Easing Training Process of Rectified Flow Models Via Lengthening Inter-Path Distance
Authors: Xu Shifeng
Yanzhu Liu
Adams Wai-Kin Kong
Venue: ICLR 2025 Spotlight (2025)
Keywords: Rectified Flow Models
Training
Easing
Distance-Aware Noise-Sample Matching
DANSM
Primary area: generative models
TL;DR: easing the training process of rectified flow models by better noise-sample pairs in each training epoch

Abstract: Recent research pinpoints that different diffusion methods and architectures 
trained on the same dataset produce similar results for the same input noise. 
This property suggests that they have some preferable noises for a given sample. 
By visualizing the noise-sample pairs of rectified flow models and stable diffusion models in two-dimensional spaces, 
we observe that the preferable paths, connecting preferable noises to the corresponding samples, 
are better organized with significant fewer crossings comparing with 
the random paths, connecting random noises to training samples. 
In high-dimensional space, paths rarely intersect. 
The path crossings in two-dimensional spaces indicate the shorter inter-path distance 
in the corresponding high-dimensional spaces. 
Inspired by this observation, we propose the Distance-Aware Noise-Sample Matching (DANSM) method 
to lengthen the inter-path distance for speeding up the model training. 
DANSM is derived from rectified flow models, which allow using a closed-form formula to calculate the inter-path distance. 
To further simplify the optimization, we derive the relationship between inter-path distance and path length, 
and use the latter in the optimization surrogate. 
DANSM is evaluated on both image and latent spaces by rectified flow models and diffusion models. 
The experimental results show that DANSM can significantly improve the training speed by 30\% $\sim$ 40\%
without sacrificing the generation quality.

Published as a conference paper at ICLR 2025
EASING TRAINING PROCESS OF RECTIFIED FLOW MODELS VIA LENGTHENING INTER-PATH DISTANCE
Shifeng Xu!
Yanzhu Liu²
Adams Wai-Kin Kong'
'College of Computing and Data Science, Nanyang Technological University, Singapore
shifeng001 (0 e ntu. edu. sg, Instiiutte for Infocomm Research (I2R) & Centre for Frontier AI Research, A S*SAR,, Singapore liu.yanzhuein lttpe:iuuuirra.ssttttesee a-star wew.t.reeuuueeeeee.e..eee.eee.e.
adamskong@nt.. edu, sg
ABSTRACT
Recent research pinpoints that different diffusion methods and architectures trained the same dataset produce similar results for the same input noise. This property suggests that they have some preferable noises for a given sample. By visualizing the noise-sample pairs of rectified flow models and stable diffusion models in two-dimensional spaces, we observe that the preferable paths, con- necting preferable noises the corresponding samples, are better organized with significant fewer crossings compared with the random paths, connecting random noises to training samples. In dith--mimnsional space, paths rarely intersect. The path crosingg in two-dimensional spaces indicate shorter inter-path distance in the corresponding high-dimensional spaces. Inspired by this observation, we pro- pose the Aettance-Aware Noise-Sample Matching (DANSM) method to lengthen the inter-path distance for speeding up the model training. DANSM is derived from rectified flow models, which allow using closed-form formula calcu- late the inter inter-path distance. To further simplify the optimization, we derive the relationship between inter-path distance and path length, and use the latter in the optimization surrogate. DANSM evaluated on both image and latent spaces by rectified flow models and diffusion models. The experimental results show that DANSM can significantly improve the training speed by 30% 40% without sacrificing the generation quality. Code:
INTRODUCTION
Diffusion- generative models, such as diffusion models (Ho et al., 2020 Nichol & Dhariwal 2021 (Song et al. 2021a Dhariwal & Nichol 2021 Rombach al.) 2022) and rectified flow models
(Liu et al.) 2023a b Liu 2022 Lipman et al. (002)). have garnered considerable attention due to their high-quality generation and broad range of applications. Training diffusion-based generative models is in fact establishing a mapping between the noise space and sample space. Recent re- search discovers that when training on the same dataset, different diffusion methods using different architectures result in similar mapping. In other words, given the same input noise, the trained models generate similar resultant samples. Fig. demonstrates such mappings. This phenomenon,
RFM
RFM
RM
Figure 1: Images generated by well-trained rectified flow model (RFM) LLu et al. 2022 and diffu- sion model (DM) (Ermongroup 2021). Images in the same group are derived from the same noise.

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Images random neises
Images rardom nosses
Labents & "andom nosses
Laterts rendom npises
Sandben
preferatie
preferable
preferable
preferable
Sample Tarleatte
(a) CIFAR-10 images
(b) Bedroom images
(c) Latents of Bedroom
(d) Latents of SD
Visualization of samples and noises using t-SNE (Van der Maaten & Hinton 2008). Each sample and its corresponding noise are connected with dashed line. The lines in the top row are messy with multiple intersections, while the bottom row illustrates well-organized lines.
Figure 2:
known as "consistent model reproducibility" in Zhang et al. (2024). has been discussed in some prior works (Song al., 2021b, Liu al.. 2023a). They attempt understand this phenomenon from the denoising score matching perspective (Vincent 201T).
