilla counterparts, our method achieves higher PSNR and SSIM, and lower MSE and LPIPS, with particularly notable improvements (PSNR improved by 1.22) and RF-Solver (PSNR improved by 1.55). Additionally, unconditional setting for each group, where our method maintains strong performance even in the absence of prompts. These results demonstrate the effectiveness of our approach in stabilizing the inversion process, enhancing reconstruction quality, and reducing the likelihood of falling into low-density regions. Additionally, we present the reconstruction error in Appendix 10
24.20
84.00
26.00
03.00
23:60
62.50
23.40
81.50
23.00
22.80
00.50
Heun
Whole Edited
22.38
evaluate the
25.20
25:18
22.31
25:16
22.30
25:14
22.25
22.20
0.90
0.91
0.92
0.03
2.94
0.95
0.96
0.07
0.95
Figure Analysis experiment of the parame- ter in mimic-CFG. PSNR and SSIM improve with moderate correction, while over correction (small w) harms both background preservation and editing quality.
Qualitative comparison.
Fig. demonstrates qualitative comparison the inversion and reconstruction task on RF-
Solver and FireFlow, both with and without our method. Our approach significantly improves construction quality in both conditional and unconditional settings. In the unconditional case, base- line models without our method exhibit noticeable degradation in detail and structural fidelity.

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5.3 EDITING
Quantitative comparison. To validate the effectiveness of our method, we demonstrate the quanti- tative results for text-driven image editing on PIE-Bench in Tab. 2 Integrating our method with all baseline samplers results in clear improvements in background preservation metrics such as PSNR and SSIM, while maintaining a competitive CLIP similarity. For instance, our method significantly improves both CLIP similarity on II-Inversion while maintaining and slightly enhancing back- ground preservation. We also compared the same baselines with and without our method across different NFE. The results show that the baseline with our method achieves similar performance with significantly lower NFE (see in Appendix [.9). These results indicate that our method effee- tively corrects velocity, balancing background preservation and editing quality, while significantly reducing the number of sampling steps.
Qualitative comparison.
In Fig. 2] we compare three methods with and without our methods. We can see that the baseline methods often have poor background preservation low editing quality. These problems usually need more sampling steps to improve. In contrast, our editing method fixes these issues while using same or fewer sampling steps.
Here, to further demonstrate the practical effectiveness of our method in real-world settings, we present additional qualitative comparative results editing tasks, as shown in Fig. 3 The first three images are from PIE-Bench, while the last one is a real-world photo. The results demonstrate that our method better facilitates the model's adaptation both benchmark and real-world image editing.
5.4 ANALYSIS EXPERIMENTS
Analysis of w. In Fig 4 we demonstrate the result of the analysis experiment of the parameter w in our proposed m C-CFG method. The result indicates that wr 0.94 provides the best balance between background preservation and editing quality, As increases, both PSNR and SSIM improve. However, as the velocity correction effect weakens (when w decreases), the editability worsens.
Analysis on the choice of norm.
Tab. shows the analysis experiment on the norm used in the with such term, using the L, norm, and using the L2 norm. The evaluation conducted on PIE- Bench with PSNR, SSIM, and CLIP similarity for both the whole image and the edited region. The L, norm achieves slightly better performance on PSNR, SSIM and CLIP similarity for the edited region, indicating it provides more robust regu- larization while maintaining local consistency.
Table 3: Analysis Experiment the choice of norm. The L1 norm achieves slightly better per- formance. Best results are in bold and identical results are underlined.
lpp We compare the setting
Method
PSNR
SSIM
Whole
Edited
wfo nerm
23.66
82.76
25.20
22.37
L Lr-mmm
23.67
82.77
25.20
22.38
1 norn
23.65
82.75
25.18
22.32
6 CONCLUSION
In this paper, we introduce Proximal-Mean Inversion (PMI), a velocity correction method for RF- based models that mitigates disturbances in the velocity field and reduces accumulated errors during inversion. Furthermore, to prevent the corrected trajectory from deviating into low-density regions, we constrain each correction step within a spherical Gaussian, with the radius derived by us based on instability analysis. Building on this correction concept, we also present mimic CFG, a lightweight correction strategy for editing tasks, which guides the predicted velocity towards the running average direction to achieve an improved balance between background preservation and editing fidelity. Both methods training-fire free and plug-and-play Extensive experiments PIE-Bende demonstrate that our approaches significantly enhance reconstruction and editing quality, reduce sampling steps compared to hate-of-the-an RF-based inversion methods.
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ACKNOWLEDGEMENT
This work was supported by the National Natural Science Foundation of China (No. 6250070674) and the Zhejiang Leading Innovative and Entrepreneur Team Introduction Program (2024R01007).
ETHICS STATEMENT
Data and Privacy. All experiments in this paper are conducted on publicly available datasets, including PIE-Bench, which consists of benchmark images for inversion and editing evaluation. These datasets contain personally identifiable information (PII), sensitive user data, or proprietary content. We strictly follow the licenses of all datasets and models employed, ensuring that our usage complies with both legal and ethical norms.
Research Purpose and Potential Risks. Our proposed methods, Proximal-Mear Inversion (PMI) and mimic-CFG, designed to improve the stability and controllability of rectified-flow (RF)- based generative models during inversion and editing. The contributions are methodological and theoretical, aiming to reduce approximation errors and enhance reconstruction fidelity and editing quality. Nevertheless, generative models can be misused in ways that cause harm, such as gener- ating disinformation, creating manipulated media without consent, reinforcing existing biases in training data. We acknowledge these risks and emphasize that our work is intended for academic research and responsible downstream applications only. We explicitly discourage malicious use of the proposed methods, particularly in domains where manipulation may infringe on human rights, amplify stereotypes, or spread harmful misinformation.
Fairness and Bias. Since our work builds upon pre-trained generative models, it may inherit dataset biases or artifacts embedded in these models. Our methods do not introduce additional bias mitigation, but by improving inversion and editing stability, they may reduce certain ertor-induced distortions in downstream tasks. We encourage further studies to combine our correction strategies with fairess--aware training evaluation pipelines ensure more equitable outcomes.
