Balanced Conic Rectified Flow
Authors: Shin seong Kim
Mingi Kwon
Jaeseok Jeong
Youngjung Uh
Venue: NeurIPS 2025 poster ()
Keywords: Generative models
Flow model
Optimal transport
Curvature
Ordinary Differential Equation (ODE)
Primary area: deep_learning
TL;DR: A rectified flow-based generative model that improves generation quality and straightens ODE solution trajectories by incorporating real images via slerp into the reflow training process.

Abstract: Rectified flow is a generative model that learns smooth transport mappings between two distributions through an ordinary differential equation (ODE). The model learns a straight ODE by reflow steps which iteratively update the supervisory flow. It allows for a relatively simple and efficient generation of high-quality images. However, rectified flow still faces several challenges. 1) The reflow process is slow because it requires a large number of generated pairs to model the target distribution. 2) It is well known that the use of suboptimal fake samples in reflow can lead to performance degradation of the learned flow model. This issue is further exacerbated by error accumulation across reflow steps and model collapse in denoising autoencoder models caused by self-consuming training.

In this work, we go one step further and empirically demonstrate that the reflow process causes the learned model to drift away from the target distribution, which in turn leads to a growing discrepancy in reconstruction error between fake and real images. We reveal the drift problem and design a new reflow step, namely the conic reflow. It supervises the model by the inversions of real data points through the previously learned model and its interpolation with random initial points. Our conic reflow leads to multiple advantages. 1) It keeps the ODE paths toward real samples, evaluated by reconstruction. 2) We use only a small number of generated samples instead of large generated samples, 600K and 4M, respectively. 3) The learned model generates images with higher quality evaluated by FID, IS, and Recall. 4) The learned flow is more straight than others, evaluated by curvature. We achieve much lower FID in both one-step and full-step generation in CIFAR-10. The conic reflow generalizes to various datasets such as LSUN Bedroom and ImageNet.

Balanced Conic Rectified Flow
Shin seong Kim Yonsei University
Mingi Kwon Yonsei University kwonmingi hereiiingg@yyniiiiemmmm .cc.se kr
Jaeseok Jeong Yonsei University
tl cltyd12@yonsei.. ac. kr
jete_jeong@yonsei. ac kr
Youngjung Uh" Yonsei University
yj uh@yonsei. ac kr
Abstract
Rectified flow is a generative model that learns smooth transport mappings between distributions through ordinary differential equation (ODE). The model learns a straight ODE by reflow steps which iteratively update the supervisory flow. It allows for a relatively simple and efficient generation of high-quality images. However, rectified flow still faces several challenges. 1) The reflow process is slow because requires a large number of generated pairs model the target distribution. 2) It is well known that the use of suboptimal fake samples in reflow can lead performance degradation of the learned flow model. This issue is further exacerbated by етrоr accumulation across reflow steps and model collapse denoising autoencoder models caused by self-consuming training. In this work, we go one step further and empirically demonstrate that the reflow process causes the learned model to drift away from the target distribution, which in leads growing discrepancy in reconstruction error between fake and real images. We reveal the drift problem and design a new reflow step, namely the conic reflow. It supervises the model by the inversions of real data points through the previously learned model and its interpolation with random initial points. Our conic reflow leads to multiple advantages. 1) It keeps the ODE paths toward real samples, evaluated by reconstruction. 2) We use only a small number of generated samples instead of large generated samples, 600K and 4M, respectively. 3) The learned model generates images with higher quality evaluated by FID, IS, and Recall. 4) The learned flow is more straight than others, evaluated by curvature. We achieve much lower FID in both one-step and full-step generation in CIFAR-10. The conic reflow generalizes various datasets such as LSUN Bedroom and ImageNet. The project page is available at htttt::/ggraisaaat github. Wtt/CCrrectifiii flox_project.page/
Introduction
demonstrates state-of-the-art image generation with fewer sampling 15 AI] k-rretified flow involves k reflow steps that make ODE paths smooth and straight 227 This allows the model to generate high-quality images simply and efficiently in just one or a few steps. Intriguingly, all rectified flow models (Flux, SD3, and AuraFlow) achieve ofate-of-the-art quality with 1-rectified flow and require about 30 NFEs (number of function evaluations)
Rectified flow 9
28 24
25
23
steps than diffusion models 6
Cooresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

X,
drifeed!
drifted
drifted!
Bectifie flw
Linor interpolation Injoctory
Re-wiring (Za)
(a) Linear trajectory
(b) I-Rectified flow learned by (X, X))
(c) Linear trajectory Z, =Z (1-02
(d) 2-Rectified flow
X, =x (1-0%%
leurned by (Zp Z,)
Figure I: Problem of rectified flow. (a) By randomly pairing data Xo wo and X1 ~ TI, linear interpolation trajectories defined. (b) The 1-rectified flow connects Xo and X1 with learned velocity field which is potentially inaccurate. After the 1-rectified flow, the trajectories are rewired to avoid crossing. (c) The trajectories for reflow are defined linear interpolation trajectories between Zo and the generated Z = v (o). Note that Z is drifted away from Tj. (d) Consequently, the 2-rectified flow has velocity field drifted away from X1.
In this paper, we identify key limitations of k-rectified flow stemming from its reflow procedure, and propose to use real images and their inversions, combined with Slerp- barrp-baedddde perturbation loss, to address them. Our core insight that the degradation in performance cannot fully explained by enror accumulation process itself induces a drift away from the target distribution, leading to a measurable discrepancy between the model's behavior on real and fake samples, which we later analyze through reconstruction error. 1) The flow drifts away from the real distribution. For example, in training 2-rectified flow, random noise vectors and their generated images from the 1-rectified flow are reused supervisory targets, which causes the ODE paths to diverge from real data. We empirically show that this results in reconstruction performance gap between real and fake samples, and that Slerp- sasp--aaedd supervision for real samples helps maintain alignment with the true distribution. 2) The number of fake samples required for reflow is prohibitively large. Our method leverages real samples and their conic neighbors to guide the flow with far fewer generated pairs, while maintaining competitive or superior performance. This efficiency makes the method scalable and less dependent on synthetic supervision. 3) The reflow process degrades image quality in full-step generation. By providing trajectory-level supervision rooted in real data geometry, our method enhances generation quality across 1-step, few-step, and full-step regimes, and mitigates the degradation in standard reflow training.
vanishing weight norms alone [18 [56]. Instead, we observe that the reflow
As a result, we successfully demonstrate better performance than existing k-rectified flow models. On CIFAR-10, 10, we reduce the FID (Fréchet Inception Distance, [4] of the existing 2-rectified flow from 12.21 5.98 while using only 7.2% of the generative pair, and show that the curvature of the ODE paths became straighter. We also introduce new method for calculating curvature, which explains the time distribution sampling method. We provide various ablation studies and show that even with simple fine-tuning, the performance of existing k-rectified flow can be significantly improved.
2 Rectified Flow
Rectified flow 227 is generative model that solves an ordinary differential equation (ODE) to induce transition trajectory between two given data distributions mo and TT. Data X₂ mo and X₃ Ti define linear trajectories X, = (1 X) + tX, for t E [0, 1]. as illustrated in Figure Ih. Then, rectified flow v is an ODE on time t parameterized by 0:
dz, di
uo (Z.,)) ( - E( Xo)XX, = Z,)
We omit 0 for brevity. Liu et al. 27 propose simplified mean squared error (MSE) loss for an ODE neural network to train velocity field v: R - R as follows:
arg min P. X1 Xo - o(xx + (1 1) (11]
(2)
With / ~ Uniform((0,) 1)). image generation tasks, Xo ~ To and X, क, are random noise and real images, sampled from a Gaussian distribution and the data distribution, respectively. Once the model has learned the velocity field, the rectified flow rewires the trajectories a non-crossing manner due to the inherent properties of ODEs, which enforce uniqueness and smoothness in

---

Recon and P- Reeeon Errors for 2.3-rectifie flow
2-RactFled Flow S-Rectifted flow
Fake
Real
Rocossmation Sample
T
π
Secved lampling singling Antattiies
Deference
Figure 2: (a) 2-rectified flow overfits fake samples. Following the reverse and forward 2-rectified flow, fake images inherently return at similar images with or without perturbation at To. In contrast, real images return at different images and it is worse with perturbation, implying overfitting. (b)
Reconstruction discrepancies emerge between real and fake images due the use of fake-only pairs.
trajectory evolution, preventing paths from intersecting in the continuous- dynamics, as depicted in Figure [p. It constitutes a I-rectified flow model denoted by Z RectFlo=((Xx,X)))
The k-rectified flow model learns a straighter sampling trajectory by repeating reflow procedure k times follows. Following Z* from Z/ induces generated pair (Zg Zt) where (Z). Z (Xo, X1). It redefines the linear interpolation trajectory Z as shown in Figure Dp. Then, fine-tuning velocity field using Eq. with (Z). z instead of (Xo. X1) constitutes Z z = RectFlow(( ( 2)
z) 17 for E [0, 1],
According optimal transport theory,
45
10
coupling the generated pairs (Zo, v (o))
ensures that the interpolation trajectory preserves the marginal distributions of the original and target domains, as well as the linear interpolation trajectory between them [21 21. k-rectified flow has superior quality of few-step sampling by straighter sampling trajectory, as shown in lT..
3 Improved Techniques for Reflow Step
In this section, we discuss the generated fake pairs in the original rectified flow and their problems. Then, introduce real pairs and balanced-conic reflow which directly supervise the flow to reach the real data distribution. Finally, we provide detailed training configurations.
3.1 Reflow steps drift the flow away from the real distribution.
As shown by Liu al. (27]. the trajectory Z between the generated pairs in the reflow process becomes smoother and straighter with each iteration of reflow, because the ODE induces a determin- istic smooth solution while preserving the same marginal distribution as the original trajectory. This straightened path is essential for generating high- nigh-quality images with a small number of sampling steps rather than SDE-based generative models [15 37 35). We use subscript F to denote the fake pairs from the original rectified flow as follows:
(Z 2i (Z) z(t)
Where Zá ~ To and - = v v(zt) >
P To simplify notation and avoid confusion, we will denote the fake pair as (Zo.F, Z1.F) when we do not need to consider the reflow step and, denote the k-th order of the rectified flow as (2k p p..
Beyond known issues such as error accumulation and model collapse 18 56], we provide empirical evidence that reflow steps cause distributional drift, observed through growing reconstruction discrepancies between real and fake samples. We provide empirical evidence of the accumulating drifts and suboptimality of the learned rectified flow. Figure 2h illustrates faithfulness and continuity" of a 2-rectified flow on both fake samples and real samples. As expected, fake images are mostly
reconstructed by inversion and generation following the 2-rectified flow, i.e., Zı = v(v (())) Also, an inversion of a fake sample and its perturbation land at similar images, i.e., u(w (100-((2)) (2,))
Forr brevity, denote the forward generation process at the t-th sampling step Xu + fo vx (X., dt :=
of Xo) and hackward process X, + fo of (Xx,) dt == o o"1(),, where n- -B.
We use the notion of continuity as in Lipschitz continuity: the generated images should be similar with the similar latents.
3