The consistent model reproducibility phenomenon suggests that given clean or generated sample, there exists some preferable noises, from which the well-trained models can generate the sample. Fig. [2 visualizes the clean samples, preferable noises, and random noises through t-SNE (Van der Maaten & Hinton| 2008). The preferable noises (the green dots in the second row) are obtained from well-trained rectified flow models and stable diffusion model through sample-to-noise process. The gray lines in the first row represent the random paths connecting the random noise and clean samples of training process, and the blue lines the second row denote the preferable paths linking preferable noises and their corresponding samples. Fig. shows clearly that the random paths used in model training are very messy with many intersections, but the preferable paths obtained from fully training models the training process and significantly slow it down. This aligns the analysis Li et al. (0024). Notice that in a high-dimensional space, intersections of path very rare. The intersections in the two-dimensional visualization imply that the paths are closer in those regions in the original high dimensional space. Inspired by the difference between the messy random path pattern used in training process and the well-structured preferable path pattern from fully trained model, we propose the Mestance-Aware Noise-Sample Matching (DANSM) method to improve training speed. In this work, we concentrate on rectified flow models in DANSM derivation, because its paths are straight-line segments, which provide an effective closed-fore form solution for the inter-path distance calculation. We further deduce the negative correlation between inter-path distance and path length, and use it in the DANSM optimization surrogate. Although DANSM is derived from rectified flow models, also test it on diffusion models, because of their popularity. Extensive experiments conducted on image and latent spaces of rectified flow models and diffusion models. The results show that DANSM significantly enhances the training process up 30%~40%
well-organized. We argue that the complex random path patterns hamper
The contributions of this work are summarized as follows:
Inspired by the difference between random paths used in training and preferable paths from well-trained models, we propose DANSM method aiming to increase the inter-path distance and enhance the training process.
Based on the straight paths of rectified flow models, we first derive closed-from formula to calculate inter-path distance. Furthermore, we prove the negative correlation between inter-path distance and path length, and use the latter in the optimization surrogate of the DANSM method.
Extensive experiments are conducted on image and latent spaces of rectified flow models and diffusion models, which demonstrate DANSM's outstanding capabilities for speeding up the training process.
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2 PRELIMINARIES AND RELATED WORKS
Without loss of generality, let æ be data sample from distribution No. be noise from standard Gaussian noise (0,, I), and & E [0, 1] be the timestep. z and & can be transformed into each other with This paper assumes continuous transformation between a and 2, x((0) = ax(), where (0)) = x and (1) 2.
2.1 RECTIFIED FLOW MODELS
Rectified Flow Models (RFM) utilize Ordinary Differential Equations (ODE) to approximate the straight paths between noises and samples. They offer unique solution generative modeling from the perspective of optimal transport (Villani 2009: 2021). In RFM, each point on the noise sample
path is linear interpolation between them.
(((()) dt
(1() (1 t)x tz,
E [0, 1].
It is worth noting that the expression of (+()) in Eq. is straightforward and its gradient remains stable. In fact, Esser et al.] (2024) also mentioned that RFM provides better theoretical properties and conceptual simplicity compared to classic diffusion models (Song et al.) [021 Ho al. 2020).
When training RFM, the model is expected to drive the flow to follow the interpolation between and z. It can be achieved by solving simple least squares regression problem (Liu al. 2023a):
CRFM (I,, 2, 2,t:0 f-
[I[(=
x) Onrss 12)
(2)
where U(0,1) 1) is the uniform distribution between and 1.
In Eq. [ the sample x and noise randomly paired for loss calculation. However, mentioned in the introduction, well-trained RFM have some preferable noises for a given sample. This implies that in most cases, the sample not paired up with its preferable noise, which can significantly hinder the model training process. This paper aims to explore this issue and propose an optimization solution to address
2.2 CONSISTENT MODEL REPRODUCIBILITY
In the realm of diffusion-based generative modeling, the consistent model reproducibility phe- nomenon has been discussed by several works. The concept of "uniquely identifiable encoding", as discussed in [Song et al. (202Tb) (page 7), suggests that the encoding (sample) for an input (noise) uniquely determined by the data distribution. Zhang al. (2024) introduce the concept of "consis- tent model reproducibility" and state that when trained on the same dataset, various diffusion- models generate similar data samples. However, they did not utilize it ease the model training process.
Theoretically, the consistent model reproducibility phenomenon can be understood through the lens of denoising score matching (Vincent 20111), interpreted via Tweedie's formula. Here, 20 is clean sample with noise, and z(1) is noisy variable with noise level d; The objective of model training is to estimate (x) [z/(1)], which represents the mean of all clean samples that could produce (t()) when perturbed by noise of level de. Since the clean samples used for training are fixed, the mapping between (=()) and (0) should remain consistent. We would like to highlight that the aim of this paper not to analyze consistent model reproducibility. However, it inspires us to
maximize the inter-path distance to speed up the training.
METHOD
In this section, we first provide a closed-form formula to calculate the inter-path distance for RFM. Next, explain how inter-path distance influences the quality of training data and the overall training process. Finally, we introduce the DANSM method, which lengthens the aater-path distance, thereby accelerating the model training process.
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3.1 PTTER-PATH DISTANCE
In RFM, the path between noise and sample is represented by a linear interpolation between them. When calculating the inter-path distance, the timestep / should be taken into account, as it determines the proportion of noise and sample at each point along the interpolation. Therefore, the distance between two paths can be considered as function of t. Given two noise-sample pairs, (21, 21) and (z2, 22), where 21 and are noises and and 22 are samples, the paths are defined with timestep t as follows:
{ ri (1 t) + 1z
E [0,
(3)
(1 ()x² + L22
Let V = 82 Iı and U 22 21. The distance between ri and r2 is an interpolation of and U: fri (1) = [1(1 1)V + 1.. Since & E [o, 1], the minimal value occurs at timestep t":
80
V (V-U) (V U) (V
t" if E [0,
where 8=
(4)
otherwise
The detailed deduction of Eq. 4 is given in appendix A.1 Computing e requires one vector sub- traction, two dot products and one division, and calculating fri.r((t) (t) requires additional one vector addition, two scalar multiplications and one norm computation. Directly using fra.ra (t) to compute all possible inter-path distances of n paths is inefficient due the complicated operations. There- fore, in this paper, the distance of two paths w.r.t. timestep t is defined as the minimal distance on timestep :":
dist(r1. 12) = min friez (t) = (t"). [E
(5)
3.2 INTER-PATII DISTANCE MATTERS IN TRAINING PROCESS
In the training process of RFM, the training data are actually the points in the noise- soise-sample paths. The quality of these paths is significantly influenced by the inter-path distances. To elucidate this, we consider an extreme case on RFM models. Consider two paths, P1 = (21, and P2 = (22, 22), that intersect at specific timestep t² t", the state of the noisy sample satisfies such equation:
xt(() (1 )=1 + t1
- t2) + 2-
However, when training the model 0 on (f') and t", the expected outputs differ based on the paths being followed as shown in Eq. 7 This discrepancy confuses the model training, different paths lead to different gradients. To avoid such conditions, sufficient inter-pate path distance is necessary.