Environmental and Computational Impact. All experiments were performed on a single NVIDIA A100 GPU with 40GB memory. Compared training large generative models from scratch, our approach is training-free and thus imposes a relatively small computational and en- vironmental footprint. We further optimized our experiments minimize unnecessary runs and repeated evaluations, Nevertheless, we recognize the broader sustainability concerns of large-scale machine learning and encourage future research to adopt resource-consciou practices.
Contribution to Society. We believe that this research advances fundamental understanding of inversion and editing in generative models, offering new tools for trustworthy and efficient appli- cations. Potential positive applications include image restoration, creative content generation with proper consent, and controllable editing that preserves structural fidelity. We expect these contribu- tions to support the responsible development of Al systems that are transparent, reproducible, and beneficial to society.
Conflicts of Interest. The authors declare conflicts of interest.
REPRODUCIBILITY STATEMENT
We have taken multiple steps to ensure the reprodueibility of our work. First, we provide complete algorithmic descriptions of both Proximal-Mean Inversion (PMI) and mimi C-CFG in See. 4 Ad- ditional theoretical details, such as proofs of stability conditions and error analysis, are included in Appendix[A] Second, we report all experimental settings, including dataset specifications, evaluation metrics, and hyperparameter ranges, in Sec. and Appendix C with detailed analysis experiments provided in Appendix B The computing infrastructure, including GPU, CPU, memory, and soft- environment, is documented in Appendix C.I Finally, to further facilitate reproducibility, provide anonymized source code as supplementary material, which implement our methods reported in the paper.

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A MISSING PROOFS
A. 1 CLOSE FORM SOLUTION OF EQUATION 12
We first restate the equation as follows:
argmin (F((..)(( (V--
Vik
(A.1)
s.t. VtA. =
where and Vt, denote the corrected and current velocities at time (gs respectively, and is the radius of the spherical Gaussian centered at Vıg. The optimal solution can be derived as follows.
First we reparameterize the constrain condition
Vt. rd,
(A.2)
where ||d]2 1, and d is the direction from Vt, to v, substituting this into the original objective, we obtain:
argmin
(v VeN)
Vig
argmin VF((,, (Vi, rd v
(A.3)
V4,
argmin )"
|
We now construct the Lagrangian function:
[(d, - = -VV((v X(I|
(A.4)
then we have the KKT condition
VC VV= 2Ad
| =1 X >
(A.5)
Setting VL = yields:
d rVF(v..
(A.6)
Substituting into the constraint ||d|]] and solving for X, we get:
A VF(v. II
(A.7)
2
Thus, the optimal solution for is:
VF(ve)
(A.8)
[VF((v.))
Therefore, the closed-form solution of Equation is:
VF(V Vt. VF(V,.))) !I
(A.9)
which equals to:
VF(v.,) Ru Vo. [VF(v..))
(A.10)
A.2 THE RANGE OF r DURING THE INVERSION.
Inspired by Zhang et al. 2025) and Yang et al. (0224), want reduce the probability of the model falling into the low density regions by using the spherical Gaussian. Next we give the proof of Proposition
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Proposition 1. Suppose the inverted latent vector 21 E R"" follows Gaussian distribution
21 ~ NO, d²1),
and define
r= Vno.
where n. is the latent dimension and > is the scaling factor. According to the instability theorem, the radius that can make the sampling points fall within the high-density region satisfies
T V 2n 3v2n
At. T
2m
+
where T is the total time, At; = 1 - 4-11 and r; denotes the step-wise correction radius at the i-th time step.
Proof. We first numerically calculate the variance ā of 21
2 1)) |z,1?
(A.11)
n
where 21 and ži denote the flattened representations of 21 and 2, respectively, According to the theorem, we have
8²n 2 2n 3v2n
(A.12)
Since / o, this can be equivalently written
22 > 2n 2n.
(A.13)
Therefore, we obtain
V 2n 3/2n.
(A.14)
Suppose that Ve > 0 and set = > 0. When
At.
0 T
< 88
(A.15)
we have
78;
(A.16)
Define the local radius
r; 18; +
(A.17)
Then r; r8; 0, and explicitly
At: 2n
2n - 3
+
(A.18)
T
Hence, for every discrete index i, the lower bound on the radius remains strictly larger than rs; while approaching rs; as E 0.
A.3 ERROR ANALYSIS
Assumption 1, The velocityfunctionn up(:, is Cu-Lipschitz.or VT, iee, given a T,
[[vo(G,, ( 1 - Cills Call,
VG1,22
Assumption 2. The velocity function vo(C.-) is Cg-Lipschitt. VC, i.e., given S.
[[vo(C, -o m2) || S C2l - -T2ll VT, T2-
Proposition 2. Suppose that the velocity field is Lipschitz, and there is constant L s.f. ||z,, Zt,-, ll2 < Lllt 1-11ll, Vt, t- E [0, 1). Then for any two consecutive steps t,-1 and t, the local error of inversion and reconstruction using our method is O(at)), achieves the same local truncation error the standard Fuler method, where Al; -4-1-
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Proof. We consider the recursive relationship of the one-step error term:
|zz, 2z, 1l - |z,,_, :1.-11l
+ (t:
v0
(2,, (..-1:
(A.19)
(V((t:)
+ (t: ti-1)l |[FF(V 11-
First, expand the velocity field differences:
[o t2) U0 -1, t(-1)
|o0 (4) 10 (2 t 1i-1)
(A.20)
+ (oo t(-1) - uy (2 14-1))l
|||||-
From the Lipschitz condition Asm and Asm we obtain:
||00 Zi,,, 1--1) - 2 2 1=-1) _ Cillz,, - -
(A.21)
|100 (x) uo (Z), ti 1 1)
- C2llt; tz-1ll.
such that
[100 (4)
t--1)
C2ll1 4/-11 Cil 12, 2 + |||||
C2St, Ci Zt. 20,-,1 |1 + |||||
(A.22)
$ C2At, + Ci((zz, - 2:,-,11 + || - _ II) + |r]]
C2Al: + CiA4; + Ci 111, - 21: 11 + ||lll
We define that 3 |z]._, - ||. the local error ofl - -2,11
glocal = (t: - 14-1) va (Zr, f4) 2 (2.... t4 14-1)11-
(A.23)
and substitute them back into the original inequality:
glocal < (C1 + C2) + CIEE + |r||t,
(A.24)
We can avoid the accumulated error ε, then we have the local error:
glocal < (C1 + C2) ||]]
(A.25)
Now we substitute the r in Prop
2 without we have
glocal < (C) + C2) + II 2n + 332n Al:
||
T
2n + 3yn
(A.26)
(C1 +
At?