---

I-Rectifie
ReReiilled Bow
3-Recilled
KL Divergence over Reflow steps
5.5
5.0
2 Divergence
4.5
4.0
4-Rectifed Bow
5. S-Rocilled Bow
6. c-Rocillied Bow
A
3.5
3.0
2.5
2.0
1.5
KL Divergence
Reflow step
Figure 3:
(a) As reflow steps increase, generated samples diverge from the target distribution, (b)
This drift is further evidenced by the rising KL divergence from the real data distribution.
(Z) +zz) where we perturb it with = 0.05. In contrast, real images lose the main object by
inversion and generation following the 2-rectified flow, i.e., X1 * v( =(v-1(X1)) (X1)). Furthermore, real
images are vulnerable to perturbation on their inversion, i.e., (X,)) F w(r ) + ez). To
evaluate the faithfulness of rectified flow to dataset X, we measure the error between the sample via the flow:
and its reconstruction 4 33
Levon (X) = Ex [|| v(u (z))l()))
Instead of Lipschitz continuity, practically evaluate the continuity of a rectified flow near samples from a dataset X by measuring a perturbed reconstruction error:
Ihp 1-reccon (X,s) Ee
| (x) e2)ll2,
where is the strength of perturbation. The lower perturbed reconstruction error near the real samples T.2 T.p-ccon (X,) indicates the more continuous generative model near the real samples 4 Figure 2 compares I.gcoon of real and fake samples, and I.g p-rcoo near real and fake samples. I.2 I.g.0 is higher at the real samples than the fake samples. indicates that the 2.3-rectified flow drifts away from the real samples. Furthermore, LE p-recco is lower near the fake samples than the real samples. It indicates that the 2-rectified flow suffers from crossing between real samples.
Critically, the original reflow accumulates the drift over recursive reflow steps. This drift is an innate phenomenon because the supervision from shifted distribution does not steer the flow toward the real distribution. Figure provides empirical evidence of the accumulating drift in a toy two moons dataset 32.. The successive reflow steps cause the fake (yellow) data diverge further from the real (blue) target distribution. Furthermore, Figure 3h illustrates the progressive increase in KI. divergence between the fake distribution and the target distribution, providing clear evidence of this phenomenon We provide solution to mitigate this issue in Section 3.3 and 3.4
3.2 Real pair
The previous subsection has unveiled the pitfall of supervision using fake pairs: samples from the domain distribution, e.g., Gaussian, and their codomain following a rectified flow. Instead of the fake pairs, propose to use the real samples and their inverse following reverse rectified flow, defined by:
Real (o, X1) (x1)
(6)
with X, ~ π. where v is the I-rectified flow and we abuse the term real pair although the Zo,n is not real. As in the original rectified flow, where it optionally provided, it is safe and easy to reverse rectified flow without stochasticity because it inherently produces deterministic solution, and using real images does not contradict the original purpose because the noise ez (Xi) is generated using v -1 To avoid confusion, from now on, we will refer to (Xa, u (X,)) as a fake pair (generated pair), where ((X,) is a fake (generated) image, and (Xx)) X1) real pair, where X1 is real image.
Wee measure the reconstruction and perturbed reconstruction error with 1 1-step Fuler sampling.
'The KL divergence measured between Gaussian mixture approximations of the fake and real samples.

---

~ N)
(-= v(Zel)
Zv (Xx))
X
Powwer semping
Fall par
Boverse sumpling
Beel parr
(a) Original rellow
Boctied tow
(b) Conic reflow
Figure 4: Illustration of original fake pairs and our real pairs. (a) The original rectified flow supervises 2-rectified flow with fake pairs (Zo, no vo(Za)): (b) Our conic reflow supervises 2-rectified flow with real pairs (17 (X1), X1) and their conic neighbors.
3.3 Conic reflow
Building upon the basic pairing of real samples with their reverse noises, introduce conic reflow, which expands their influence on the domain distribution to their neighboring areas, as shown in Figure When train the model, noise n and a randomly sampled noise e ~ [(, 7) with the interpolation ratio Ç:
spherical linear interpolation (Slerp) between the reverse
Alerp(Zo,n: c, sin((1 ()
sin(SØ)
sin(Ø)
Zo
sin(d) C,
where e- arccos( Zo n: e) denotes the angle between Zo, n and e. Then define conic inverse from a real sample X1:
e,
1X, + ()slerr(2oo
(8)
where Zo "o (X)), and E [0, 1]. During training, we sample and ( multiple times to let the target flow stochastically cover the nearby domain. As the collection of the paths over multiple iterations looks like cone, we name our method conic reflow. The schedule of interpolation weight Ç is deferred to Section 3.5 Our training objective with conic reflow is:
i = arg min
(' 0
[R] X1 Alerp(Zo,re e, () (Conic(Xx,,,G,0)) de
Where t ~ exp((0, IJ), slerp schedule([O, 1]), ~ (O I), and w, is weighting function (default=1). Slerp commonly used for interpolation in the noise space of generative models, preserves vector magnitudes on the Gaussian hypersphere and enables smooth semantic transitions In case, Slerp serves geometry-aware regularizer that improves the alignment between numerically inverted real samples and the underlying data manifold. Rather than simply reusing generated samples. we supervise the model using real samples and their perturbed inversions to reduce reconstruction loss discrepancies observed during reflow. This approach not only mitigates distributional drift but also aligned with adversarial robustness strategies studied in inverse problems and reconstruction-bue based neural networks [. From this perspective conic reflow provides localized supervision that encourages stability and continuity around real data points in the noise space.
47 16.
3.4 Balanced conic rectified flow
We design new reflow procedure which consists of our conic reflow and the original reflow. For each training iteration, we design different training schemes for real pairs and fake pairs. We alternate between conic reflow steps with real pairs (Za.n, X1) and original reflow steps with fake pairs. It encourages the trajectories to head toward the exact real distribution while fake pairs ensure the entire domain distribution receive supervision. Fake Pairs: For fake pairs, proceed with the reflow process exactly as it done in the original rectified flow model. Its training objective Ô as follows:
arg min E [] F Zo.F V0 (1 1) =)
(10)
5

---

cancc sarb
Noise Scale
Total Reflow Noise Schedule
Original leter
Dura
Bassite
Copert
Not Scade
Orginall falon
Txpestin Rod Far
Cemplle ti
No of 2-Bee Fied Riw
VD 2.43
Training stops
1500
Training steps
1930
Traiing staps (10*)
Towining steps (LO7)
Figure 5: (a) Slerp noise schedule for single conic reflow and (b) Total slerp schedule. (c) Curvature and IVD during training, and (d) From 2- to 3-rectified flow.
with t exp([O, 1J). The entire training objective of our method as follows for the given fake oo FL.F) and real pair RZ X1):
min [[] (2(F-vo(ZeF))
vol (2F()))) + Xieat (X1- slerp(Zo,as (Conic( X1, s.0))
(11)
where Xfake and Xreal are indicator functions for given index subsets Useal and Utaee such that Urea U Unke = N. Then, for E N:
if E Unake
{
if E Ureal
Xfake
0
else
Xreal
else.
(12)
The notation Ç, Conic(.). and follows the definitions in Section 3.3
3.5 Detailed training schemes
In this section, we provide a more detailed explanation of our proposed training schemes, including visualizations of the Slerp scheduling and maximum Slerp noise magnitude. The maximum magnitude of the Slerp noise, Smax (0, 0.5] selected in accordance with our intuition that is the point where the discrepancy between the perturbed reconstruction errors of real and fake samples is maximized. After a warm-up phase of training steps, we compute Suax using 10,000 images each from the real and fake. The value is selected based on the following hyperparameter <max:
max
max (E(0.0.5)
E X
[lleo (Slerp(za,n, - xx|12 - ||v (Slerp(zo, =||2]
(13)
Z~Z
where (O0 I), with 50, = (x) and 20,F U/g F= denoting the inverse noise estimates for the real and fake samples, respectively. Figure 5[a) shows a single conic reflow noise schedule in training steps (For CIFAR-10 and ImageNet, we set (max to 0.13 and 0.23, respectively). Each conic trained to progressively reduce the noise scale over time. Specifically, the Slerp noise schedule
<((') is defined as ((t') { T+erz, t' E 1J, where t' = 1 corresponds to the start of training and t' = to the end. This design follows the intuition from traditional diffusion models [[5], where noise is progressively reduced to guide samples toward realism. To strengthen the real image trajectory during training, periodically update the real sample pairs used in the reflow process. Figure b) illustrates an example of conic reflow with updated real pairs, assuming 220K total training steps for visualization. When the number of updates is 2K, we schedule the maximum noise magnitude mmux by scaling it according to the pattern [K, K-1, 1, 2..., K], assigning <max/ the smallest value and çuax to the midpoint. The noise increases linearly in the first half and decreases symmetrically in the second half. The full pseudocode for our training method is provided in Appendix K
2r
Experiments
We conducted experiments to evaluate the effectiveness of our method. Our findings demonstrate: Superiority over original reflow terms of (1) Quality of the results, (2) Straightness of the flow, (3) Mitigation of distribution shift, well (4) Ablation study, (5) Generalization to other datasets.

---

(FR
1T
Inception (IS) Score
Ont-Step Generation (Euler solvee, N=1)
9.25
I-Rectifedd Flow
(.15 378 (6.18)
2-Rectifeed Flow
Original (ioil))
8.08 (9.00)
12.21 (4.55)
rectified
distiiiours)
Ours (+Doell)
8.79 (9.11)
5.98 (4.16)
5.50
3-rectited flawlours)
3-distitours)
Rf-- 23
8.87
4.43
2-rectifidd flow(baseline
2 (isti((aaeelnn))
(rours)
3-rectmeed tlowbbaseline
distiicasseline)
Ohiginal (4 (+Donil)
3 Reciifed Fiow
8.47 (8.79)
(5.21)
100
16
16a
10'
Ours (+Dimill)
5.84 (8.96)
5.48 (4.88
FID (Fréchet Inception Distance) Score
Full Simulation (Rumge Kutta (RK45),
Adapiive
2-rectifled flowlours)
2-distitours)
Rectitted Flow
127
9.60
2.58
10-
3-rectified flawlours]
3.distilours)
Recrifeed Flow
2-rectifled flowbbaseline
2dstii(beeellne)
3-rectified faawlbaslline
s-disttchaselinn)
9.30
2 R-Rectjhed Fiew Original
104
9.01
3.96
Ours
9.14
370
10a
Table 1: One-step and full- simulation comparison of 2.3
Euler steps (log Scale)
Figure CIFAR-10 generation quality across Euler steps.
Rectified Flows on CIFAR-10.
Experimental setup Most our experiments are conducted on CIFAR-10 (20]. The IVD, curva- ture, reconstruction, and perturbed (0.05e, 1-step) reconstruction error values reported were computed using 10,000 random samples, with the expectation taken over these samples. We employ Scipy's 's (665 5(4) Runge-Kutta method with adaptive step size and step count determined by specified tolerances, following the same parameters 37 Further details on the training configurations are provided in Appendix[1
4.1 Image quality
Our method achieves better FID and IS scores across all sampling steps, i.c., 1-step. few-step. and full-step generations, as shown in Table and Figure Notably, significantly fewer than the 4M fake pairs used in the original rectified flow, demonstrating the efficiency of our approach. Furthermore, our method outperforms the original in generation quality with RK sampling, even with a lower NFE. We also apply our method to Rectified++ [24) and compare generation quality. While RF++1 uses 800K synthetic pairs, our variant uses 600K fake pairs and 50K real images, achieving better FID. This indicates that our method generalizes well to reflow-based generative models. Detailed settings for RF++ are provided in Appendix J
only 300K fake pairs,
Additionally, our method produces images with superior quality even when applying the same distillation in the original rectified flow [27) as compared to solid (ours) and dashed (original) lines, and star (ours) and rectangle (original) in Figure These results suggest that our method produces more favorable initial velocity field than the original rectified flow. A detailed comparison of precision and recall scores is provided in Appendix B Furthermore, we show in Appendix D that fine-tuning the original rectified flow using only real images and their reverse noise pairs effectively reduces reconstruction discrepancy and quickly improves generation quality. Additional performance comparisons more complex datasets and higler-resolution images are presented in Sections 4.5 and 4.6 To further contextualize the performance of reflow-based methods, we also report the unconditional generation quality of pretrained diffusion models CIFAR-10 in Appendix N
4.2 Straightness
We evaluate trajectory straightness using curvature, following prior works [27 23 Straighter trajectories reduce discretization error under few step solvers, improving sample quality [3 For any continuously differentiable process Z {2.} the curvature is measured by
/
E [Zi 2o) 2, |
S(Z)
dt
(14)
Additionally, it is known that S(Z) indicates exact straightness.
Relationship between curvature and initial velocity delta (IVD) While curvature captures trajectory straightness, it does not fully explain 1-step sampling quality, which depends solely on the initial velocity. We propose Initial Velocity Delta (IVD) to directly evaluate the accuracy of the
7