Expected model prediction: 0 (z((1),)) ,,1)
[1 x1
if training on path Pı
22
52
if training on path P3
To analyse how the inter-path distances impact model training, we delve deeper into the inter-path distances. For better elaboration, we define the terms as below.
preferable path: The path between sample and its preferable noise, which is obtained by sample-to-noise process (the inverted sampling process) on fully trained model.
random path: The path between sample and any random noise.
average distance: For n paths, there are c = nnn- - 1)/2 path pairs, resulting in e inter-path distances. The average these distances termed the average distance of the n paths. minimal distance: For n paths, each path has nearest path, yielding minimal distances. The mean of those distances called the minimal distance of the paths.
training loss: For RFM, the ground-truth gradient is as in Eq. [ and the predicted gradient is denoted as g. The squared le norm [| x) ğll is called training loss.
As shown in Fig. of Sec. 5.1 when comparing by both average and minimal distances, the preferable paths exhibit greater inter-path distances than random paths. Additionally, as the inter- path distances increase, the training losses decrease, indicating an improvement in the quality of noise-sample paths. This aligns with our argument that shorter inter-path distances imply lower path quality, thereby making the training process difficult.
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3.3 DISTANCE-AWARE
MATCHING
DANSM's problem setting is defined as follows. Given a set of n sample points and a set of n noise points, the goal is establish one-to-one mapping between the two sets, resulting in paths. Let Pi.j be the path from the i-th noise point to the j-th sample point. The objective of DANSM is to maximize the inter-path distances of those paths while ensuring the process can be completed within in reasonable amount of time. In this paper, the sample (or noise) set size is referred as match-size, which serves as key parameter. The objective is defined as:
max 2
(00))
n(n-1) i=1 j=i+1
where o is a permutation with input from 2, ... n, and (ist(pi.o (i) ) is the distance be- tween the paths Pi. and Pj.o()), as specified in Eq. [ It worth noting that the noise set has the same size as the sample set, ensuring that all noises are utilized in the training process without any being discarded. This is crucial because if the noise set contains more points than the sample set, certain noises will inevitably be excluded from the model training. Due to the relationship between inter-path distance and path length, the unused noises are those that are farthest from the samples, meaning certain regions in the noise space remain untrained. This lack of training in specific noise space areas can negatively impact the overall generation performance.
Although the objective of the DANSM method is clear, calculating the inter-path distance for multi- ple paths in high-dimensional space is computationally expensive, especially for a large number of paths. To address this issue, we identified a surrogate approach to manipulate the inter-pate patt distance without calculating it. This approach is elaborated in Sec.
4 OPTIMIZATION SURROGATE
To further analyze the inter-path distance, we introduce the path length, which is the length of the path between noise and sample. In this section, we prove the negative correlation between inter-path distance and path length. Building on this insight, we propose an optimization surrogate of DANSM, which aims to shorten the path lengths, thereby increasing the inter-path distances accordingly.
4.1 NEGATIVE CORRELATION BETWEEN INTER-PATI DISTANCE AND PATH LENGTH
Note that the noises and samples of RFM have the same dimension and therefore, they can be considered from the same high-dimensional space Rd which encompasses both noise space and sample space. We elaborate the correlation between inter-path distance and path length in Rd The path of RFM is the line segment between noise point and sample point, while the inter-path distance defined in Sec. 3.1
We begin with simple scenario containing only two paths. Let the sample points be I1 and 22, and the noise points be 21 and 22. Consequently, the paths between samples and noises have two cases:
Pi = (21, ), P2 = (22, 22) and Q = (21, Q2 = (22, Following Sec. 3.1 we define the vectors U = 22 21. V = Z2 21 and V = Z2. Notably, V = V and ||
V where
|| means the length of vector or line segment. These variables are illustrated in Figs. 3a
and 3b
As shown in Sec. 3.1 the inter-path distance between P1 and P2 is interpolation of U and V. Similarly, the inter-path distance between QI and Q2 an interpolation of U and V. These relationships shown in Figs. [c 3d Let Leef 7 and Lgcf = 7. Since V - V, y and are supplementary angles for each other. Using the law cosine, we derive:
co8 7 = ||U||2 + |IV IU - VII
1|4a + 1l 122 - (lp.l + 12
(9)
|V
21 IV
where P1 = I - 21, P2 = 22 - 22. 91 = x2 - 21 and 92 = 21 22. The detailed derivation of Eq.
9 is provided in the appendix A.2 In the same way, have:
|0||2 | 2 IU VI
|p 1l + 12 (119g 12 + [|gal?2) -c08 7. (10)
cos 7 0os77
2|10 |V||
2|10
[|
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P1
Q2
f
(a) Case
(b) Case
(c)
Distance in case 1 (d) Distance in case 2
Figure 3: Inter-path distances in a simple scenario of two sample points and two noise points.
Algorithm Surrogate method of DANSM
Input: noise set 2 of size and sample set X' of size n.
Output: a noise-sample path set P of size n.
1: Initialize path set p
2: for each sample x in do
3: From Z, find who has shortest path length with
search all in Z > each noise is used only once
4: Remove noise from
5: Add path 2(2,2) into P
6: end for
This reveals the relationship between the path lengths and the inter- path distances. For case as shown in Figs. 3a and 3c] the inter-path distance between Pi and P2 is ||c d]|, where ed 1 ef and d is the foot of the perpendicular. Similarly for case 2, as illustrated in Figs. 3b and 3d the
inter-path distance of Qı and Q2 is ||c hill Assume that |||] [| + ||2]| 2 || 1 42 12 have
cos > and cos < 0, indicating that ||c dll > ||c- hll. This conclusion is proved in appendix A.3
Based on the above deduction, we can conclude that between two samples and two noises, the paths with shorter lengths result in longer inter-path distance, while the paths with longer lengths lead to shorter inter-path distance.