T
O(At))
The Proposition 2
indicates that our method will not increase the error compared to the baseline.
A.4 ANALYSIS ON THE EXPANSION ORDER OF F(v)
To better understand the effect of using higher-order approximations in PMI, we analyze the Taylor expansion of the proximal objective F'(v) around the current velocity Vty-
Second-order expansion. The second-order Taylor expansion of F(v) around Vt, is given by:
F(v) ~ F(v.) + VF(v (v
V..) +
2 (v (v-
T (V - -v.).
(A.27)
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Following the constrained formulation in Eq. A.1
obtain:
1 J(v - v. - Vs,). VF((v,)) - vo,.)
arg min |d
(A.28)
Since F consists of an & regularization term and quadratic term, its Hessian satisfies
(=v(v)))= = I
(A.29)
where the & term contributes zero curvature almost everywhere. Eq. A.28 arrive at:
Plugging this structure into
1 1| Ve, 12. arg min VF(v.) vo..) 2A
(A.30)
| =1
Using the spherical constraint reparameterization
Vta +rd
|d]l2 = 1,
(A.31)
we obtain
arg min VF((v)) d p2
(A.32)
|||]==1
2X
Since the second term constant w.r.t. the optimization reduces
arg min (V [
(A.33)
which is identical to A.3 This shows that the optimal direction of the update remains unchanged even when second-order information is included. Thus, the PMI update rule under second-order expansion is equivalent the first-order case.
Higher-order expansions. For n 2 3, the Taylor expansion of contains higher-order derivatives:
V"F(v
(A.34)
However, F(v) is composed of an & term (piecewise linear) and a quadratic term (constant Hes- sian). Therefore,
V"v((v.))= = 0, Vn 3.
(A.35)
This implies that all higher-order Taylor terms vanish identically. Consequently, third-order and higher-order expansions produce the same optimization objective the second-order case, and thus yield exactly the same closed-form update direction.
B EXPERIMENTS
B. ANALYSIS EXPERIMENT À ON RECONSTRUCTION
To investigate the optimal choice of X for the inversion and reconstruction task, we conduct an analysis experiment under the conditional case. We aim to select İ value that yields higher PSNR and SSIM while achieving lower LPIPS and MSE. As shown in Fig. B. the choice of the trade- off factor À significantly influences the performance increases, both PSNR and SSIM generally improve, with = 10 achieving the highest values. Mcanwhile, LPIPS and MSE exhibit decreasing trend, and À 10 yields the lowest values across both metrics. These results indicate that À = 10 provides the best balance between preserving perceptual quality and ensuring pixel-level fidelity.
the inversion and reconstruction task. As
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91.29
PSNR
30.48
SSIM
91.28
30.47
91.27
30.46
30.45
91.26
30.44
91.25
30.43
91.24
30.42
91.23
30.41
91.22
2
4
6
8
10
LPIPS
18.10
53.3
MSE
18.05
53.2
18.00
53.1
17.95
53.0
17.90
17.85
52.9
0
2
4
6
8
10
Figure B.1: Analysis experiment the effect of the proximal operator parameter Increasing leads to improvements in PSNR and SSIM, while excessively large values result in overcorrection, compromising editing quality.
B.2 ANALYSIS EXPERIMENT OF À ON EDITING
We study the effect of the hyperparameter À in the proximal operator for editing tasks. As illustrated in Fig. B.2, we vary from 0 and report PSNR. SSIM, and CLIP similarity (Whole and Edited). As X increases, both PSNR and SSIM exhibit slight improvements, indicating that stronger proximal correction enhances image fidelity. However, the CLIP similarity results suggest that overly large degrade editing performance: Whole similarity drops gradually, and Edited similarity becomes less stable, reflecting weakened editability and overcorrection. These results demonstrate a trade-off' between reconstruction accuracy and editing effectiveness. We observe that À 1 offers good balance and adopt as the default setting for editing tasks for FireFlow.
We also study the effect of the hyperparameter À the performance of the PMI solver when used with other solvers, including Euler, Heun, and RF-Solver. As shown in Fig. B. Fig. B.4 and Fig. B.5l the optimal À for Fuler, Heun, and RF-Solver are 8, À 1, and X T, respectively.
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Zoom: & SSIM (A E0,0 11)
23.670
23.672
02:80
23670
23,003
82.77
23668
23,660
23.008
02.78
23,855
52.75
23.002
10.74
02:76
10
0.0
1.0
Zoom: Whole & Edited E(O.
Whole Edited
25.23
22:36
22.36
25.21
22:35
25.22
22.34
25.20
75.21
25.10
22.32
25.20
22.32
Figure B.2: Analysis experiment on the effect of the proximal operator parameter with the solver FireFlow. Increasing À leads to improvements in PSNR and SSIM, while excessively large values result in overcorrection, compromising editing quality.
Zoom: PSNR & SSIM (A E (0, 1])
22,050
22:050
SSIM
78.94
22.040
22.035
78.90
22.035
22.080
22.080
78.85
78.86
10
0.0
0.5
1.0
Edited [
Whole Edited
19.64
21.90
19.85
23.58
39.63
21.84
21.33
19.62
2193
21.92
19.61
19.63
21.51
15.00
21.50
39.59
Figure B.3: Analysis experiment on the effect of the proximal operator parameter with the solver Euler.