---

initial velocity and its impact on 1-step generation. The calculation method for IVD is provided in the equation below:
IVD (o) -0 - 2o) Zco
(15)
Curvature and initial velocity delta. Our approach demonstrates improved trajectory straightening and better preservation of the initial velocity direction. As shown in Figure []c), our method consistently achieves lower curvature and IVD values than the original, indicating a more stable trajectory even with fewer fake pairs. Furthermore, Figure highlights that applying additional reflow step from 2-rectified to 3-rectified flow further reduces both curvature and IVD, reinforcing the effectiivnees of our training method.
4.3 Reconstruction with perturbation to address drift from the real distribution
We empirically show that incorporating real data pairs via Slerp-based supervision in the reflow step significantly improves the preservation of real image trajectories. As illustrated in Figure aa). the original rectified flow exhibits a clear reconstruction error gap between real and fake images, whereas our method progressively narrows this gap during training. Figure (b) highlights two key aspects. First, in terms of reconstruction error, our method better preserves real trajectories and mitigates overfitting fake samples. Second, the lower perturbed reconstruction error suggests that the model more accurately aligns the velocity field around real images, leading to improved robustness against perturbations.
4.4 Ablation study
We report results from an ablation study on various settings of our proposed framework, comparing four configurations: (1) without Slerp noise; (2) reflow using just single real pair; (3) our full method; and (4) the original 2-rectified flow. The comparison focuses on 1-step generation quality, curvature, IVD(Initial Velocity Delta), reconstruction error, and perturbed reconstruction error.
4.4.1
Benefits of incorporating real data via slerp-based perturbation training
Model
FID
IS
Curvature
IVD
Recon Real Recon Fake Perturbed Real Perturbed Fake
Original
12.21 8.08 0.002837 0.295078
0.033668
0.024106
0.035481
0.026270
Ours* (Slerp conic)
5.98
8.79
0.002295
0.253334
0.019404
0.023139
0.022382
0.025206
Fixed Real
8.59
0.002313
0.242444
0.020227
0.020607
0.022890
0.022914
No Slerp
6.60 8.57 0.002322
0.240884
0.023380
0.020154
0.026063
0.022496
Table 2: An ablation table comparing various 2-rectified models across multiple metrics: FID, IS (I-step), Curvature, IVD, and errors (Recon Real/Fake, Perturbed Real/Fake).
Benefits of adding noise via Slerp Adding noise via Slerp avoids trajectory crossover, preserving straighter path relative to real images. This leads to improved trajectory quality and enhances 1-step sampling efficiency. Moreover, using Slerp results in lower reconstruction error for real images compared to not using it. This demonstrates that Slerp- based noise perturbation helps stably maintain the trajectory and local neighborhood structure of reverse noise corresponding to real images. In particular, it plays key role in reducing the reconstruction discrepancy we observe between real and fake samples during reflow, thereby effectively mitigating the distributional drift that occurs later iterations.
L2 Recon Error
L2 P-Recon Error
Recon and P-Recon Errrs and Differences
2-RF ginal
2.RF Ours
3-RF Original
Error
3-RF Ours
Difference of Recon error
Difference of P-Recon emor
Training steps (10*)
Training steps (10*)
La Faico
L, Real
Fake
Real
Figure 7: Reconstruction and perturbed reconstruction error across training iterations.

---

Effect of real pair injection and refreshing Incorporating real pairs into training improves trajectory fidelity and 1-step generation quality by anchoring learning closer to the target distribution. Notably, fixing the real pair only once leads to inferior performance, while periodically refreshing reverse noise achieves better FII,, IS, and lower reconstruction emors. This highlights the importance of continuously updating real-sample guidance throughout training to maintain alignment with real data trajectories.
Additional experiments, including k = 4 generation quality, precision/recall, low fake pair settings, and real-only supervision, are provided in Appendices C and
4.4.2 Slerp noise patterns and lerp
In this section, we show that our Slerp-based reflow method consistently improves 1-step generation quality, even when the Slerp noise pattern varies. To support this claim, we train 2-rectified flow models using different Slerp schedules, including patterns where the noise gradually increases, gradually decreases, or first increases and then decreases. We also include a baseline where Slerp replaced with Lerp for comparison. All experiments are conducted with batch size of 256, and training is performed for 300K iterations. Each setting uses 300K fake pairs and 60K real pairs for training on Cifar 10. Other configurations remain identical to those described in Section and Appendix
Category
Method
Schedule Type
IS (T)
FID (L)
Ours
CR
/ /K -> C - (.tax /K max. m K Cuua
8.72
6.63
Slerp
Strictly Increasing
8.48
6.64
Strictly Decreasing
max, maax - yuax /K
8.45
6,70
Lerp
Linear Interpolation
(1- . - (Xi) + C. [
8.46
7.50
Table 3: Each method uses Slerp or Lerp-based noise trajectories. The best and second-best values bolded and underlined, respectively.
As shown in Table 3] observe three key findings. First, Slerp-based schedules consistently outperform Lerp in both FID and IS, suggesting that Slerp more effectively preserves the reverse noise trajectories of real images. Second, the strietly increasing noise schedule yields better performance than the decreasing one, indicating that progressive noise injection improves training stability. Third, our method achieves the highest IS while maintaining comparable FID to the best baseline, demonstrating improved sample diversity without compromising generation quality.
4.5 More complex dataset (Unconditional generation on ImageNet 64x64)
On ImageNet 64 6 77, our method consistently improves unconditional generation quality over the original model. Using the same setup CIFAR-10, we train I-rectified flow for 700K steps with batch size 256. The original uses 1M fake pairs, while our method uses 600K fake and 60K real pairs. As shown in Table 4 Although 60K real images may be insufficient to fully cover the target distribution of ImageNet compared to CIFAR-10, 10, our method still yields substantial gains in FID and Recall. This suggests that even limited real data can provide strong guidance in mitigating distributional drift. Moreover, it improves reconstruction error, perturbed reconstruction error, recall, and precision (see Table 5 and Appendix F), demonstrating its ability to reduce distributional drift even on large-scale datasets.
4.6 High-resolution image generation
In this section, we assess the generalizability of our method on the LSUN Bedroom dataset [52 at resolution of 256x256. We use the same hyperparameters, time schedule, and EMA settings as in the experiments by Liu al. [27]. We use 60K fake pairs and 5K real pairs, while the original uses 120K fake pairs. Despite GPU limitations on the larger LSUN dataset, our method consistently outperformed the original in image quality. Figure X shows superior few step generation quality, and with adaptive step sampling (RK45), our approach achieved comparable quality with significantly fewer fake pairs than the original. Further details on the training configurations are provided in Appendix [ For comparisons under different random seeds, refer to Appendix !

---

Metric
FID
1S
Model
Solver
Precision
Recall
Solver
RK-45
RK-45
Euler
Original
Fuler I 1-step
0.4129
0,4604
1-rectified flow
23.1
369.8
12
1.1
Ours
Euler 1-step
0.3812
0,5325
2-rectified
31.2
39.7
Original
RK-45 RK-45
0.4717 0.4432
0.5264
Ours
28.2
37,8
11.4
10.3
Ours
0.6032
Table 4: FID and IS across rectified flows
Table 5: Precision and recall on ImageNet.
using RK-45 and Euler solver on ImageNet.
(Original/Ours)
Precision
Iistep Euler
139.98 26.34
0.0290 / 0.4822 0.0220/ 0.2274 0.4525 / 0.4703
RK
24.76 / 24.14
0.2386 / 0.2388
IVD, Recon, P-Recon error (Fake/Real)
IVD
Lrecon
L
Original
1.1790 0.9103
0.0822 /0.1147
0.0820 0.1164
Ours
0.0487 0.0405
0.0486 / 0.0407
A Meccon
L recon
Original
0.0325
0.0326
Ours
0.0083
0.0079
Euler 1-step
Euler 2-step
RK
Figure 8: Visual and quantitative comparison on LSUN. Left: 2-row layout showing original (top) and ours (bottom) for each solver (I-step, 2-step, RK). Right: evaluation metrics.
5 Related work
Recent efforts to improve rectified flow models focus on modifying the time distribution, loss functions, or model architectures 24 Some works incorporate real data pairs introduce discriminator- based regularization to reduce the effect of out-of- distribbbtt samples [18). Others analyze model degradation through the lens of denoising autoencoders, attributing performance drop to the vanishing of weight norms caused by repeated training synthetic pairs [56]. Perfilow takes different approach by constructing piecewise linear intermediate paths to stabilize reflow training
(50) In contrast these works, our method introduces perturbation-ban supervision using real data inversions.
Conclusion and future work
We propose balanced conic reflow, simple yet effective method that addresses key limitations of traditional rectified flow by incorporating real pairs through Slerp-based perturbation strategy.
Our approach improves generation quality across multiple settings, while requiring significantly fewer synthetic samples. Its plug-and-play nature makes it compatible with various flow-based generative models, such as InstaFlow and SD3 28 9. Future directions include extending our method with additional loss functions or integrating it with diffusion-basee synthetic supervision Moreover, our strategy may complement recent frameworks such as Rectified Diffusion 48). which demonstrate that reflow-like improvements are possible even without retraining u-prediction- based flow matching models, where conic reflow can further help preserve real image trajectories and reduce reconstruction discrepancy without relying on linear interpolation paths discriminator-bast based supervision.
24 18
Acknowledgment
This work partly supported by the National Research Foundation of Korea RCS-2023-0022306 and an ITP grant (RS-2020-I1201 3200111233 RS-2002-000339662 funded by the Korean government (MSIT).
10

---

References
[1] Jonas Adler, Emil Kobler, Sebastian Lunz, Carola-Bibians, Schönlieb. and Simon Arridge. Task adapted reconstruction for inverse problems. In International Conference on Machine Learning, pages 74-84. PMLR, 2021.
[2] Luigi Ambrosio, Gianluca Crippa, Camillo De Lellis, Felix Otto, Michael Westdickenberg. Luigi Ambrosio, and Gianluca Crippa. Existence, uniqueness, stability and differentiability properties of the flow associated to weakly diffcrentiable vector fields. Transport equations and multi-D) hyperbolic conservation laws, pages 3-57, 2008.
[3] Velin Antun, Francesco Renna, Clarice Poon, Ben Adcock, and Anders C Hansen. On instabilities of deep learning in image reconstruction and the potential costs of ai. Proceedings of the National Academy of Sciences, 1 17(49))30088-30055 2020.
[4] Emmanuel J Candes, Justin K Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccrraee measurements. Communications on Pure and Applied Mathematics: Journal Issued by the Courant Institate of Mathematical Sciences, 1207-123,, 2006.
[5] Hongrui Chen, Holden Lee, and Jianfeng Lu. Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learning, pages 4735-4763. PMLR, 2023.
[6] Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34:176 14:17695177709 2021.
[7] Jia Deng, Wci Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenct: A large-scale hicrarchical image database. In 2009 IEEE Conference on Computer Vision and Puttern Recognition (CVPR), pages 248-255. IEEE, 2009.
[8] P Kingma Diederik. Adam: A method for stochastic optimization. (No Title), 2014.
[9] Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling rectified flow transformers for high-resolution image synthesis. In Forty-first International Conference on Machine Learning, 2024.
[10] Alessio Figalli and Federico Glaudo. An invitation to optimal transport, wasserstein distances, and gradient flows. Springer, 2021.
[11] R Flamary, N Courty, D Tuia, and A Rakotomamonjy. Optimal transport for domain adaptation. IEFE Trans. Pottern Anal. Mach. Intell, 2016.
[12] Zhengyang Geng, Ashwini Pokle, William Luo, Justin Lin, and J. Zico Kolter. Consistency models made easy, 2024. URL httpss//s//aaxi org/ hbrg/abb//4000 14548
[13] Zhengyang Geng, Mingyang Deng. Xingjian Bai Zico Kolter, and Kaiming He. Mean flows for one-step generative modeling, 2025. URL hattss:///arrttee org/ 2205. 13447
[14] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by two time-scale update rule converge to a local nash equilibrium. Advances neural information processing systems, 30, 2017.
[15] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33:6840-6855 2020.
[16] Young Kyun Jang, Dat Huynh, Ashish Shah, Wen-Kai Chen, and Ser-Nam Lim. Spherical linear interpola- tion and text-anchoring for zero-shot composed image retrieval. arXiv preprins arXiv:2405.00577 2024.
[17] Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion based generative models. arXiv preprint arXiv:2206. 00364, 2022.
[18] Beomsu Kim, Yu-Guan Hsieh, Michal Klein, Marco Cuturi, Jong Chul Ye, Bahjat Kawar, and James Thornton. Simple reflow: Improved techniques for fast flow models. arXiv preprint arXiv:2410.07815. 2024.
[19] Dongjun Kim, Chich-Hsiee Lai, Wei- Wei-Hsiang Liao, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yutong He. Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory models: Learning probability flow ode trajectory of diffusion. arXiv preprint arXiv:2310.02279, 2023.
22