4.2 SURROGATE METHOD OF DANSM
As proved in Sec. 4.1 the path length is negatively correlated with the inter-path distance, indicating that decreasing the former will increase the latter accordingly. This relationship can be utilized in the surrogate method of DANSM. When assigning noises samples, we aim to shorten the noise- sample path lengths, which in turn lengthen the inter-path distances. The minimization objective is defined in Eq. with lower computational complexity than the maximization objective Eq,
12
min a
Σ
|P:a(t))l-
(11)
72
To further reduce processing time, a greedy algorithm employed to pair up noise and sample. Specifically, for each sample in X, selects the nearest available noise from for pairing. The detailed steps are shown in Alg. []] In this way, we do not need to calculate the exact inter-path distance and significantly reduce the computation time. This surrogate method is simple, straight- forward, and most importantly, fast. Its Euclidean distance calculation is highly suitable for GPU parallelization, allowing be executed efficiently.
A recent work, Immiscible Diffusion (Li ct al. 2024), employs similar approach to our surTo- gate method for assigning noises to samples. It aligns with the goal of our DANSM method, but its algorithm functions as another surrogate of DANSM and has not been evaluated on RFM. It is grounded in physics intuition and analogy, lacking a rigorous mathematical foundation. While its noise and image batches have the same size, it does not provide clear rationale that why "equal- sized" batches are necessary. It focuses on shortening noise-sample path lengths but fails to establish the negative correlation between path length and inter- path distance. In summary, although Immis- cible Diffusion identifies an effective method, does not offer deeper explanations for its findings. Further comparison is given in Sec. 5.6

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Inter patth distance
Photerable eble path Kando path
0.22
Epochcled
0:35
Epech:120
Treiniing kett
0.330
Opsch:000
Training lecs
6.30
Spoch:500
tpoch.d0 500
12,991
0.22
0.25
0.20
0.20
Minimal
Average
18.10 38.52 18.54 38.06 18.60 38,00 38.32 inter-path distance
34.00 34.33 34.15 36.15 36.00 35 34.30 34.35 Averege inter-peth distercce
(a) Inter- Inter-path distance
(b) Minimal inter-part path distance
(c) Average inter-path distance
Figure 4: Inter-path distances of CIFAR-10 data from well-trained RFM. (a) Minimal and average
inter-path distances of the preferable and random paths. (b).(c): Training losses vs. distances.
18.25
973
33.800
Epoch 300 Epech-503
distance
18.300
23.000
Training
18:000
13.405
23,22
90.75 19.00 00.23 90.00 19.75 90.33
40.75 10.50 10:25 1020 56.15 56.0 56.5 Peth length
55.75 03.50 85.25 50 39.25 5157 99.25 Path length
Path length
(a) Minimal inter-path distance
(b) Average inter-path distance
(c) Training loss
Figure 5: Path length analysis from well-trained RFM on CIFAR-10. 10. (a), (b): Relationship between path length and inter-path distances. (c): Relationship between path length and training loss.
5 EXPERIMENTS
To verify the effectiveness of the DANSM method on RFM, experiments are conducted on both image space and latent space. For image space, three datasets CIFAR-10 (Krizhevsky et al. 2009), ImageNet64 (Deng et al. 2009), and LSUN Bedroom YYu et al. 2015) are tested. They have image size of 32x32, 64, and 256 256, respectively. For latent space, the autoencoder of Stable Dif- fusion (SD) (Rombach et al. (022) is utilized for the encoding and decoding between images and latent variables. The latent variables serve as clean samples in the training and sampling processes. Once sampling is completed, the generated latent variables are decoded into real images for evalua- tion. Throughout the experiments, the noises (O, I), the image pixel values are normalized to the range of [-1, 1], and the latent variables are kept as their original values without scaling.
5.1 INTER-PATH DISTANCE MATTERS IN TRAINING PROCESS
In this paper, we propose ease the training of rectified flow models lengthening the inter-path distance. Therefore, when conducting the experiments, we firstly validate the importance of inter- path distance in the training process. We compare the inter-path distance of preferable path and random path in Fig. 4a The definitions of preferable and random paths can be found in Sec. 3.2 Based on the well-trained RFM (Liu al. 2022) for CIFAR-10, 10, the average inter-path distance of preferable paths is 35.5, which is greater than that of random paths - 33.9. Similar results are disclosed for minimal inter-path distance. Moreover, Figs. 4b and 4c how training loss evolves as the inter-path distance changes: with distance increasing, the training loss of RFM model decreases accordingly. These results offer the basis that inter-patt path distance plays a critical role in reducing training loss and improving the model training process.
5.2 NEGATIVE CORRELATION BETWERN -NTER-PATH DISTANCE AND PATH LENGTH
As discussed in Sec. 4.I path length is negatively correlated with inter-path distance, which is why we use path length as the optimization surrogate. In this section, we validate this relationship through experiments with RFM on CIFAR-10 dataset. As shown in Fig. 3 when path lengths de- crease. the related minimal and average inter-path distances increase, and the training loss decreases accordingly. The decreasing training losses shown in Fig. [c highlight that shortening the path lengths improves the training data quality, which can, in turn, ease the model's training process.