B.3 ANALYSIS EXPERIMENT OF ON EDITING
We demonstrate the result of the analysis experiment of the parameter w in our pro- posed mimic-CFF method, As shown in Fig. [B6]] the result indicates that w = 0.94 provides the best balance between background preservation and editing quality for Euler and Heun, 0.92 provides the best balance between background preservation and editing quality for RF-Solver
B.4 ANALYSIS EXPERIMENT OF w ON STABLE DIFFUSION-3.5
We demonstrate the result of the analysis experiment of the parameter in pro- posed mimic-CFG method. As shown Fig. [.7] the result indicates that w = 0.94 provides the best balance between background preservation and editing quality
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Zoom: CI0,
POWR
26.04
96.92
26.00
26.02
86,90
26.00
25.08
25.08
84
25.96
-2.97
1.0
1.0
Zoom: Whole Edited (A E (0)
22.54
22.30
20.07
22.53
22.53
22.52
20.05
22.50
22.51
20.04
Whole Edited
10
0.5
1.0
Figure B.4: Analysis experiment on the effect of the proximal operator parameter with the solver Heun.
SSIM
PSWR 5GM
26.91
67.82
26.90
26.68
87.75
26.88
$7.80
87.76
26.84
$7.60
26.87
87.74
26.88
67.72
24.15
1.0
Zoom: Whole Edited (A E 13)
22.7
22.64
22.6
20.75
22.5
20.50
22.4
20.25
22.62
22.61
22.
22.2
Whole Edited
19.30
19.75
20.2
220
22.58
20.0
10
0.0
0.5
1.0
Figure B.5: Analysis experiment on the effect of the proximal operator parameter with the solver RF-Solver.
B.5 ANALYSIS EXPERIMENT OF À ON STABLE- DIFFUSONIF
We demonstrate the result of the analysis experiment of the parameter in our pro- posed PMI method. As shown Fig. B.8 and Fig. B.9 the result indicates that provides the best performance reconstruction task, and X provides the best performance on editing task.
B.6 ANALYSIS EXPERIMENT
We demonstrate the result of the analysis experiment of the parameter in proposed PMI method. As shown in Fig. 10 the result indicates that provides the best balance between background preservation and editing quality.
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Euller
23.44
F2B
22.2
120
20.4
22.0-
7330
-525
22:4
0.82
6.94
6:34
25. 84-
2300
25.23
Edied
25. 60
2305
23.20
2236
2238
25/6
22 760
22
2234
25.76-
23.01
25.47
23.00
23.22
25.35
+22 725
0.0
0.44
0.56
0.0
0.34
0:22
0.22
0.34
0:32
Figure B.6: Analysis experiment on the effect of the mimic-CCr CFG parameter with three different solvers, including Euler, Heun, and RF-Solver.
Table B.1: Comparison on different velocity computing scheme. The results show that PMI mimi C-CFG achieve better results across all metrics in the editing task, especially in terms of CLIP similarity. Best results are in bold.
Structure Distance
Background Preservation
CLIP Similarity
Method
NFE
PSNR
LPIPS!
MSE
SSIM
Whole
Edited
FireFlow
29.39
22.40
136.49
79.35
80.78
25.35
22.52
EMA
28.82
22.74
129.36
78.59
82.97
25.39
22.54
18
+ Ouus
28.01
22.79
128.55
78.02
81.76
25.50
22.68
18
B.7 DIFFERENT VELOCITY AVERAGING SCHEME
To further stabilize the velocity accumulation used in our trajectory correction module, we introduce an Exponential Moving Average (EMA) formulation as an alternative the simple summation- based mean velocity adopted in the Eq. equation g
The original scheme accumulates the velocity contribution across steps as
Vt. Vt2-1 + AtgVEL
which may propagate noise due to the unweighted influence of all past increments. To obtain smoother estimate of the running velocity, we adopt EMA, which gradually down-weights older increments via exponential smoothing.
Let us = At&Ve. denote the velocity increment at step k. The EMA update is defined as V n (1 a:) vez
where a E (0, 1) controls the smoothing strength.
We compare EMA with the original integral- -ased averaging strategy in Table B.1 Without any parameter tuning. EMA already achieves improvements over the FireFlow baseline, indicating that amoothing-baset averaging can be beneficial even with minimal design effort. Our final method, which uses the proposed integral-based correction, achieves the best results across most metrics. This suggests that velocity formulation provides more effective control over trajectory stability. We leave a more systematic investigation of EMA and additional averaging strategies for future work.
B.8 COMPARISON ON DIFFERENT EDITING METHOD
Tab. B.2 presents the comparison results between mimic-CFo and PMI in the editing process. The results show that while mimi C-CGG slightly outperforms PMT in terms of PSNR and SSIM,
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PSNR SSIM
81.6
20.5
81.5
20.4
88.4
20.3
81.3
20.2
81.2
20.1
81.1
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
21.9
24.6
Whole
Edited
24.5
21.8
24.4
21.7
24.3
21.6
24.2
21.5
24.1
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
Figure B.7: Analysis experiment on the effect the 3.5 with solver FireFlow.
parameter w on Stable Diffusion-
Table B.2: Comparison on different editing method. The results show that mimic- outper- forms PMI across all metrics in the editing task, especially in terms of CLIP similarity. Best results are bold.
Method
Structure Distance kp>
PSNR
LPIPS' 2106
Background Preservation
MSE! 104
SSIM!
210
Whole
Similarity
Edited
NFE
Time(s)
haseline
29.
22.40
136.49
79.35
80.78
25.35
22.52
18
28.5
22.74
130.80
80.09
81.50
25.34
23.57
18
5,2
+mimic-CFG
significantly surpasses PMI in both CLIP Similarity scores, demonstrating its superior effectiveness in enhancing editing performance.
B.9 QUANTITATIVE COMPARISON ON EDITING
Tab. B.3 demonstrates the comparison results on PIE-Bench with different NFE. We can observe that method achieves results comparable the baseline at higher NEE, even with significantly lower NFE. For example, Heun combined with our method achieves better structural consistency with fewer steps and lower NFE. Similarly, FireFlow benefits from correction, showing higher
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20.92
82.40
20.92
82.39
20.91
82.38
20.91
82.37
20.91
82.36
20.91
82.35
20.91
82.34
20.90
PSNR
82.33
SSIM
20.90
82.32
2
8
10
LPIPS
128.10
MSE
188.40
128.00
188.20
127.90
188.00
127.80
187.80
127.70
187.60
0
2
4
8
10
Figure B.8: Analysis experiment on the effect of the PMI parameter À Stable Diffusion-3.5 with solver FireFlow (Reconstruction).