---

[20] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, University of Toronto, Toronto, ON, Canada, 2009. URL lattps://wwx.cc ca. toronto.ooooo..o edu/ ktriz/learning-eatteeee features- TR. pdf Technical report.
[21] Thomas G Kurtz. Equivalence of stochastic equations and martingale problems. Stochastic analysis 2010, pages 113-130, 2011.
[22] Tuomas Kynkäänniemi, Tero Karras, Samuli Laine, Jaakko Lehtinen, and Timo Aila. Improved precision and recall metric for assessing generative models. Advances in neural information processing systems, 32, 2019.
[23] Sangyun Lee, Beomsu Kim, and Jong Chul Ye. Minimizing trajectory curvature of ode-based generative models. In International Conference Machine Learning, pages 18957-18977 PMLR, 2023.
[24] Sangyun Lec, Zinan Lin, and Giulia Fanti. Improving the training of rectified flows. arXiv preprint urXiv:2405.20322 203300 2024.
[25] Hangyu Li, Xiangxiang Chu, Dingyuan Shi, and Lin Wang. Flowdreamer: exploring high fidelity text-to-3d generation via rectified flow, 2024. wttptpttttttt......a//paa/a//att
05008
[26] Yaron Lipman, Ricky T. Q Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling, 2023, URL htttps://arxiiii org/ abs/2210 2210/2220000... 02747
[27] Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.0300 2022.
[28] Xingchao Liu, Xiwen Zhang, Jianzhu Ma, Jian Peng, et al. Installow: One step enough for high- quality diffusion-based text-to-image generation. In The Twelfth International Conference on Learning Representations, 2023.
[29] Cheng Lu and Yang Song. Simplifying, stabilizing and scaling continuous-timn consistency models, 2025. URL hhttts::///xtiiii /ttp:////axivv/// org/ abaaabb//441 2410.110888
[30] Eric Luhman and Troy Luhman. Knowledge distillation in iterative generative models for improved sampling speed. arXiv preprint arXiv:2101.02.3 2021.
[31] Weijian Luo,, Tianyang Hu, Shifeng Zhang, Jiacheng Sun, Zhenguo Li, and Zhihua Zhang. Diff-instruct: A universal approach for transferring knowledge from pre-trained diffusion models. Advonces in Neural Information Processing Systems, 36, 2024.
[32] Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss. Vincent Dubourg, Jake Vanderplas. Alexandre Passos, David Cournapeau, Matthieu Brucher. Matthieu Perrot, and É.douard Duchesnay. Scikit-learn: Machine learning in python. Journal of Machine Learning Research, 12:2825-28 28300, 2011.
[33] Saiprasad Ravishankar, Jong Chul Ye, and Jeffrey A Fessler. Image reconstruction: From sparsity to data-adaptive methods and machine learning. Proceedings of the IEEE, 108(1):86-110) 2019.
[34] Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling diffusion models. arXiv preprint or Xiv:2202. 00512, 2022.
[35] Jiaming Song. Chenlin Meng. and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02500 2020.
[36] Yang Song and Prafulla Dhariwal. Improved techniques for training consistency models. arXiv preprint 10. 189 2023.
[37] Yang Song, Jascha Sohl- -ick-Dics,,ein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.1355. 2020.
[38] Yang Song. Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. arXiv preprint arXiv:2303.01469, 2023.
[39] Hans Stetter et al. Analysis of discretization methods for ordinary differential equations, volume 23. Springer, 1973.
[40] Alexander Tong. Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-basor generative models with minibatch optimal transport, 2024. URL htttps://arxivs .rgg 2b22//2002202
23

---

[41] Belinda Tzen and Maxim Raginsky. Theoretical guaranlees for sampling and inference in generative models with latent diffusions. In Conference on Learning Theory, pages 3084-3114. PMLR, 2019.
[42] Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-hased generttive modeling in latent space. Advances in Neural Information Processing Systems, 34:1 14:111277-1302 2021.
[43] Francisco Vargas, Pierre Thodoroff, Austen Lamacraft, and Neil Lawrence. Solving schrödinger bridges via maximum likelihood. Entropy, 23(9):1134, 2021.
[44] Cedric Villani. Optimal transport: old and new. Springer, 2009.
[45] Cedric Villani. Topics in optimal transportation. American Mathematical Soc, 2021.
[46] Pauli Virtanen, Ralf Gommers, Travis Oliphant, Matt Haberland, Tyler Reddy, David Cournapcau, Eygeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, et al. Seipy 1.0: fundamental algorithms for scientific computing in python. Nature methods, 17(3):261-273 2020.
[47] Clinton Wang and Polina Golland. Interpolating hetween images with diffusion models, 2023. URL astpss:/openreeviee. net/ppff/dd-LDD0yxxxxx OpenReview preprint.
[48] Fu- Yun Wang, Ling Yang. Zhaoyang Huang, Mengdi Wang, and Hongsheng Li. Rectified diffusion: Straightness is not your need in rectified flow. arXiv preprint arXiv:2410.07303, 2024.
[49] Yilun Xu, Ziming Liu, Max Tegmark, and Tommi Jaakkola. Poisson flow generative models. Advances in Neural Information Processing Systems, 35: 1672--6666,, 2022.
[50] Hanshu Yan, Xingchao Liu, Jiachun Pan, Jun Hao Liew, Qiang Liu, and Jiashi Feng. Perflow; Piecewise rec- tified flow universal plug- Boug-and-pll play accelerator, 2024. URL |RL //arxiv
07510
[51] Tianwei Yin, Michaël Gharbi, Richard Zhang, Eli Shechtman, Fredo Durand, William T Freeman, and Tacsung Park. One-step diffusion with distribution matching distillation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6613-6623, 2024.
[52] Fisher Yu, Ari Seff, Yinda Zhang. Shuran Song. Thomas Funkhouser, and Jianxiong Xiao. Lsun: Con- struction of large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv: 030.0033,,, 2015.
[53] Hongkai Zheng, Weili Nie, Arash Vahdat, Kamyar Azizzadenesheli, and Anima Anandkumar, Fast sampling of diffusion models via operator learning. In International conference on machine learning. pages 42390-42402. PMLR, 2023.
[54] Linqi Zhou, Stefano Ermon, and Jiaming Song. Inductive moment matching, 2025. URL https
25503333 2503.07565
[55] Mingyuan Zhou, Huangjie 7heng, Zhendong Wang, Mingzhang Yin, and Hai Huang. Score identity distillation: Exponentially fast distillation of pretrained diffusion models for one-step generation. In Forty-first International Conference Machine Learning, 2024.
[56] Huminhao Zhu Fangyikang Wang, Tianyu Ding, Qing Qu, and Zhihui Zhu. Analyzing and improving model collapse in rectified flow models. arXiv preprint arXiv:2412.08172 2024.
[57] Yuanzhi Zhu, Xingchao Liu, and Qiang Liu, Slimflow: Training smaller one-step diffusion models with rectified flow, 2024, URL https://aaa http//axxvv.rrr/bbb htsp///a/2222.... 14711 12718
Additional References
[A1] Stanczuk, Jan Pawel. Georgios Batzolis, Teo Deveney, and Carola-Bibiane Schönlieb. Diffission models encode the intrinsic dimension of data manifolds. International Conference on Machine Learning (ICML), 2024.
[A2] Mingi Kwon, Shin seong Kim, Jaeseok Jeong. Yi Ting Hsiao, and Youngjung Uh. Tefg: Tangential damping classifier- free guidance. 2025. URL http://aaaxi..or///b/////.....1
[A3] Dhariwal, Prafulla and Alexander Nichol. Diffussio models beat GANs on image synthesis. Advances in
Neural Information Processing Systems (NeurPPS), 34:8780-8794, 2021.
24

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: We have summarized our contributions at the end of the introduction.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize other settings.
It is fine to include aspirational goals as motivation long it is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
See Appendix
Guidelincs:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
authors are encouraged to create separate "Limitations" section their paper.
The paper should point out any strong assumptions and how robust the results are violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well- spec--sifciition asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets with few runs. In general, empirical results often depend implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures hecause fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If l'applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 'tent acknowledged in the paper. The authors should that individual actions in favor of transparency play an important role developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
their best judgment and recognize
Theory assumptions and proofs
Question: For each theoretical result. does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [NA]
Justification: The paper does not present formal theoretical results proofs. Instead, focus empirically analyzing and mitigating distributional drift during the reflow process.
Guidelincs:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
25

---

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core
paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: We describe the full training mechanism of our method, including hyperparameter settings, in the Section [35 and Appendix [ Additionally, provide the pseudo algorithm and detailed training configurations for each dataset in the Appendix
Guidelines:
The answer NA means that the paper does not include experiments.
the paper includes experiments, No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data provided or not.
If the contribution is datasct and/or model, the authors should describe the steps taken make their results reproducible verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, contribution novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data often good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (c.g., in the case of a large language model), releasing of model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require relcasing code, the conference does require all submissions provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily reproduce that algorithm.
algorithm, the paper should make clear how
(b) If the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results way to reproduce the model (e.g.. with open-source dataset instructions for how construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access the model is limited in some way (e.g., to registered users), but it should be possible for other rescarchers to have some path to reproducing or verifying the results.
Open access to data and code
the data and code, with sufficient instructions faithfully reproduce the main experimental results, described supplemental material?
Question: Does the paper provide open
Answer: [Yes]
Justification: We provide a partial implementation with core components in Appendix M and plan to release the full codebase and instructions by the camera-ready deadline.
Guidelincs:
The answer means that paper does not include experiments requiring code.
Please see NeurIPS code and data submission guidelines http:::://iip.cc/publil for more details.
guides/Code Stte///oooSuumuiiiiinonoe
While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
26

---

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines htttps://nisa.cc/ppin
Buidee//DodeSuumiissioooPPo
for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for new proposed method and baselines. If only a subset experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release amonymized versions (if applicable).
Providing as much information possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify training and test details (e.g.. data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: We provide clear guidelines for choosing the perturbation magnitude in the Section 3.5 and describe all model configurations including the optimizer in both the Experiment Setup Section|4 and the Appendix J
Guidelines:
The answer means that the paper does not include experiments.
The experimental setting should be presented in the core the paper to level of detail that necessary to appreciate the results and make sense them.
'Te full details can provided either with the code, appendix, or as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate informa- tion about the statistical significance of the experiments?
Answer: [Yes]
Justification: We explicitly state the number of data points used to compute the mean values of curvature, initial velocity delta, and reconstruction error in Section I [and Appendix The magnitude the epsilon used to calculate the perturbed reconstruction error also clearly described. Additionally, we specify the number of synthetic samples used for evaluating generation quality metrics such as recall, precision, and FID.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error hars, confidence intervals, statistical significance tests, least for the experiments that support the main claims the paper.
factors variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
It is OK to report I-sigma error bars, but one should state it. The authors should preferably report 2-sigma error bar than state that they have 96% CL. if the hypothesis of Normality of errors not verified.
For asymmetric distributions, authors should be careful not show in tables or figures symmetric error bars would yield results that are out of range (e.g. negative error rates).
error bars are reported tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
27