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Table FID., comparison during RFM training at different epochs and DANSM match-sizes ("ms"),
CIFAR-I0 (10-step sampling)
ImageNet64 (5-stop sampling)
Bedroom (10-step sampling)
spoch
100
200
300
400
500
<poch
40
80
120
160
200
epoch
20
40
60
buseline
24.1
19.9
18.2
17.2
16.5
baseline
74.2
68.4
64.3
60.8
57.1
baseline
84.3
68.0
34.7
ms=1,000
20.9
16.6
14.8
14.2
13.5
ms=100
74.1
67.5
61.2
57.6
54.7
ms=100
69.6
67.3
28.9
9.7
72.2
58.1
19
17
37
110:15
15
15
Epoch
Training time (hours)
FID epoch
(b) FID training time
(c) Baseline
(d) ms:100
(e) ms: 10K
Figure 6: Comparisons between bascline RFM and DANSM with various match-size("mig on latent space of LSUN dataset by 10-step sampling. (c)-(e) Example images generated by different training
models at the same training time (15.4 hours). A.5 includes more comparisons.
5.3 THE PERFORMANCE OF DANSM IN IMAGE SPACE
When training in image space, the DANSM method exhibits superior performance in RFM models compared to the vanilla baseline. To quantify this performance improvement, we evaluate the mod- els at various stages during the training process. At specific epochs, when the model has not yet fully converged, we generate samples using the model and compute FID scores compare the effective- ness of DANSM against the baseline. Tab, Trporrs FID scores across different epochs (in columns) and match-sizes (in rows) different datasets. The last two rows of the table are DANSM models saved out on the specific epochs. The DANSM method consistently achieves lower FID scores com- pared to the baseline, with the lowest FID scores highlighted in bold font. On CIFAR-10, DANSM with match-size 10,000 yields best performance, reducing the FID from 16.5 to 9.7. Similarly, on Bedroom dataset, the DANSM method achieves lower FID scores than the baseline method, demon- strating its effectiveness on large images.
5.4 THE PERFORMANCE DANSM IN LATENT SPACE
We evaluate the DANSM method in the latent space using autoencoder of SD for image synthesis. In the experiments, the dimension of latent space is 4x32x32, and the latent variables are encoded from 50,000 LSUN Bedroom images with the resolution 256x256. To evaluate the effectiveness of the DANSM method, we train the models from scratch with different match-sizes. At specific epochs during the training process, latent samples are generated by the model and decoded into real images by the decoder of SD to calculate FID scores. Additionally, evaluate the improvement in training efficiency. The FID scores over epoch and training time are given in Fig. 6 As shown in the figure, DANSM achieves same FID in significantly less number of epochs and shorter training time for RFM, where match-size 10K has the best performance.
track the training time to
5.5 TRAINING PROCESS ACCELERATION
The DANSM method aims to lengthen the inter-path distance, which eases the training process and reduces training time. However, DANSM itself introduces computational overhead as well, which needs to be carefully managed. A critical consideration whether the performance gained from DANSM can justify the additional computational overhead it introduces.
To evaluate above points, the experiments performed on latent space of LSUN Bedroom images, which are constructed in the same way as Sec. 5.4 Throughout the training process, we record the training time and calculate the FID scores of 10-step sampling specific epochs. As illustrated in
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20
Matcheine:10 10
DASS
20
Match 200. 1.000.000
Training time (hours)
DANCH
90
Traiking bast 250
10
30
50
50
15
20
Thaniing spocts 500
20
Natchsiou-100
20
Batch wow.h:hhooooo...o 0,0000
20.5 23.4 23. 2L1
15
10:1
60
1,000
10,000
Training time (hours)
Training time (hours)
Match-size
(a) FID of match-size 10 & 100
(b) FID of match-size IK & 10K
(c) Training time match-size
Figure 7: Comparison of DANSM and Immiscible Diffusion on the latent space of LSUN Bedroom.
Table 2: Path lengths and inter-pan path distances across different match-sizes ("ms"). The "length" is path length, and "baseline" refers to random paths of RFM without DANSM. Meanwhile, "dang" and "dmmin" represent the average inter-path distance and minimal inter-path distance, respectively.
images (3 xx32xx32
Bedroom latent variables (4 x 32 32)
Redroom (3 256
mis
baseline
100
1K
10K
100K
baveline
100
1K
10K
100K
baxeline
10
100
1K
length
62.04 61.73
61.36
61.06
60.83
60.64
83.47
82.97
82.40
81.92
81.53
81.19
502.57 502. 18 50188 281.63
502.89
dang
34.24
34.36
34.45
56.77
57.03
$7.28
281.82
282.05
282.24
17.99
18.02
18.06
18.10
18.13
18.17
47.65
47.88
48.17
48.41
48.62
48.79
192,43 192.46
192.54 192.64
Fig. 6b the DANSM method significantly reduces training time compared to the baselines. For example, when baseline RFM reaching the FID) score of 15.3, the DANSM method with match- size 10,000 cuts training time from 25.5 15.4 hours, saving 39.6% of the total training time. demonstrates that the DANSM method can reduce training time by a large margin.
5.6 COMPARISON WITH IMMISCIBLE DIFFUSION
Immiscible Diffusion Li al. 2024 has the same goal as DANSM but utilizes the Hungarian algo- rithm (Kubn 1955) as its optimization method. Given n noises and samples, Immiscible Diffusion incurs a time complexity of (n²) the noise-sample matching procedure while DANSM obtains more efficient (n²) complexity. For fair comparison, two identical RFM models are trained from scratch for 500 epochs on the latent space of LSUN Bedroom by the two methods. Throughout the training process, FID scores calculated for 10-step sampling results at specific epochs: 100, 200, 300, 400, and 500, and total training time is tracked to analyze the overhead introduced by the noise-sample matching. The dots in Fig. represent the epochs at which FID scores are calculated. As shown in the figure, DANSM achieves comparable FID scores with Immiscible Diffusion but requires less training time, especially with match-size of 10,000. Further analysis is provided in A.4
5.7 ABLATION
Ablation studies are conducted the only hyperparameter of DANSM match-size. Tab. shows how the match-size influences the path lengths and inter-path distances (both average and minimal) across three datasets: CIFAR-IO, Bedroom latent, and Bedroom image. For CIFAR-IO, the path length starts at 62.04 and decreases gradually as match-size increases to 100K, where the path length becomes 60.64. The average inter-path distance (daug) and minimal inter-path distance (dman) follow slightly upward trend, with daeg increasing from 33.97 to 34.45, and dmin from 17.99 to 18.17. The similar trends are shown for Bedroom latent and Bedroom image datasets. should be highlighted that even slight shortening in path length, slight increasing in inter-path distance, can contribute to speed up the training process significantly.