Table B.3: Image editing comparison PIE-Benci with different NFE. The results show that method brings baseline metrics closer to those of high NFE baselines while reducing NFE, effec- tively accelerating the model. Best results are in bold and identical results are underlined.
Method
Structure
Background Preservation
CLIP Similarity
NFE
Distance x104
PSNR
T.PIPS) 210>
MSF 2x10
SSTM!
Whole
Edited
Euler
50.08
20.57
187.47
135.03
76.02
25.73
23,00
50
Ours
$2.09
20.36
193.21
121.59
76.11
25.89
22.79
40
33.27
22.27
+ Ours
28.21
23.01
132,49
71.83
81.64
25,22
22.37
48
RI-ennersoon
63.41
18.03
252.89 89
21.44
62.65
25.02
22.56
56
Ours
64.34
18.03
252.85
21.44
62.94
25.38
22.76
52
RI Solver
33.71
21.29
156.51
96.24
79.63
25.29
22.61
60
FireFlow
34,30
21.91
147.68
96.71 98.27
80.16
25.56
22.67
26
+
31.87
22.31
80.84
25.71
22.98
PSNR and better editing fidelity with reduced computational
Overall, approach achieves
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Zoom: [0.,
PSNR
81.36
20.315
81.38 20.915
20.910
61.35
-83.36
$1.34 20.905
20,900
82.32 20.900
81.32
Zoom: Whole & Edittd E0,0
24.29
Whole
21,890 24.20
21.890
Edito
W
24.27
21,860 24.27
21,860
24.25
20, 07,87 24.26
23.875
24.25
21.870
0.0
0.5
1.0
Figure B.9: Analysis experiment on the effect of the PMT parameter À on Stable Diffusion-3.55 with solver FireFlow (Editing).
Zoom: PSNR & SSIM (e (0,1
23.870
10:75 25.850
82.75
23.565
23.660
62.73
23.655
23,640
82.74
23.650
52.71
23.635
62.73
PSNR
23.640
S5IM
0
16
0.0
0.1
6.5
1.0
Edited (0,
Whole Edited
25:220
25.215
22.38
25.23
25.210
25.22
25.21
22.35
25.200
22.28
25:195
22.35
25.38
22.26
23:190
22.34
Figure B.10: Analysis experiment the effect of the PMI parameter FireFlow.
Table B.4: Editing performance on SDXL-Turbo. Our proximal correction significantly improves PSNR and SSIM while achieving the best CLIP similarity both whole and edited regions.
Background Preservation
CLIP Similarity
Method
PSNR
LPIPS 4
MSE
SSTM x002
Whole f
Edited
GNRI
21.96
126.63
93.27
75.00
24.90
21.98
Burbo-Edit
22.26
107,30
99.96
80.05
25.85
22.74
Direct Inv
22,46
105.53
79.96
80.22
25.64
22.68
DDIM
22.00
118.56
90.16
76.00
24.38
21.48
DDIM Ours
22.49b.49t
105.17 13.91
84.01 6.158
80.27 4.777
25.62 .241
22.70 1.242
a more balanced trade-off between background preservation and editing quality while reducing the required sampling steps.
To verify the generalization capability of our method, we applied our proximal correction a widely adopted diffusion model architecture. As shown in Tab. B.4 incorporated method into the
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Table B.5: Reconstruction performance of SD3.5 with FireFlow.
Method
PSNRT
LPIPS 100 +
MS +
SSIMx100 1
FireFlow
20.91
128.16
189.79
82.32
FireFlow Ours
20.92
127.89
187.86
82.35
Table B.6: Editing performance of SD3.5 with FireFlow.
Method
PSNRT
LPIPS +
MSEx10+ 4
SSIM x x t
Wholet
Edited
FireFlow
20.57 20.67
150.73
137.14
81.52 81.76
24.51
21.71
FireFlow Ours
148.63
135.43
24.58
21.83
DDIM sampler on the SDXI--Turb.. For comprehensive comparison, we also experimented with several ofate-of-the-arr methods on SDXL-Turbo. including DDIM Song al. (2022). GNRI [N itt al. (2025) Turbo-Edit Wu et al. (2024), and Direct Inv Ju et al. (2024). The results demonstrate that method can be effectively applied diffusion-basee models.
We also applied our proximal correction to Stable Diffusion 3.5. As shown in Tab, B.6and Tab, B.5 method can be effectively applied to other RF-based model.
B.10 RECONSTRUCTION ERROR
FireFlow
FireFlow Ours
60
Euler
Euler+Ours
50
1.0
11
12
33
14
35
18
17
1.5
1.9
20
Step
Figure B.11: Reconstruction error (RMSE) across different sampling steps. Our method consistently reduces error compared to FireFlow and Euler.
Fig. B.1l compares reconstruction errors under different solvers. FireFlow shows relatively stable errors, while FireFlow Ours achieves consistently lower values across most steps. Euler suffers from етror accumulation, especially at small step counts, whereas Fuler Ours effectively sup- presses this growth and maintains lower reconstruction error throughout.
C EXPERIMENTAL SETTINGS
COMPUTING INFRASTRUCTURE.
The experiments in this paper were conducted using the following computing infrastructure:
Hardware:
-PUU: A single NVIDIA A100-PCIE-40GB with 40GB memory.
CPU: 10 vCPUs, Intel Xeon Processor (Skylake, IBRS).
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Memory: 72GB RAM.
Software:
Operating System: Ubuntu 22.04.
Python: Python 3.12.
PyTorch: PyTorch 2.5.1.
CUDA: CUDA 12.4.
C.2 INVERSION AND RECONSTRUCTION
We follow the official open-source implementation of FireFlow For Euler, Heun, RF-Solver and FireFlow, the editing configuration see Tab. B.7
Table B.7: Hyperparameter setting in Reconstruction task.
Argument Solver
Euler
Heun
RF-Solver
FireFlow
--gui guidance
2
2
2
2
-sttrt layer index
20
20
20
20
--end layer incex inject
37
37
37
37
1
1
1
reuse_v
1
editing.strategy
replace_v 1.0,1.0,1.0
replace_v 1.0,1.0,1.0
replace_v 1.0,1.0,1.0
replace_v
-av-rratio
1.0,1.0,
Quantitative comparison. For Tab. results are obtained using Flux.1. 1. use 30 Euler steps for the baseline, and 12 steps for Heun, RF-Solver, and FireFlow.