---

Answer: [Yes]
Justification: We provide the details of our CPU and GPU resources the appendix A Additionally, the number of training iterations, batch size, and the number of fake and real samples used in each experiment specified in the experiment section aand Appendix
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary failed experiments that didn't 't make into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https: btttpa:/neeripsscoo wwtperippp.cccpuliiicccc httppp/c///tchaettueelliii.eee.e
Answer: [Yes]
Justification:
We confirm that the research conducted in this paper adheres to the NeurlPS Code of
Ethics.
Guidelines:
The answer means that authors have not reviewed the NeurlPS Code of Fthics.
the authors answer No, they should explain special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [No]
This work analyzes distributional drift in reflow and proposes technical solution for flow-based generative models. Since it does not involve application specific deployment or sensitive content, we found no broader socictal impacts to discuss.
Justification:
Guidelines:
The answer NA means that there is societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has societal impact why the paper does address societal impact.
Examples of negative socictal impacts include potential malicious unintended uses (e.g.. disinformation, generating fake profiles, surveillance), faimess considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy consider- ations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point out. For example, the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
legitimate to point out that an improvement in
The authors should consider possible harms that could arise when the technology is being used intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
there are negative societal impacts, the authors could also discuss possible mitigation strategies (c.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitor- ing misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency accessibility ML).
11. Safeguards
28

---

Question: Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g.. pretrained language models, image generators, scraped datasets)?
Answer: [NA]
Justification: This work does not involve any models or datasets that pose foreseeable risk of misuse or harm.
Guidelines:
The answer NA means that the paper poses no such risks. Released models that
high risk misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors take this into account and make best faith effort.
12. Licenses for existing assets
Question: Are the creators original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: We used the official repositories of existing flow models 27 and clearly cited the datasets used for training (CIFAR-10 (00), LSUN [52]. ImageNet (7)). We also specified the version
of the Runge- Kutta sampling tool (from SciPy ) used during sampling [46
Guidelines:
The answer means that the paper does not use existing assets.
authors should the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from particular source (e.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should he provided. For popular datasets, wwwpppaiipahseeeccc wnwerswwthccce.. /oan/aatsett has curated licenses for some datasets. Their licensing guide can help determine the license a dataset.
existing datasets that are re-packaged, both original license and license of derived asset (if it changed) should provided.
this information not available online, the authors are encouraged reach out the asset's creators.
13. New assets
Question: Are new assets introduced the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: We do not introduce any new assets. All experiments are conducted using widely used
public datasets such as CIFAR-10, 10, ImageNet, and LSUN Bedroom 20
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the datasct/codd/mmode as part of their submis- sions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and sereenshots, applicable, as well as details about compensation (if any)?
Answer: [NA]
29

---

Justification:
This paper does not involve crowdsourcing research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then much detail possible should be included in the main paper.
According to the NeurIPS Code of Ethics, workers involved data collection, curation, other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals equivalent for research with human subjects
Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Question:
Answer: [NA]
Justification: This paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing rescarch with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects rescarch. If you obtained IRB approval, you should clearly state this the paper.
recognize that procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applica- ble), such as the institution conducting the review.
16. Declaration LLM usage
Question: Does the paper describe the usage of LLMs if it is important, original, non-standard component of the core methods in this rescarch? Note that if the LLM is used only for writing. editing, or formatting purposes does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Answer: [NA]
Justification: We did not use LLMs for any part of the core methodology or scientific contribution of the paper.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs any important, original, non-standard components.
Please refer to our LLM policy (https://neure htt.///rrrrpreeeeoooeeeeeeee///e LLM) for what should should not be described.
30