Furthermore, to better disclose the mechanism of DANSM, we report the t" timestep (Eq. 4 that DANSM finds. As explained in Sec. 3.1 t* is the timestep at which the closest inter-path dis- tance occurs. t 0 corresponds to clean sample and f = 1 represents pure noise. Meanwhile, different datasets have different data distributions, which leads to different t" values. For example, the CIFAR-10 10 image pixel values scaled the range [-1, 1). resulting in mean-varte vrean-variance values of ( (-0.053, 0.253). In contrast, the latent space variables in SD exhibit mean- variance values of (0.13, 0.68). Their 1" values are compared in Tab. 3. which also shows the contrasts of path lengths

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Data of preferable and random paths on different datasets. The terms "dayg" andd "dmin" refer to the average and minimal inter-pan patt-path distances, respectively, while "Fayy" and " min represent the timesteps where the average and minimal distances occur.
Table 3:
SUK CIFAR- 10 images (3 x2x2)
50K Bedroom latent (4x332 3x32XX x.2)
10K SD latent (4 x64 X64)
length dang
faug
longth
daup
fang
longth
dave
faug
preferable random
0.1645
0,0075
55.10
68.76
0.2004
129.27 165,91
0.2882
115.08 0.1559 98, 0.2993
62.04 33,97 0.1916 17.99
0.0557
83:47
55.74
0,3803
47.65
0,2870
110.93 0,3761
32
Baseline
34
Baseline
10
ms: 10
ms: 100
ms: 100
1K
mi: IK
29
me: 10K
ms: 10K
28
910:2
26
200
300
400
500
600
5
8
Epoch
Training time (hours)
(a) FID vs. epoch
(b) FID vs. training time
(c) Baseline
(d) ms:100
(e) ms:10K
Figure DM on latent of Bedroom by 10-step sampling: comparison between baseline and DANSM with various match-sizes ("ms"). (c)-(e) images are generated by models at the same training time (5.37 hours) but with different training methods.
and inter-path distances between preferable and random paths. These results show that the prefer- able paths have shorter lengths, greater average and minimal inter-path distances, and lower t values compared random paths. Another notable finding is that values are small in both preferable and random paths, indicating the closest inter-path distance happens near the clean sample.
5.8 GENERALIZATION TO GENERAL DIFFUSION MODELS
DANSM is proposed based on rectified flow models. However, it can be generalized to diffusion models (DM) (Song et al., 2021a) without any extra adaptation. On CIFAR-10 10 and latent Bedroom datasets, DANSM outperforms baseline in DM model. Tab. shows that on CIFAR- 10, it yields best performance with match- size 1,000, reducing the FID) from 93 to 58. Moreover, Fig. 8 shows the improvement on latent Bedroom. To reach FID of 27.8. the baseline requires 8.55 hours whereas DANSM with match-size 100 takes only 5.37 hours (as shown in Fig. 8b), resulting in a 37.2% reduction in training time.
Table 4: FID, comparison be- tween different epochs of DM with DANSM.
DM with DANSM CIFAR-10 (3-step sampling)
10
epock
200
400
600
800
1,000
buseline
120
106
99
94
93
ms=1,000
62
ms=5,000
142
108
89
79
CONCLUSION
In this paper, we aim to ease the training process by increasing inter-path distance. Using the straight path property of rectified flow models, we first derive closed-form formula to calculate the inter- path distance and propose our method, DANSM. Furthermore, we derive the negative correlation between inter-path distance and path length. Based on this relationship, we use path length as a sur- rogate of DANSM. Although DANSM is developed based on rectified flow models, the experimental results show that provides excellent speed up for both rectified flow models and diffusion models. DANSM is simple, scalable, and fast, leading to substantial improvements in training efficiency.
ACKNOWLEDGMENTY
This research is supported by the National Research Foundation, Singapore and Infocomm Media Development Authority under its Trust Tech Funding Initiative. Any opinions, findings and conclu- sions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore and Infocomm Media Development Authority.
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REFERENCES
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Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans image synthesis. In Ad- vances Neural Information Processing Systems, 2021.
Ermongroup. Source code and well-trained model checkpoints of DDIM. https://github. Com/e htmm://rmmnrrou/ddd..... 2021.
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.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, 2020.
Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical Report TR-2009, University of Toronto, 2009.
Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 1955.
Yiheng Li, Heyang Jiang, Akio Kodaira, Masayoshi Tomizuka, Kurt Keutzer, and Chenfeng Xu. Immiscible diffusion: Accelerating diffusion training with noise assignment, 2024. URL https:
// aax.iv.rrg.........a..........oe httpg///b////4..1222
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Cedric Villani. Topics in optimal transportation. American Mathematical Soc., 2021.
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A APPENDIX
A.1
INTER-PATH DISTANCE DEDUCTION
Given two noise-sample pairs, (1) and (22, the paths are defined by the timestep as follows:
ri (1 t)) 12g (12 = (1 - 1) + lz2
E 1].
(12)
Furthermore, the distance between 71 and r2 is defined as the Euclidean distance between the points corresponding to t:
(t) = ||r m ll
[[ t) t22 (1 (1-t)a: f)x1 - tzill2
(13)
[I (2 x1) + t(22 21)l22
In the following derivation, we simplify the notation by using f(t) to present (t). Let V = 32 - 2i and = 22 21, we get new expression of f(1):
f(t) = ||V + t(U - V)
||(1 t)V +U/12
(14)
Its derivative w.r.t. timestep is:
(1(1) (t)
(V + ((--V)) (U--v))(UVv) VV-
(15)
+ tUil2
As [(1) is concave parabola, its minimal value occurs when the derivative f'(i) 0.