Qualitative comparison. For Fig. we set the same number of function evaluations (NFE) for both conditional and unconditional reconstructions. All methods, with and without our velocity correction, are evaluated using step 8.
C.3 EDITING
We again follow the official FireFlow implementation. For Euler, Heun, RF-Solver and FireFlow, editing configuration see Tab. B.8
Table B.8: Hyperparameter setting Editing task.
Argument/ / Method
Euler
Heun
RF-Solver
FireFlow
--guidance
2
2
2
2
-start,layer.indes
0
0
0
0
--nnd lade
37
37
37
37
-inject
2
2
2
2
reuse v
1
1
editinggstrategy
replace-v 1.0,1.0,1.0
replace-v 1.0,1.0,1.0
replace.v
replace_v
1.0,1.0,1..
1.0,1.0,1.0
Qualitative comparison. For Fig. RF-Inversion is evaluated with step = 26 (Ours) and step = 28 (Vanilla). RF-Solver uses step 12 (Ours) and step 15 (Vanilla). FireFlow uses step 10 (Ours) and step = 12 (Vanilla).
C.4 ABLATION STUDIES.
Ablation study of w. For ablation study of w, FireFlow chosen as the baseline with step = 12 for all experiments in Section 5.4
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Ablation study on the choice of norm. For ablation study the choice of norm., FireFlow is chosen as the baseline with step = 10 for all experiments in Section 5.4
Algorithm 3 PMI: RF-Solver Case
Algorithm 4 mimic-CFG: RF-Solver Case
Input: velocity vg. image Zo. schedule T = tto...., EN
Input: velocity vg. image Zo.
schedule T = (fo ty weight w.
Output: 21
Output: 2g
1: Xdist = 0
1: 21 = PMI (Ug, Zu: t)
2: for = to N do
2: Xdist 0
3: Al li+1 li
3: for i N to do
4:
Vin u0 (t: (z 11)
AI li-1
5:
"t = Z4: + A Vt:
5: VE: = Ug /2, 44)
6:
Vi
RFS = 00 L mid 1 (t141 + +;))
Z 2t, At Vt,
7:
Xdist + Atv. RFS
7:
V RFS uo &
(li+1 4:)
8:
VtL E to Xdist
8:
Xdist + At RFS
FF (v
9:
V2:
Xdist
9:
V RES rt.
F RFS )||2
10:
Vi 11 V.
10: a.. = Ve. Vts
11:
(1 W) + v RFS
At/2
VL. VV.
11:
Z44+1 Ze, + At Vi. + + 1 at:
12: a,
A1/2
12: end for
13: return Z
13: Zt.-1
Zt. + At Atv,, + at
14: end for
15: return 20
Algorithm 5 PMI: Heun Case
Algorithm 6 mimic-CFG: Heun Case
Input: velocity vo. image Zo. schedule T = {fo. , |N
Input: velocity rg. image Zo.
schedule T (to...., (N weight w.
Output: 21
Output: 20
1: Xdist = 0
1: 21 = PMT (Uo,20,1)
2: for to N- do
2: Xdist
3: At tit1 ti
3: for i N to 1 do
4:
Vt. uol Zt. (Z ti)
4: At ti-1
5:
zt.
mid =
2 Vt
5:
= vo (2 tt
mid 2r. At.
6:
Ve
heun uo mii (t1+1 + t4)
6:
z
7:
Xdist + At houn
7:
v t,
heun tig mid
(t+11 + t4)
8:
Vt. h+t1 to Xdist
8:
heun Xaist + Al V
V F v heun
9:
Vt, = 24 1-v Xdist
9: VE. heun rt:
vhoun padi
VF
v heun
10:
Vi:
[ IE Vedit
2611 Zz; A V4
11: end for
11: Vt. (1 w)v p.
+ vt
heun
12: 2 21, + Vive.
12: return Z
13: end for
14: return 20
C.5 HIGHI-OREER ALGORITIIMS
additionally provide the full implementations of PMI and mimic-CFG under several commonly used ODE solvers beyond the Euler case. These vari- ants share the same correction structure while differing in the underlying base velocity estimation. Specifically, Algorithm 5 and Algorithm gorithm correspond to the RF-Solver version, and Algorithm together with Algorithm g show
For better reproducibility,
6 present the Heun implementation, Algorithm 3 and Al-
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the FireFlow integration. We hope these complete solver-specific implementations facilitate further research and replication of our results.
Algorithm 7 PMI: FireFlow Case
Algorithm 8 &immic-CFG FireFlow Case
Input: velocity function ug (.), image Zo. time schedule T = {to .....,t IN 1}.
Input: velocity function (-(.). image Zo, time schedule T {to .,..., IN = 1}, interpolation weight w.
Output: 21
1: Xdist = 0
Output: 20
2: for 0 to N do
21 PMI (vo. Zo.
3: i0 then
2: Xdist 0
4: @-mid uol Ze. 1,)
3: for N to do
5:
else
4: if i= then
6:
previous
5: i-mid = uo (z tx)
7:
end if zt. mid
6: else
8:
Zt, + At Vigid
7:
Vi prevvious
9:
Ve: Ug (2 (l.41 + 4))
8:
end if mid 1;
Xdist (t++1 ve,
9:
2r, + 3 At P.
10:
Vr. uo Lt, mid (t:-1 + +t))
11:
Ve, t: to
Xdist
Xdist
(ti Vi.
(V(v.))
12:
Ve. = =V,,- - rt. [V
12:
vedit
Xdist
4:11 In
13: yprevious V2
13:
VProj
|| vedit
Vi, Vidii
14: 21141 2c, (t,+1 t.)&u
15: end for
14:
Vr. (1 w) Vproj + w V1;
16: return 21
15: vprevious Vt.
16: 2t,-1 = St. + (1:-1 11.
end for
18: return 20
D ADDITIONAL QUALITATIVE COMPARISON
D.1 QUALITATIVE COMPARISON FOR EDITING
We also demonstrate more editing modalities in PIE-Bench, as shown in Fig. D.I our method helps baselines achieve better result Change Content, Change Pose, and Change Material. The in Fig. i.Ilis is generated from RF-Solver, second row from FireFlow, and the third row from Euler.