Reviews:
[{"review_id": "9O2WAuJzyh", "reviewer": "Reviewer_AnKE", "summary": "This paper first demonstrates that the original reflow process in Rectified Flow relies on large amounts of generated data and can cause drift from the target distribution. To address this, the authors propose Balanced Conic Rectified Flow, which incorporates real image inversions and spherical linear interpolation. This approach achieves better performance while requiring significantly fewer generated samples.", "questions": "See Weakness. Also, \n- The arrangement and legends in Figures 3a–b and 5b could be improved for better readability. \n- The first footnote on page 3 is somewhat confusing. It refers to \"the t-th sampling step\", and adds the velocity at time $t$ directly to $X_0$ or $X_1$, which does not make sense. It seems that the process being described should involve integration from $t = 0$ to $t = 1$, or vice versa, rather than referring to a single \"t-th sampling step\".", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:  \n- The proposed method outperforms reflow and rf++, with experiments on diverse datasets including CIFAR-10, ImageNet, and LSUN.\n- The drift analysis is insightful and serves as a good motivation. \n\nWeaknesses:\n- The paper analyzes drift using 1-step Euler sampling; however, Rectified Flow models are not designed for 1-step generation, and fake samples for the reflow process are generated via multi-step integration. This raises concerns that the measured drift may reflect Euler integration error rather than actual model drift, potentially biasing the conclusions. Clarification or validation with a multi-step analysis would strengthen the argument. \n- The proposed method introduces several hyperparameters, including the maximum Slerp interpolation weight $\\zeta^{max}$, the repairing step, and the mixing ratio between real and fake pairs in balanced conic reflow. However, the paper lacks sufficient analysis of how sensitive the model's performance is to these hyperparameters. It is unclear whether the reported improvements are robust or heavily dependent on careful tuning. \n- While the paper empirically demonstrates that using Slerp-based perturbations around real image inversions helps mitigate drift, it does not offer a theoretical explanation for why this technique effectively addresses the issue.", "was_revised": true, "final_justification": "The authors have adequately addressed my concerns regarding [W1] the use of 1-step Euler sampling and [W2] the stability of the hyperparameter settings. I appreciate the clarifications and will maintain my current borderline accept score.", "rebuttal": "We appreciate the reviewer’s agreement with our motivation and contributions, especially the drift analysis and improvements over existing reflow methods.\n\n## W1 \nWe appreciate the reviewer’s concern and the opportunity to clarify our choice of analysis. We suggest that 1-step reconstruction is an appropriate and informative diagnostic for drift (i.e., bias), for the following reasons:\n\n1. **Support from prior work**: Prior studies on exposure bias in diffusion models (e.g., [1], [2]) emphasize that the early steps of sampling are where the most critical errors or biases arise. Our use of 1-step evaluation is consistent with this literature.\n\n2. **Reduced solver accumulation error**: Multi-step sampling introduces numerical errors that accumulate with each step, potentially masking model-induced error. 1- or 2-step evaluation avoids this confounding factor and more cleanly isolates the model’s contribution to reconstruction performance.\n\n3. **Direct signal of drift** (bias toward fake samples): Since the reflow trajectory heavily depends on the model’s predicted initial velocity, a 1-step Euler reconstruction error directly reflects the alignment of the model’s estimated vector field with the true reverse direction.\n\n4. **Statistical robustness**: We report reconstruction errors averaged over repeated evaluations on 10k samples using the same solver settings. The consistent error gap between real and fake samples likely reflects intrinsic model behavior, not solver stochasticity.\n\n5. **Multi-step support in supplement**: In Supplementary Figure 5, we also report 2-step Euler reconstruction error on ImageNet. The gap between fake and real reconstruction errors remains, and our proposed conic method demonstrably narrows this gap.\n\n[1] Elucidating the Exposure Bias of Diffusion Models ICLR, 2024\\\n[2] Input Perturbation Reduces Exposure Bias in Diffusion Models ICML, 2023\n\n## W2\nWe appreciate the reviewer’s concern regarding hyperparameter sensitivity.\n### 1.We provide several ablation studies on different Slerp settings.\nRegarding the mixing between real and fake pairs, we provide an ablation study covering:\n  - Real-only pairs (Original Rectified flow using only Real pair),\n  - Real + Slerp-perturbed pairs,\n  - Real + Fake + Slerp pairs,\n\n|Setting|1-step IS|1-step FID|RK-step IS|RK-step FID|\n|-|-|-|-|-|\n|Real only (no Slerp)| 8.21|6.93|8.65|5.13|\n|Real only (+ Slerp)| 8.53|6.71|9.02|4.47|\n|Ours (real + fake + Slerp)|**8.79**|**4.16**|**9.30**|**3.24**|\n\ndemonstrating that performance improves consistently when our full method is applied (Supp. Fig. A4).\n  \n\nFor **Slerp perturbation patterns**, we provide detailed comparison in **Table 3**, showing that the interpolation method yields consistent quality across different variants.\n- Ours Slerp Pattern: $\\zeta_t=\\zeta_{\\text{max}}/K \\rightarrow\\zeta_{\\text{max}} \\rightarrow \\zeta_{\\text{max}}/K$ \n- Strictly Increasing:$\\zeta_{t}=\\zeta_{\\text{max}}/K \\rightarrow\\zeta_{\\text{max}}$  \n- Strictly Decreasing: $\\zeta_{t}=\\zeta_{\\text{max}}\\rightarrow\\zeta_{\\text{max}}/K$\n\n|Category|Method|IS ↑|FID ↓|\n|-|-|-|-|\n|Slerp|Ours|**8.72**|**6.63**|\n||Strictly Increasing|8.48|6.64|\n||Strictly Decreasing|8.45|6.70|\n|Lerp| Linear Interpolation |8.46|7.50|\n\n\n### 2. $\\zeta_{max}$ is adaptively selected\n\nWe would like to clarify that:\n\n$\\zeta_{max}$ is not manually tuned but **adaptively selected** based on the **maximum p-reconstruction error** between fake and real samples (eqn 13 in Sec3.5).\n\nTo further address the reviewer’s concern, we will additionally report generation quality across varying fake-to-real pair ratios, comparing both the baseline and our method.\n\n|#Fake Pairs|NFE|IS↑|FID↓|\n|-|-|-|-|\n|Original 4M (+Distill)|1|8.08 (9.01)|12.21 (4.85)||\n||110|9.24|3.36| \n|Ours 600K(+Distill)|1|**8.79 (9.11)**|**5.98 (4.16)**|\n||104|**9.30**|**3.24**| \n|Ours 3M(+Distill)|1|8.69 (8.92)|6.03 (4.72)| \n||104|9.20|3.35|\n|Ours 60K|1|8.60|7.06|\n||104|9.17|3.98|\n\n**Key takeaways:**\n1. Conic ReFlow improves 1-step performance across all settings.\n2. 600K achieves the best trade-off between efficiency and performance.\n3. Too few fake pairs (e.g., 60K) results in poor full-step generalization.(For detailed analysis, see Appendix E.)\n\nThese results collectively indicate that our approach maintains robust performance across a range of hyperparameter configurations, and the reported improvements are not heavily reliant on specific tuning choices.\n\nWe hope this alleviates the reviewer’s concern.\n\n## W3\n\nThank you for the valuable comment.\n\nWe argue that **Conic ReFlow effectively mitigates drift** (i.e., bias toward fake samples), and provide the following rationale:\n\n1. Generative models inherently cannot perfectly learn the ideal target distribution.\n2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.\n3. A noticeable gap in recon and p-recon error is observed between fake and real images.\n4. This gap indicates that the model is **biased toward fake images** and lacks robustness around real samples.\n5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.\n6. As a result, the error gap between real and fake samples is reduced, the degradation in full-step quality is mitigated, and our reflow method has been empirically shown to outperform in various metrics such as recall, precision, curvature, and IVD.\n\nImportantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [1–5], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [6–9].\n\n\n[1] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations  \n[2] Anomaly Detection with Robust Deep Autoencoders  \n[3] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection  \n[4] Šmídl et al., Anomaly Scores for Generative Models (2019)  \n[5] Adversarial Examples for Generative Models, ICLR Workshop (2020)  \n[6] Input Perturbation Reduces Exposure Bias in Diffusion Models, ICML, 2023  \n[7] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection  \n[8] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning, NeurIPS, 2018\n[9] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.\n\n## Q1\nThank you for the helpful suggestion.  \nWe acknowledge that the readability of Figures 3a–b and 5b can be improved and will revise the arrangement and legends in the final version as follows to enhance clarity:\n\n- For **Figure 3(a)**, we will move the legend to the top center and increase the font size of both the legend and the values above the bars.  \n- For **Figure 3(b)**, we will remove the duplicated legend and relocate the remaining one to the top right, increasing the font size to improve readability.\n\nAppreciate your attention to detail.\n\n## Q2\nThank you for the helpful suggestion. We agree that the current description is misleading, as it omits the integration process and may imply a direct addition of velocity at a single time step.\nWe will add that integration over time is required in the final version.\n\nWe would be happy to elaborate over the discussion phase"}, {"review_id": "fAIK5uilZT", "reviewer": "Reviewer_MDqu", "summary": "This article reveals that the reflow process causes the learned model to drift away from the target distribution and designs a new reflow step, namely the conic reflow, to address this drift problem. It supervises the model by the inversions of real data points through the previously learned model and its interpolation with random initial points. Conic reflow achieves much lower FID in both one-step and full-step generation in CIFAR-10. The conic reflow generalizes to various datasets such as LSUN Bedroom and ImageNet.", "questions": "1、Please refer to the question in the weakness section.\n\n2. The ablation study of selecting slerp or conic hyperparameters is not sufficient. The metric differences between different hyperparameter combinations are too small.\n\n3. I wonder if the conic reflow costs more compute resources than the original Rectified Flow?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "# Strengths\n\nThe article is well written and organized. The article proposes a new type of reflow, named conic reflow. It keeps the ODE paths aligned with real samples, as evaluated by reconstruction. It uses only a small number of generated samples, 600K and 4M, respectively, instead of the large ones. The learned flow is straighter than others, as evaluated by curvature. Experiments demonstrate that learned model generates images with higher quality, evaluated by FID, IS, and Recall.\n\n\n\n# Weaknesses\n\n1. The proposed method lacks a thorough theoretical analysis of the conic flow and its potential to alleviate drift issues. Additionally, the paper does not include a comprehensive analysis for selecting slerp or conic hyperparameters theoretically or empirically. \n\n2. The paper compares performance only with the original rectified flow and does not include a comparison with recent flow matching models or diffusion models.\n\n3. The empirical improvements for the conic reflow in CIFAR-10 are too small, and it does not adequately verify the effectiveness of the conic reflow. The author should include more empirical results (such as higher-resolution ImageNet, FFHQ, AFHQ, or other classes of LSUN).\n\nBased on the above content, I recommend that the author submit additional experimental results for review. If you can provide additional theoretical analysis for the conic reflow, it will also help change the perceptions of the paper.", "was_revised": true, "final_justification": "I have reviewed the author's rebuttals to all the reviewers, and they have addressed my concerns.", "rebuttal": "We thank the reviewer for their careful reading and appreciation of our key ideas, including the conic reflow design, drift mitigation, and strong empirical performance across datasets.\n\n## W1\n\nThank you for the valuable comment.\n\nWhile we do not provide a formal theoretical analysis, **Conic ReFlow is logically designed to mitigate drift** (i.e, biased toward fake images). Its effectiveness is supported by empirical results, and its rationale can be reasonably grounded in existing literature on reconstruction-based robustness and reduces Bias.\n\n1. Generative models inherently cannot perfectly learn the ideal target distribution.\n2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.\n3. A noticeable gap in reconand p-recon error is observed between fake and real images.\n4. This gap indicates that the model is **biased toward fake images** and lacks robustness around real samples.\n5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.\n6. As a result, the error gap between real and fake samples is reduced, the degradation in full-step quality is mitigated, and our reflow method has been empirically shown to outperform in various metrics such as recall, precision, curvature, and IVD.\n\nImportantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [1–5], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [6–9].\n\n\n[1] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations, CoRR, abs/1802.00187, 2018  \n[2] Anomaly Detection with Robust Deep Autoencoders, KDD 2017  \n[3] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection, BMVC 2015  \n[4] Anomaly Scores for Generative Model, arXiv:1905.11890, 2019  \n[5] Adversarial Examples for Generative Models, ICLR Workshop, 2020  \n[6] Input Perturbation Reduces Exposure Bias in Diffusion Models ICML, 2023  \n[7] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection, CVPR, 2024  \n[8] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning, NeurIPS, 2018\\\n[9] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024\n\n## W2\n\nThank you for the detailed feedback.\n\nOur paper includes comparisons with recent diffusion-based models in Appendix M to further contextualize the performance of reflow-based methods. It is mentioned in L210 and Section 4.6.\n\nWe suggest that including comparisons with more recently developed flow-based generative models would help readers better understand the overall generative quality landscape. We will provide FID and IS scores of recent flow models, including [1–4], in Appendix M of the final version.\n\n[1] Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport, TMLR , 2024.\\\n[2] Zhu, H., Wang, F., Ding, T., Qu, Q., & Zhu, Z. (2024). Analyzing and Improving Model Collapse in Rectified Flow Models. arXiv preprint arXiv:2412.08175.\\\n[3] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.\\\n[4] Simple ReFlow: Improved Techniques for Fast Flow Models, ICLR, 2025.\\\n\n## W3\n\nWe appreciate the reviewer’s request for additional high-resolution experiments.\n\nWe would like to note that we have already included high-resolution results on the LSUN Bedroom dataset in our current submission, covering FID, precision, recall, $L_2^{recon}$, and $L_2^{p\\text{-}recon}$ (see Figure 8). We also provide qualitative comparisons across random seeds in Appendix (Figures A6–A10), demonstrating the robustness and effectiveness of Conic ReFlow.