VT(V U)
i (V - UV
(16)
A.2 DEDUCTION OF EQ.
Using the law of cosine, derive cos 7 as:
|1 | UU
VII2
Cos 7
2|| [V]
20V
21|011 IIVI
2(22 - 21 (x) x1)
211U11 |
2z2222 + 22 2zzT1 221
(17)
2 VII
o9 27 + x3 + 23 222011 22102 (af + 29 + +m3 22 222
221
211111 - V|
| - 21 21 + ||||| - 22 2 - (1|:0) - 21 1| + || =21l²)
2|107| V|
1 2| + |2 1 (1]p1] 1| + [p2)l2)
2 00 [V]]
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d
d
-
U
V
C
v
U
C
g
g
h
(a) Point d locates in line segment cf
(b) Point locates out of line segment cf
Figure 9: Comparing the inter-path distances between case and case of Fig.
A.3 COSINE VALUE vs. INTER-PATHE DISTANCE
In this section, compare the inter- path distances of case 1 and case from Fig. of Sec. The basic setup is illustrated in Fig. 9a Case involves the triangle Aefe, where Lfee ed 1 ef and point is the foot the perpendicular. Similarly, case 2 involves the triangle Aef g. where Lfeg 7. ch 1 fg and point is the foot of the perpendicular, Given that the vectors |.. and both V and V lie on the same straight line. Consequently. the two triangles have equal areas, Aefe = Aefg. On the other hand, as concluded in Sec.
V = = we have [|V||
< 0 and cos 7 > O. Therefore, the angles 7. All these variables are shown in Fig. The lengths of the line segments have the following relationship:
Acfg Lefe
{>2
( ||c - h|l 11g - fll = |cc dl - fll
(18)
[g fl > le (c-11 fll
||cc h|l < Ic lddl]. dll.
Meanwhile, since y is an acute angle, the perpendicular foot d may occupy different positions relative the line segment e/. could lie on e/, as shown in Fig. 9a fall outside of ef. as illustrated in Fig. 9b In the latter scenario, the inter-path distance is ||c ell. Notably, ||c All < ||c dll than that in case 2.
c - ell. thus confirming that the inter-path distance case is greater
A.4 COMPARISON WITH IMMISCIBLE DIFFUSION
The key feature of Immiscible Diffusion (Li et al. 2024) is the noise-sample assignment by "one line code", which stems from Scipy (Virtanen et al. 2020 library. The code snippet is:
scipy opt imi ze.
()
However, the above code operates only at the CPU level and is incompatible with GPU acceleration. This limitation prevents it from leveraging parallel processing techniques. making it unsuitable for efficiently handling large-scale inputs. Therefore,
the match-size increases, such as 10,000 in Fig. 7c the computational overhead of noise-sample assignment (referred to overhead) rises significantly, becoming major bottle neck of the training process. Unfortunately, the Immiscible Diffusion paper evaluates their generation quality soicly based on the number of training epochs (as shown in their figures 4, 6, and 8), without considering the overhead. Our DANSM method takes computational overhead into account, ensuring thorough and fair evaluation by comparing generation quality based on total training time, including the overhead (as in Figs. 6b and 8b).
It is worth noting that in Fig. 7c Immiscible Diffusion requires much more time than DANSM with match-size of 10. The reason is closely related the batch size used in the experiment, which is set to 250. Consequently, for each batch, Immiscible Diffusion performs 25 CPU-based operations
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to optimize the noise-sample matching. These repeated CPU calls become the primary source of overhead, impacting the overall computation efficiency.
Additionally, present FID comparison results based on training epochs. This evaluation excludes the computational overhead the noise-sample matching procedures and focuses solely on the effects of data optimization. As shown in Fig. 10] both methods exhibit comparable effectiveness in optimizing noise-sample matching.
20
20
Match-size:10
DANSM
Match-size:1,000
DANSM Immiscible
DiMusion
100
200
300
400
500
100
200
300
400
500
Match-size:100
20
Match-size:10.000
EID
15
15
100
200
300
100
500
100
200
300
400
500
Epoch
Epoch
Figure 10: FIDI comparison between DANSM and Immiscible Diffusion for RFM models on the latent space of the LSUN Bedroom dataset. The FID values are computed based on a 10-step sampling process, Both methods demonstrate similar FID scores across various match-sizes.
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A.5 VISUAL COMPARISON OF DIFFERENT METHODS TRAINED ON LATENT OF BEDROOM
(a) Vanilla
ms:100
(c) ms:10% 10K
(d) Vanilla
(c) ms:100
(f) ms:10K
Figure 11: RFM on latent of Bedroom. Visual comparison of images generated by RFM models.
The models are trained on the same latent space using different methods: vanilla RFM (columns and d), DANSM with match-size 100 (columns b and e), and DANSM with match-size 10K (columns e and f). Different rows are generated by different sampling steps: 3-step (rows and 5), 4-step (rows 2 and 6), 5-step (rows 3 and 7), and 10-step (rows 4 and 8).
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A.6 VISUAL COMPARISON OF DIFFERENT METHODS TRAINED ON IMAGENET
Vanilla ms: 1s:
ms:1K
Vanilla ms:100
ms:1K
Vanilla ms:100
ms:1K Vanilla ms: 100
ms:1K
Figure 12: Visual comparison of ImageNet images (64x64) generated by RFM models. The models are trained on the same dataset using different methods: vanilla RFM, DANSM with match-size 100 ("ms: 100"), and DANSM with match-size 1000 ("ms:/K") Different rows are generated by different sampling steps: 3-step (rows 1, 5, and 13), 4-step (rows 2, 6, 10 and 14), 5-step (rows 11 and 15), and 10-step (rows 4, 8, 12 and 16).