E THE USE OF LARGE LANGUAGE MODELS (LLMS)
In accordance with the ICLR guidelines, we report the usage of large language models (LLMs) in the preparation of this paper. LLMs were employed solely as a writing assistance tool for language polishing, grammar correction, and improving readability. They were not used for research ideation, experimental design, data analysis, or the generation of scientific content. All technical ideas, the- oretical developments, proofs, and experimental results presented in this paper are the work of the authors. The authors take full responsibility for the final content of the submission.
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Source
Bascline
Ours
source prompt:: photo
field
target prompt: photo of bronze horse in the field
source prompt: detailed ol painting
tanget prompt:
detalled painting a laughing beoubifll women with
source prompt a fox welking in the snow with head down
Figure D.I: Qualitative comparison of editing results PIE-Bench. We considered more edit- ing modalities in PIE-Bench, including Change Content, Change Pose, and Change Material. Our method help baseline achieve better results.
31


Reviews:
[{"review_id": "iQPV4I751k", "reviewer": "Reviewer_TwVa", "summary": "This paper proposes Proximal-Mean Inversion (PMI) to improve the stability and accuracy of inversion in Flow Matching models. The key idea is to perform a proximal correction of the predicted velocity field by guiding it toward a running average of past velocities, thereby mitigating accumulated approximation errors during inversion. The method also constrains updates within a theoretically derived spherical Gaussian region, ensuring stability in high-dimensional latent spaces. In addition, the authors propose mimic-CFG, a lightweight editing strategy that interpolates between the predicted velocity and its projection on the average direction, effectively balancing structural consistency and editability.", "questions": "* The stabilization of inversion and editing velocities is a promising direction. Could this proximal correction mechanism be extended to other tasks, such as inverse problem solving or classifier guidance stabilization (as in [3])?\n* Eq. (9) defines a proximal objective with a simple average of velocities. What would happen if this were replaced by an exponential moving average (EMA) or momentum-based scheme? Would it improve convergence or maintain better global consistency over long trajectories?\n* Could the authors discuss the potential interaction between PMI and high-order solvers like Adams-Bashforth-Moulton or used by RF-Solver or FireFlow? It would be insightful for readers to understand how the entire workflow evolves.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper introduces a theoretically grounded proximal correction framework supported by rigorous stability and error analyses..\n* The PMI formulation is elegant and practical, addresses the instability and accumulated inversion errors in flow-based generative models.\n* Extensive quantitative and qualitative evaluations on PIE-Bench demonstrate consistent improvements across multiple baselines, in both inversion and editing tasks.\n* The paper is well written, clearly organized, and easy to follow.", "weaknesses": "* Although the focus is on inversion-based editing, it would strengthen the paper to compare against a broader set of diffusion and flow-based editing baselines on PIE-Bench.\n* Including the full benchmark results or additional comparisons with diffusion-based methods (e.g., DDIM inversion variants) would provide clearer context.\n* Missing relevant recent works such as InfEdit [1], which explores inversion-free diffusion model based editing, and FlowEdit [3] / FlowChef [2], which also employ flow steering without inversion. These could better situate PMI’s contribution in the broader editing landscape.\n* The proximal objective (Eq. 9) and its averaging strategy appear empirical. It would be interesting to analyze alternative formulations. For instance, exponential moving averages or adaptive weighting of past velocities.\n* While mimic-CFG’s interpolation weight (w = 0.94) is empirically validated, additional discussion on its generalization or task dependency would help establish robustness.\n\n[1] “Inversion-Free Image Editing with Natural Language,” CVPR 2024.\n\n[2] “FlowChef: Steering of Rectified Flow Models for Controlled Generations,” ICCV 2025.\n\n[3] “FlowEdit: Inversion-Free Text-Based Editing Using Pre-Trained Flow Models,” ICCV 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JxIqlJphbw", "reviewer": "Reviewer_miCb", "summary": "The paper introduces Proximal-Mean Inversion (PMI), a novel, training-free gradient correction method to enhance the stability and accuracy of RF-based generative model inversion. The motivation of this paper is the problem where the existing inversion methods suffer from approximation errors that accumulate across timesteps. PMI addresses this by guiding the perturbed velocity field toward a running average of past velocities using a proximal update. The paper also proposes mimic-CFG, a lightweight velocity correction that interpolates between the current velocity and its projection onto the historical average, balancing editing effectiveness and structural consistency. The proposed method demonstrates good results from the expriments.", "questions": "see weakness section.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The proposed methods are techically sounds and supported by theoretical proofs.\n2. The approach achieves state-of-the-art quality on PIE-Bench with fewer sampling steps and no additional NFEs, accelerating the model.\n3. Mimic-CFG provides an efficient guidance mechanism for editing on top of CFG, balancing structural consistency and editing control according to the experiment results.", "weaknesses": "1. The performance of the proposed methods seems to be dependent on hyperparameter selection, and could have potential for overcorrection. Over-correction (small $w$) harms both background preservation and editing quality, despite better SSIM/PSNR improvements. Similarly, for the proximal operator parameter $\\lambda$ in editing, large values can lead to overcorrection, compromising editing quality.\n2. How sensitive is the performance to the hyper-paraetmer across different RF models?\n3. While the method is integrated into several solvers (Euler, Heun, RF-Solver, FireFlow), the key ablation studies (on $\\lambda$ and $w$) are primarily conducted using one base method (Fig. 3). This limits confidence in how well the optimal hyperparameters and correction strategies generalize across different flow solvers or data distributions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z2R4Uqz3oh", "reviewer": "Reviewer_eSEF", "summary": "This paper proposes Proximal-Mean Inversion (PMI), a training-free, gradient-based correction method for stabilizing velocity fields during inversion in rectified-flow (RF)-based generative models. The paper further introduces mimic-CFG, a velocity correction mechanism for editing tasks inspired by classifier-free guidance. Both methods aim to address approximation errors and instability in existing inversion approaches, improving reconstruction quality and editing fidelity. Experiments illustrate the effectiveness of the proposed methods.", "questions": "See weakness", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The motivation of this paper is clear. The inaccuracy of inversion is a noticeable problem of RF-based methods, where many prior works has been attempting to eliminate this. The proposed method aims to address this problem through gradient correction, which is intuitive and promising.