\n\nHowever, further experiments on other high-resolution datasets such as LSUN-Church, AFHQ, or ImageNet 256 are realistically infeasible within the rebuttal period due to the following constraints:\n\n1. The original Rectified Flow repository only provides 1-Rectified Flow checkpoints for these datasets, requiring us to re-train 2-Rectified Flow baselines from scratch for fair comparison.\n\n2. High-resolution datasets of this scale require at least 120K fake samples to produce meaningful comparisons under the reflow paradigm.\n\n3. Given our limited GPU resources, training with batch size 16 would take **approximately 4 weeks** to reach convergence and obtain statistically meaningful results.\n\nWe sincerely regret that we are unable to include these additional results in the rebuttal. However,** we plan to report full evaluations on LSUN-Church and AFHQ datasets in the camera-ready version**, including comparison with the original Rectified Flow and release of corresponding model checkpoints.\n\nWe hope this addresses your concerns and clarifies the practical limitations involved.\n\n## Q2\n\nWe appreciate the reviewer’s concern regarding hyperparameter sensitivity.\n\n### We provide several ablation studies on different Slerp settings.\nFor **Slerp perturbation patterns**, we provide detailed comparison in **Table 3**, showing that the interpolation method yields consistent quality across different variants.\n- Ours Slerp Pattern: $\\zeta_t=\\zeta_{\\text{max}}/K \\rightarrow\\zeta_{\\text{max}} \\rightarrow \\zeta_{\\text{max}}/K$ \n- Strictly Increasing:$\\zeta_{t}=\\zeta_{\\text{max}}/K \\rightarrow\\zeta_{\\text{max}}$  \n- Strictly Decreasing: $\\zeta_{t}=\\zeta_{\\text{max}}\\rightarrow\\zeta_{\\text{max}}/K$\n\n|Category|Method|IS (↑)|FID (↓)|\n|-|-|-|-|\n|Slerp|Ours|**8.72**|**6.63**|\n||Strictly Increasing|8.48|6.64|\n||Strictly Decreasing|8.45|6.70|\n|Lerp| Linear Interpolation |8.46|7.50|\n\nFuthermore, appendix E Table A4 compares the IS/FID performance when using only real pairs, using both real and fake pairs, and applying our conic reflow method.\n  - Real-only pairs (Original Rectified flow using only Real pair),\n  - Real + Slerp-perturbed pairs,\n  - Real + Fake + Slerp pairs  \n\n|Setting|1-step IS|1-step FID|RK-step IS|RK-step FID|\n|-|-|-|-|-|\n|Real only (no Slerp)| 8.21|6.93|8.65|5.13|\n|Real only (+ Slerp)| 8.53|6.71|9.02|4.47|\n|Ours (real + fake + Slerp)|**8.79**|**4.16**|**9.30**|**3.24**|\n\nOverall, the results empirically support that conic reflow consistently outperforms standard reflow regardless of the presence of fake pairs or the specific noise injection pattern used.\n\n### 2. $\\zeta_{max}$ is adaptively selected\n\nWe would like to clarify that:\n$\\zeta_{max}$ is not manually tuned but **adaptively selected** based on the maximum p-reconstruction error between fake and real samples (eqn 13 in Sec3.5).\n\nWe acknowledge that Table 3 may have caused confusion by presenting a fixed value for $\\zeta_{\\max}$. In the final version, we will revise this to clarify the scheduling pattern:\n- For strictly increasing : $\\zeta_{t}=\\zeta_{\\text{max}}/K \\rightarrow\\zeta_{\\text{max}}$\n- For strictly decreasing : $\\zeta_{t}=\\zeta_{\\text{max}}\\rightarrow\\zeta_{\\text{max}}/K$\n\nOnce again, thank you for the helpful suggestion, and we hope this response has clarified your concerns.\n\n## Q3\nThank you for the thoughtful question.\n\nWe would like to clarify that the computational cost of conic reflow is essentially the same as that of the original rectified flow, except for the initial step of estimating $\\zeta_{\\max}$.\n\nImportantly, conic reflow is significantly more efficient in terms of total training resources. Unlike the baseline our method achieves strong performance using only about **7–10% of fake pairs**. As a result, the overall training cost is actually **lower** than that of the baseline.\n\nWe hope this addresses your concern regarding computational efficiency, and we would be glad to provide further clarification during the discussion phase."}, {"review_id": "umQntr9AIF", "reviewer": "Reviewer_ABQ1", "summary": "This paper addresses ReFlow, which is a procedure for straightening flow trajectories by training flow models upon noise-image pairs generated by itself. While ReFlow does produce faster flow models, the generated marginal drifts away from true marginals as a result of training with fake pairs (pairs generated by solving the flow starting from noise to data). While training with real pairs (pairs generated by solving the flow starting from data to noise) can alleviate this drifting problem, one can only generate a finite number of real pairs.\n\nTo mitigate this problem, the authors propose to augment real with with **conic pairs**. Given a real pair, a conic pair is produced by spherically interpolating generated noise with random noise. ReFlow with conic pairs is shown to out-perform vanilla ReFlow on CIFAR10, ImageNet 64x64, and LSUN Bedroom datasets.", "questions": "I am willing to raise the score to *borderline reject* if the authors address concerns in\n\n- **Clarity** with further explanations or proofs,\n- **Originality** by adding proper baselines,\n\nand further to *borderline accept* if the authors address concerns in\n\n- **Significance** by demonstrating that conic ReFlow can achieve results (e.g., FID scores) comparable to those of recent ReFlow models such as RF++ and Simple ReFlow.", "limitations": "The authors discuss some limitations in the conclusion.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths and weaknesses are denoted [S#] and [W#], respectively.\n\n**Quality**\n- [S1] The paper provides thorough ablations along with experiments on standard benchmark datasets such as CIFAR10, ImageNet, and LSUN Bedroom.\n\n**Significance**\n- [S1] Conic ReFlow may be applied to larger flow models such as InstaFlow and SD3.\n- [W1] The performance of conic ReFlow is rather weak when compared to recent ReFlow models such as [2] or [3]. For instance, on CIFAR10, while conic ReFlow achieves 3.24 FID with NFE=104, [2] achieves 1.98 FID with NFE=9, and [3] achieves 3.07 FID with NFE=1. On ImageNet 64x64, while conic ReFlow achieves 28.2 FID with full simulation (NFE unreported), [2] achieves 1.74 FID with NFE=9, and [3] achieves 4.31 FID with NFE=1.\n\n**Clarity**\n- [S1] The experimental protocol is described clearly.\n\nI had difficulty following the logic at some parts of the paper. See below for details.\n\n- [W1] At lines 104-109, the authors assert that continuity near real samples imply closeness of generative and real marginals. Why should this be the case? The former property is a local one, and the latter property is a global one, and in general, one does not imply the other.\n- [W2] At lines 146-148, the authors claim \"this approach (conic ReFlow) not only mitigates distributional drift but is also aligned with adversarial robustness strategies studied in inverse problems and reconstruction-based neural networks\". Given the ambiguity described in [W1], I have difficulty seeing how conic ReFlow mitigates distributional drift.\n- [W3] In Section 4.4.1, the authors propose refreshing real pairs every fixed number of model updates. However, it is unclear why one should do so. Wouldn't it be better just to generate a large number of real pairs prior to training?\n\n**Originality**\n- [S1] This paper proposes a new type of augmentation for real pairs for ReFlow training.\n- [W1] A similar augmentation has been explored in [1] (albeit with different motivations). Specifically, [1] proposes annealing ReFlow, where, given a noise-image pair, one interpolates noise with random gaussian noise.\n- [W2] The idea of balancing training with real and fake pairs has also been suggested in [2].\n\n[1] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.\n\n[2] Simple ReFlow: Improved Techniques for Fast Flow Models, ICLR, 2025.\n\n[3] Improving the Training of Rectified Flows, NeurIPS, 2024.", "was_revised": true, "final_justification": "The rebuttal has partially addressed my concerns with\n\n- Significance [W1] with experiments on CIFAR10 (but experiments on ImageNet is missing),\n- Clarity [W1], [W2], [W3] with further discussion.\n\nHowever, some concerns with Originality [W1], [W2] still remain unaddressed:\n\n- No numerical comparisons to relevant baselines such as noise annealing in SlimFlow and real-fake pair balancing in Simple ReFlow,\n- Incremental nature of the proposed method compared to previous works.\n\nHence, I have raised the score to **borderline reject**.\n\n==========\n\nUpdated 25.08.12\n\nThe authors' latest response has addressed my remaining concerns, but **additional experiments with noise annealing and SlimFlow and real-fake pair balancing in Simple ReFlow should be included in the camera ready version, if accepted**. I tentatively raise my score to **borderline accept**.", "rebuttal": "We appreciate the reviewer for positively noting our ablation studies,  potential applicability to larger flow models.\n\n## W1\n\nWe thank the reviewer for the detailed comparison. However, we respectfully argue that the evaluation that Conic ReFlow underperforms compared to [2] and [3] is not fully justified for the following reasons.\n\nWhile the results reported in the main paper of [2] are based on a different configuration with batch size 512 and 800K iterations, we apply Conic ReFlow to RF++ under the same setting as the original Rectified Flow, using batch size 128 and 500K iterations, to ensure a fair comparison.\n\nAs shown in Table 1 and Table 6 in Appendix G, our method achieves better FID and IS scores under this aligned setting.\n\n* Same configurations\nEDM initialization, LPIPS-Huber-1/tloss, batch size 128 with 500K training itration.\n* Different configurations:\nRF++ / RF++(+Ours) uses 800K/**600K** fake pairs.\n\n|Model|NFE|IS↑|FID↓|\n|-|-|-|-|\n|RF++(+Ours)|1|**9.15**|**3.84**|\n|RF++|1|9.04|4.14|\n|RF++(+Ours)|2|**9.36**|**3.03**|\n|RF++|2|9.24|3.16|\n\nNotably, even with fewer training pairs, our method outperforms the baseline, highlighting that Conic ReFlow consistently improves upon standard reflow even when applied to stronger flow models like [2].\n\nConic ReFlow is not intended to compete with existing rectified flow techniques but rather to offer a complementary improvement. Our primary objective is to alleviate the drift issue that arises during the reflow process.\n\nIn contrast, [2] and [3] focus on overall generation performance improvements, such as EDM initialization, refined loss, and better samplers. Conic ReFlow provides a structural contribution by improving the stability during the reflow. \n\nTherefore, to ensure a fair comparison, all experiments in our paper are conducted using Rectified Flow as the baseline initialization. In addition, the reported results on ImageNet are based on **unconditional generation**, so comparing image quality against class-conditional generation, which benefits from explicit guidance, is not appropriate.\n\nWe also clarify that the purpose of **Table 4 and 5** is to empirically demonstrate that our proposed Conic ReFlow consistently improves overall generation quality (FID, IS, and recall) over the original ReFlow step when applied to the same baseline, particularly on more complex datasets. It is **not intended as a direct comparison against current state-of-the-art flow models**.\n\nOur contribution goes beyond merely improving FID scores. The primary goal of Conic ReFlow is to evaluate and enhance the straightness of the reflow path, diversity, fidelity, and mitigation of drift. To this end, we employ a variety of metrics:\n\n- Straightness, IVD\n- Precision / Recall\n- $L_{2}^{recon}, L_{2}^{p-recon}$ and these gaps\n\nOur method demonstrates consistent improvements across diverse datasets, including CIFAR-10, ImageNet 64x64, and LSUN.\n\nIn fact, the most methodologically similar approach to ours is SlimFlow [1], which also employs an Annealing ReFlow strategy using noise interpolation for performance gains:\n\n|Model|NFE|FID↓|\n|-|-|-|\n|Slim (EDM)|1|4.53|\n|Slim (1-RF)|1|5.81|\n|Ours (Dist)|1|**4.16**|\n|Ours (RF++)|1|**3.84**|\n\nConic ReFlow is a novel approach to structurally address the inherent issues of the ReFlow process. This contribution is complementary rather than competitive to prior methods, and its validity is supported by extensive experimental evidence.\n\n[1] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.\\\n[2] Improving the Training of Rectified Flows, NeurIPS, 2024.\\\n[3] Simple ReFlow: Improved Techniques for Fast Flow Models, ICLR, 2025.\n\n## W2\nWe thank the reviewer for their detailed and thoughtful questions.\n\nOur claim is that discontinuities near the data manifold can indicate potential global drift. We do not claim that  continuity is theoretically a sufficient condition for global alignment.\n\nThe connection between the local continuity and global continuity stems from the structural nature of flow models, where the global vector field is **gradually constructed through the accumulation of conditional path supervision** [1]. Specifically, recent work leverage local continuity as a means to improve global performance, including generalization guarantees under locally Lipschitz conditions [2, 8]. It is a commonly accepted view that local behavior can influence the overall quality of generation.\n\n## W3\n\nThank you for the valuable comment. We have quantitatively demonstrated that Conic ReFlow mitigates drift (i.e., bias toward fake images), and this is supported by the following evidence:\n\n1. Generative models inherently cannot perfectly learn the ideal target distribution.\n2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.\n3. A noticeable gap in reconand p-recon error is observed between fake and real images.\n4. This gap indicates that the model is **biased toward fake images** and lacks robustness around real samples.\n5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.\n6. As a result, the error gap between real and fake samples is reduced, the degradation in full-step quality is mitigated, and our reflow method has been empirically shown to outperform in various metrics such as recall, precision, curvature, and IVD.\n\nImportantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [3–7], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [8–10].\n\n[1] Flow Matching for Generative Modeling\\\n[2] De Bortoli et al., Convergence and Generalization of Score-Based Generative Models*(2023)\\\n[3] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations, CoRR, 2018\\\n[4] Anomaly Detection with Robust Deep Autoencoders, KDD 2017\\\n[5] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection, BMVC 2015\\\n[6] Šmídl et al., Anomaly Scores for Generative Models, arXiv:1905.11890, 2019\\\n[7] Adversarial Examples for Generative Models, ICLR Workshop, 2020\\\n[8] Input Perturbation Reduces Exposure Bias in Diffusion Models, ICML, 2023\\ \n[9] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection, CVPR, 2024\\\n[10] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning (NeurIPS 2018), NeurIPS, 2018\\\n\n## W4\n\nWe justify our design choice based on both intuitive considerations and theoretical motivations from recent literature:\n\n1. Reverse noise are inherently imperfect : These pairs are generated through imperfect inversion processes. If reused without refresh, they can cause the model to overfit or saturate to suboptimal or outdated couplings.\n\n2. Motivation from Minibatch and Multisample Coupling Strategies : As shown in  [1] and [2], jointly constructed couplings over minibatches or multisamples better approximate ideal transport plans than independent pairings. These methods explicitly construct $\\pi(x,y)$ on a batch to reduce training variance and improve flow straightness and transport cost.\n\n3. Bias mitigation and perturbation-aware stability : Periodic refreshing prevents bias from stale supervision and ensures that reverse noise perturbations act on diverse, up-to-date pairs—promoting stability and robustness during training.\n\n[1] Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport, TMLR , 2024.\\\n[2] Multisample Flow Matching: Straightening Flows with Minibatch Couplings, PMLR , 2023.\n\n## W5\n\nWe respectfully clarify that while our method and Slim Flow [1] both involve perturbed supervision, the motivation, design intent, and experimental behavior differ significantly:\n\nSlim Flow introduces Annealing Reflow solely to provide an efficient initialization for a smaller student model by gradually transitioning the target velocity from random to teacher model.\n\nIn contrast, our Conic Reflow is designed to improve the reflow process itself, by correcting biases introduced by imperfect real noise couplings. Our method is applicable to models of any size and is not limited to distillation settings.\n\nSlim Flow only evaluates 1-step generation quality, as its objective is to distill a teacher model's immediate behavior. It does not report or ensure performance on multi-step sampling.\n\nIn contrast, our method directly improves both *few-step and multi-step generation quality* (see Table 1), making it robust for practical generative tasks.\n\nSlim Flow lacks any data-driven or path-aware adaptation of the annealing parameter $\\beta$.\n\nConversely, our Conic Reflow includes an adaptive $\\zeta_{\\max}$ formulation (Eq. 13) that adjusts perturbation intensity, providing more principled and effective supervision.\n\n## W6\n\nWhile both methods utilize real data points, the motivations and problem setups are fundamentally different.  \nSimple Reflow employs forward pairs as a minimal remedy to mitigate error accumulation.  \nIn contrast, our method aims to address a more foundational issue by leveraging real and generated image pairs.  \nSpecifically, we mitigate instability and bias in the reflow process, construct adaptive paths, and enhance training stability.\n\n## Q1\n\n- Clarity :  \n  Through responses to W2 and W3, we provided logical explanations, experiments, and supporting evidence.\n- Originality :  \n  Through W5 and W6, we clarified the differences between our method and proper baselines, highlighting the originalit of our approach.\n- Significance:  \n  In W1, we clarified that our experiments include a direct comparison under the same configuration.\n\nWe hope these clarifications help resolve the reviewer’s concerns and would be glad to provide further clarification during the discussion phase."}, {"review_id": "KnlGbQ3j5Z", "reviewer": "Reviewer_srv3", "summary": "The authors propose conic rectified flow (RF), a new alternative to RF to mitigate the distribution drift in the successive reflow steps. The main idea of the conic RF is to design a map that transports a neighbourhood of points (rather than a single Gaussian point) to a corresponding image point $X_1$.\nThe experimenst show that conic RF is able to reduce the reconstruction error gap between real and fake images, and also achieves comeptetive results in real examples.", "questions": "## Major comments\n**Definition of $v$ is not clear:**\n- Line 72: ''coupling the generated pairs $(Z_0, v(Z_0))$\nensures that the interpolation trajectory preserves the marginal distributions of the original and target\ndomains'': Assuming $v$ is the velocity field,  I think $(Z_0, v(Z_0))$ does not preserve the marginals of the source and target distribution. In fact, $v(Z_0)$ is just the velocity field (the time argument is missing). I think it should be $(Z_0, T(Z_0))$, where $T(Z_0):= Z_0 + \\int_{0}^1 v(Z_t, t)\\; dt$. Do the authors mean to write $T$ instead of $v$?\n- Line 87: Footnote 1 suggest that the $t$-th sampling step is denoted by $v(X_0)$ which shoudl be $X_t + v(X_t, t)dt$ (Not $X_0 + v(X_t, t)dt$) as per my understadning. Is it denoting something else?  Also, $v(Z_0^k)$ seems to denote the last sampling step instead of any $t$th sampling step. The notations are hard to understand. \n\n- Shouldn't there be an integral over $\\epsilon, \\zeta$ in Eq(9)?\n\n\n**What exactly is better in conic RF than regular RF?:**\n\n- Section 3.1 argues that RF suffers from distribution drift, i.e., $T(T^{-1}(X_1)) \\ne X_1$. How is this getting resolved in the Conic RF? As far as I understand, conic RF is trying to map rather an expanded neighbourhood around $T^{-1}(X_1)$ to a single point $X_1$. But, how does it stop the distribution shift? \n\n- Prior works have shown that real data augmentation reduces model collapse. Conic RF also uses real data during training, along with fake data. It might be possible that real data augmentation is the main factor that is preventing the distribution drift, rather than the explicit design of the flow mechanism. For example, [2] shows that real data augmentation mitigates model collapse. In light of this fact, it seems that using the methodology in [2] is more practical than opting for conic RF, as conic RF is comparatively harder to train. What is the extra advantage of Conic RF? I think a discussion on this is important. Maybe adding a comparison between the method in [2] and conic RF will help.\n\n\n## Minor comments\n\n- Line 67 writes k-rectified flow, whereas Line 74 writes $k$-rectified flow. It is advisable to choose one consistent notation.\n\n- The set $\\mathbb{N}$ in Line 160 is no defined. Also, where does the index $i$ in Eq (12) occur in Eq (11)? Are you referring to the empirical version of the loss in Eq. (11)?\n\n[1] Liu, X., Gong, C., & Liu, Q. (2022). Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.03003.\n\n[2] Zhu, H., Wang, F., Ding, T., Qu, Q., & Zhu, Z. (2024). Analyzing and Improving Model Collapse in Rectified Flow Models. arXiv preprint arXiv:2412.08175.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strength\nThe idea seems novel and quite effective in practice. To generate reliable results in practice, it is essential to address and resolve the issue of distribution drift in the RF reflow steps. This paper attempts to provide a solution in that direction. The experiments presented in the paper are comprehensive and showcase the usefulness of conic RF.\n\n## Weakness \nAlthough the idea is novel, the exact benefit of the current approach over regular RF is not clear. For example, how exactly conic RF prevents the distribution drift is not clear to me immediately. All in all, I think a stronger case needs to be made in favor of conic RF to argue the significance of conic RF over other methods (e.g., RF with real data augmentation introduced in [1] )\n Also, there are some presentation issues that I have pointed out later.\n\n\n[1] Zhu, H., Wang, F., Ding, T., Qu, Q., & Zhu, Z. (2024). Analyzing and Improving Model Collapse in Rectified Flow Models. arXiv preprint arXiv:2412.08175.", "was_revised": true, "final_justification": "My concerns have been addressed properly.", "rebuttal": "Thank you for recognizing the effectiveness and novelty of our approach, as well as for appreciating the comprehensiveness of our experiments and the practical relevance of addressing distribution drift in ReFlow.\n\n## W1,2\n\nThank you for this insightful comment. You are absolutely right: in this context, $v(Z_0)$ is not meant to represent the velocity field evaluated at a single point, but rather the outcome of solving the ODE defined by the velocity field. In other words, our intention was to represent the trajectory mapping $T(Z_0) := \\int_0^1 v(z_t, t) dt$, and we used the notation $v(Z_0)$ as a shorthand for this operator.\n\nTo simplify notation and avoid introducing an additional operator like $T$, we opted for this form. However, as you point out, this may lead to confusion, especially when integration is not explicitly stated.\n\nWe also acknowledge that in Footnote 1, the integral form was unintentionally omitted, which may have compounded this ambiguity. We will revise the footnote to explicitly include the integral definition, and we will consider introducing auxiliary notation (e.g., $T(Z_0)$) in the revised version to improve clarity and prevent confusion for future readers.\n\nWe sincerely appreciate your careful reading and helpful suggestion.\n\n## W3\n\nThank you for pointing this out. Indeed, $\\epsilon$ and $\\zeta$ are sampled during training and introduce stochasticity. However, in Equation (9), we focused on clearly conveying the core structure of the conic reflow loss. For simplicity, we presented the expectation over time $t$, treating $\\epsilon$ and $\\zeta$ as fixed samples per iteration.\n\nWhile it is more mathematically precise to express the full expectation over all random variables $\\epsilon$, $\\zeta$,and $t$, omitting some expectations is a common simplification in generative modeling literature, especially when the omitted variables are re-sampled at every training step.\n\nThat said, we appreciate the reviewer’s suggestion and will denote the sampling of $\\epsilon$ and $\\zeta$ more explicit in the final version for improved clarity and rigor.\n\n## W4\n\nWe identify a meaningful gap between:\n- **Reconstruction error**:  \n$L_{\\text{recon}}^2(X) = \\mathbb{E}_{x \\sim \\mathcal{X}} \\| x - v(v^{-1}(x)) \\|^2$\n- **Perturbed reconstruction error**:  \n$L_{\\text{p-recon}}^2(X, \\varepsilon) = \\mathbb{E}_{x \\sim \\mathcal{X}, z \\sim \\pi_0} \\| x - v(v^{-1}(x) + \\varepsilon z) \\|^2$\n\nWe argue that **Conic ReFlow effectively mitigates drift** (i.e., bias toward fake samples), and provide the following rationale:\n\n1. Generative models inherently cannot perfectly learn the ideal target distribution.\n2. Original ReFlow relies only on fake samples, which boosts 1-step quality but leads to degradation in full-step generation.\n3. A noticeable gap in reconstruction and perturbed reconstruction error is observed between fake and real images.\n4. This gap indicates that the model is **biased toward fake images** and lacks robustness around real samples.\n5. Conic ReFlow anchors on the reverse noise of real samples and supervises its perturbed path directly.\n6. As a result, it narrows the real/fake error gap and alleviates degradation in full-step quality.\n\n**Conic Reflow is theoretically sound and geometrically motivated.** \nImportantly, reconstruction error is a widely adopted proxy for assessing model robustness, sampling stability, and anomaly detection capability across various generative modeling approaches [1–5], and training methods based on input perturbation are commonly used to reduce model bias or improve performance [6–9].\n\nOur method learns not just to map a point but to **cover the cone-shaped neighborhood around real data on the manifold**. This enables training of geometry-aware flows that remain robust in real data regions—something not achievable by straight paths generated from fake-only inversions.\n\n[1] Clustering and Unsupervised Anomaly Detection with L2 Normalized Deep Auto-Encoder Representations  \n[2] Anomaly Detection with Robust Deep Autoencoders  \n[3] Learning Deep Representations of Appearance and Motion for Anomalous Event Detection  \n[4] Šmídl et al., Anomaly Scores for Generative Models (2019)  \n[5] Adversarial Examples for Generative Models, ICLR Workshop (2020)  \n[6] Input Perturbation Reduces Exposure Bias in Diffusion Models (arXiv:2404.01734)  \n[7] LaRE²: Latent Reconstruction Error Based Method for Diffusion-Generated Image Detection  \n[8] PixelDP: Leveraging Adversarial Perturbations for Robust Supervised Learning (NeurIPS 2018)\\\n[9] SlimFlow: Training Smaller One-Step Diffusion Models with Rectified Flow, ECCV, 2024.\n\n## W5\n\n1. The direct performance difference between using only real+fake data ([2]) and applying Slerp can be seen in our ablation study (Section 4.4, Table 2).  \nThe comparison between** \"No-Slerp\"** and **\"Ours\"** clearly demonstrates the **impact of Slerp-based perturbation**.\n\n|Model|FID|IS|Curvature|IVD|Recon (Real)|Recon (Fake)|P-Recon (Real)|P-Recon (Fake)|\n|-|-|-|-|-|-|-|-|-|\n|Ours*|**5.98**|**8.79**|**0.002295**|0.253334|**0.019404**|0.023139|**0.022382**|0.025206|\n|No Slerp|6.60|8.57|0.002322|**0.240884**|0.023380|**0.020154**|0.026063|**0.022496**|\n\n**Strengths of Ours**\n\n1.Better generation quality.  \n2.Straighter sampling path.  \n3.Lower reconstruction and perturbed reconstruction error on real images.\n\n**Using only real pairs is not sufficient**\n1. Despite worse generation quality, IVD is lower.  \n    - This indicates that the model initially predicts a worse velocity direction compared to ours.  \n    - In other words, simply using real pairs does not sufficiently mitigate the bias toward fake samples in standard ReFlow.\n  \n2.**[2] analyzes the **self-consuming training issue** by measuring the magnitude of DAE model weights**.\n\nThis is a perspective focused on the training dynamics. In contrast, we target the distributional drift and reconstruction error directly and propose a perturbation-aware conic guidance mechanism. Thus, although both methods aim to improve reflow, our approachs are **trajectory and bias toward fake samples**, and are fundamentally different from [2].\n\n[2] Zhu, H., Wang, F., Ding, T., Qu, Q., & Zhu, Z. (2024). Analyzing and Improving Model Collapse in Rectified Flow Models. arXiv preprint arXiv:2412.08175.\n\n## Q1\n\nWe appreciate the detailed feedback.\n\n- In the final version, we will unify the notation for *k*-rectified flow using **consistent LaTeX math style ($k$)** throughout the text.\n\n\n## Q2\n\nThank you for the detailed feedback.\n- The set $\\mathbb{N}$ refers to the **total number of training steps** involving both real and fake pairs (Lines 159–160).  \nHowever, we acknowledge that the reference in Line 161 to Section 3.3 for its definition may cause confusion.  \nTo avoid confusion, we will remove the notation $\\mathbb{N}$ from the corresponding line in the final version.\n\n* Equation (11) presents the overall loss function that combines both fake and real pair supervision.\n\n* Equation (12) provides the indexing rule that determines which of the two loss terms is active at each training iteration.\n\nSpecifically:\n- If the training index $i \\in U_{\\text{fake}}$, then the first term $\\chi_{\\text{fake}} \\cdot \\left( \\dot{Z}_{t,F} - v_\\theta(Z_{t,F}) \\right)^2$ is activated.\n- If $i \\in U_{\\text{real}}$, then the second term $\\chi_{\\text{real}} \\cdot \\left( X_1 - \\text{slerp}(Z_{0,R}, \\epsilon, \\zeta) - v_\\theta(Z_t^{\\text{Conic}}) \\right)^2$ is used.\n\nTherefore, Equation (12) is not an empirical approximation of (11), but rather describes the index set over which the individual terms in (11) are selected and applied during training.\n\nWe hope these clarifications help resolve the reviewer’s concerns We hope this addresses your concern regarding computational efficiency, and we would be glad to provide further clarification during the discussion phase."}]

Author rebuttal:

Meta-review: This work shows that the reflow process can cause the learned model to deviate from the true target distribution and proposes Conic Reflow as a mitigation. Although Conic Reflow introduces additional hyperparameters, rebuttal evidence (in response to Reviewers AnKE and MDqu) suggests limited sensitivity to them. It is encouraged to include these results, along with the additional experiments suggested by Reviewer ABQ1 (e.g., applying the method to Simple ReFlow and SlimFlow), in the revision. Overall, I recommend **acceptance**.
Decision: Accept (poster)