\n- The derivation of PMI is mathematically detailed, including closed-form updates (see Proposition 1 and Appendix A.1), and analysis of error bounds are also provided (Proposition 2, Appendix A.3). \n- Experiments are comprehensive, demonstrating the effectiveness of the proposed method on various settings.\n- The paper is well-written and easy to follow.", "weaknesses": "- Some ablation studies are missed. As stated in Line 238, authors use the first-order Taylor Expansion to estimate the objective in Equation (10). Authors are expected to conduct experiments to demonstrate how does the expansion order influence the performance.\n\n- The settings of baseline methods is not clearly stated in the paper. Some hyperparameters in baseline methods are crucial to their performance (such as the feature sharing choices in FireFlow and RF-Solver). Empirically, the flaws of baseline methods in Figure 2 might be addressed through adjusting these hyperparameters. As a result, authors are expected to provide the more detailed information for fair comparion.\n\n- Personally, I think the performance improvement of PMI is marginal, especially in the image reconstruction tasks. As illustrated in Figure 1, in the conditional case, the performance of RF-Solver is good enough. Although the unconditional image reconstruction results of PMI in Figure 1 are better, I think it is not necessary to test the performance under this setting because nowadays the description of images are easy to be obtained (for example, by MLLM), which can be served as the condition.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "siNNiI86NH", "reviewer": "Reviewer_g18M", "summary": "The paper proposes Proximal-Mean Inversion (PMI), a training-free technique designed to stabilize the velocity field during the inversion process of Rectified Flow (RF) models. The core idea is to apply a proximal operator that pulls the predicted current velocity towards a running average of past velocities ($\\bar{v}_{t_{k}}$). This correction is constrained within a theoretically derived spherical Gaussian to prevent trajectory deviation into low-density regions. Furthermore, the paper introduces mimic-CFG for the editing stage, which uses an interpolation strategy (akin to Classifier-Free Guidance) between the current velocity and its projection onto the mean velocity direction to balance editability and structure preservation. The authors claim state-of-the-art performance in reconstruction and editing tasks on PIE-Bench with enhanced efficiency (fewer NFEs).", "questions": "above", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "Strong Theoretical Motivation and Novelty: The identification of accumulated approximation errors leading to velocity field instability is a crucial problem in RF inversion. The proposed solution of using a running mean velocity for proximal correction is novel and intuitively sound, as it leverages the global consistency property of the Rectified Flow trajectory (near-constant velocity field).Zero NFE Overhead: The implementation of PMI as a correction step without requiring any additional Neural Function Evaluations (NFEs) is a significant practical advantage. Many existing inversion refinement methods (like FPI) achieve accuracy at the cost of increased computational load, while PMI promises improved stability for free in terms of NFE count.Theoretical Constraint and Justification: The derivation of the Stability Condition (Proposition 1), which defines the radius $r_i$ for the spherical Gaussian constraint, is a strong point. It provides a principled way to control the correction magnitude, tethering the solution to high-density regions based on instability theory (citing Zhang et al., 2025). This moves the method beyond a simple heuristic.Comprehensive Experimental Validation: The method is evaluated rigorously across four distinct solvers (Euler, Heun, RF-Solver, FireFlow) for both reconstruction and editing tasks, covering a wide range of integration complexity. The consistent quantitative improvements in PSNR, SSIM, and LPIPS across all baselines (Table 1 and Table 2) convincingly demonstrate the general effectiveness and plug-and-play nature of PMI.Effective Editing Strategy (mimic-CFG): The mimic-CFG strategy is a clever application of the PMI principle to the editing context, successfully adapting the highly effective CFG framework using only the internal, historically averaged velocity as the \"unconditional\" signal. This efficiently addresses the trade-off between editability and consistency.", "weaknesses": "1. Reliance on Simplistic Approximation in PMI:The core update (Eq. 13) is derived by minimizing a first-order Taylor approximation of $F(v)$ (Eq. 11) constrained by an $L_2$ sphere (Eq. 12). This reduces the complex proximal minimization to a simple gradient step, making the \"Proximal\" in \"Proximal-Mean\" a somewhat generous term. The simplicity is beneficial for efficiency, but the theoretical depth of the final step is thin.\n\n2. The $\\mathcal{O}(\\Delta t_i^2)$ Error Analysis (Proposition 2) is Misleading:The paper claims the local error remains $\\mathcal{O}(\\Delta t_i^2)$, the same as the standard Euler method. This is expected, as PMI is a correction applied after the Euler step, and it does not change the core differential equation integration method. The proof is trivial and does not justify the benefit of PMI. The true benefit lies in correcting the accumulated errors by leveraging the running mean, not in improving the local truncation error bound, which is solely dictated by the Euler method's order. This proposition adds little value and distracts from the method's actual contribution.\n\n3. Lack of Sensitivity Analysis for $r_{t_{k}}$ (Proposition 1):The Stability Condition is derived based on the full-trajectory Gaussian properties of $\\hat{z}_1$. However, the final parameter choice for $r_i$ (Eq. 14) still includes an arbitrary $\\epsilon$ term (line 793, 794) and is proportional to $\\Delta t_i$. The paper provides no empirical analysis on how $\\epsilon$ is chosen or how sensitive the results are to the constant factor $\\sqrt{2n+3\\sqrt{2n}}/T$. This makes the supposedly \"theoretically derived\" radius look like a complex, slightly obfuscated heuristic, undermining the strong theoretical claim.Limited \n\n\n4. Editing Evaluation:While the quantitative results are strong, the editing evaluation is primarily focused on background preservation (PSNR, SSIM on unedited regions) and CLIP similarity (a weak proxy for true edit fidelity). A critical review requires evidence that the editing is semantically better and not just more consistent with the background. More focused metrics, such as a user study or more fine-grained editing task failures, would be required to fully validate the claimed \"enhanced editing quality.\" The qualitative results in Figure 2 show subtle improvements, but the editing changes are often minimal.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)