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A.7 COMPARISON OF DIFFERENT METHODS TRAINED ON FFHQ
Table 5: FID4 comparison between different epochs of RFM training processes FFHQ(256x256)
images, where "ms" means match-size of DANSM. The FID scores are calculated on images gener- ated by 10-step sampling process.
epoch
20
40
60
80
100
buseline
121.92
104.59
97.25
89.97
86.76
msm1.000 1,000
95.46
87.24
78.18
75.27
66.95
(a) Vanilla
ms:100
(c) ms:1000
(d) Vanilla
(e) ms:100
(f) ms: 1000
Figure 13: RFM on FFHQ. Visual comparison of images generated by RFM models. The models are trained on the same dataset using different methods: vanilla RFM (columns a and d), DANSM with match-size 100 (columns b and e), and DANSM with match-size 1000 (columns and f). Different rows are generated by different sampling steps: 3-step (rows 1), 4-step (rows 2), 5-step (rows 3), and 10-step (rows 4).
A.8 FID COMPARISON WITH FEWER STEPS
Table 6: FIDJ comparison of RFM model training different epochs and match-sizes ("ms").
RFM with DANSM on CIFAR-10
RFM with DANSM Bedroom
(2 (2-step sampling)
(2-step sampling)
epoch
100
200
300
400
20
40
60
80
haveline
ms=5.000
88.13
87.91
84.61
83.67
82.41
ms=100 100
248
203
184
173
186
ms=50.000
74.42
73.42
74.15
73.53
73.52
ms=500
164
18


Reviews:
[{"review_id": "GhuB5xAzlj", "reviewer": "Reviewer_oVk9", "summary": "This work proposes Distance-AwareNoise-Sample Matching (DANSM) to increase training speed in generative models without any loss (in fact gain) in performance. The proposed method is inspired by \"the difference between random paths used in training and preferable paths from well-trained models\".\nThe work is theoretically motivated and sound.\nThe work offers limited evaluations, however, they seem to be sufficient to make the case.", "questions": "Q1- I would appreciate a better explanation for Figure 6 (b), the plot looks a bit unclear to me. Additionally, there are lines behind the legend curbing my ability to understand the plot completely. My essential question is that unlike Figure 7, in Figure 6 (b) I do not see any gains in training times when using the proposed method. Is this understanding of mine correct?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The work is well written except for a few typos (for example, in line 198/199 \"To **analysis**\" is written instead of \"To **analyse**\", and in line 309/310 \"have not been **evaluation** on RFM\" is written instead of **evaluated**).\n\nThe idea is sound and is theoretically motivated, the results are in line with the hypothesis.\nThe implementation seems simple and straightforward. \n\nThe claims made in the paper seem to be backed by empirical evaluations.", "weaknesses": "I have 2 major points in this regard:\n1. The plots are not well made, some of the lines in the plots, for example, Figure 6(b), Figure 7(b), and Figure (b) seem to be starting from arbitrary locations. It would help to invest more time and make better-looking plots without these artifacts, or if they are not artifacts then an explanation for the same would be very helpful.\n\n2. The work lacks comparisons to \"Immiscible Diffusion\", while the paper does point out the key differences in lines 307-315, it would be interesting to see empirical evaluations in comparison to \"Immiscible Diffusion\" since the methods are closely related.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IUxNNCOSgN", "reviewer": "Reviewer_zPez", "summary": "This paper explores how different diffusion methods yield similar results with the same dataset, identifying \"preferable noises\" for samples. It introduces the Distance-Aware Noise-Sample Matching (DANSM) method to optimize training by increasing inter-path distances. DANSM significantly speeds up training by 30%–40% without losing quality, offering insights into enhancing diffusion model efficiency.", "questions": "1. Include a comparison using longer sampling steps.\n\n2. Conduct more extensive experiments on complex datasets.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The diagram is very clear, making it easy to understand how different diffusion methods yield similar results with the same dataset, identifying \"preferable noises\" for samples.\n\n2. The proposed method significantly speeds up training by 30%–40% without losing quality, offering new insights into enhancing diffusion model efficiency.", "weaknesses": "1. The FID calculations are measured with very few sampling steps, making it difficult to ensure the quality of the generated results. It would be beneficial to provide a comparison of different methods while maintaining image quality, ideally with visualizations, such as images decoded using stable diffusion.\n\n2. The experiments were only validated on CIFAR and LSUN-BEDROOM datasets. Validation on a broader range of datasets, such as ImageNet and FFHQ, would provide more comprehensive insights. Additionally, including some actual generated images for visual comparison would be advantageous.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SCIsy6EPxI", "reviewer": "Reviewer_Ly7e", "summary": "The paper reveals that different diffusion models trained on the same dataset tend to produce similar outputs when given the same input noise. This suggests the existence of \"preferable noise-sample pairs\" in the training process. The authors propose the Distance-Aware Noise-Sample Matching method to lengthen the inter-path distance for speeding up the training of diffusion-based models. The experiments show that the proposed method improves the training speed by about 30%~40%.", "questions": "Why is it necessary to learn latent-noise pairs that are well-aligned in the t-SNE space? This might suggest that straight flow matching is preferable, but the underlying reasoning is not clearly explained in the paper. Furthermore, I am confused about whether \"consistent model reproducibility\" is beneficial or detrimental for diffusion-based models. The fact that different model architectures can generate similar images from the same noise might indicate limited generation patterns due to the isotropic nature of the Gaussian noise.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper presents a framework that simplifies the optimization of inter-path distances to path length optimization, based on the observations of noise-sample pair relationships in diffusion models.\n\nThe approach leverages closed-form formulas derived from rectified flow models to enable efficient optimization without requiring architectural modifications, making it easy to integrate with existing methods.", "weaknesses": "The authors heavily emphasize \"consistent model reproducibility,\" which seems interesting but abruptly transitions to proposing a method for improving training speed. This makes the main idea of the paper quite unclear.\n\nThe presentation of the proposed method is vague, and Algorithm 1 cannot be effectively reproduced with the provided steps. I suggest adding mathematical formulations in Sections 3.3 and 4.2 to rigorously explain how the proposed model is optimized.\n\nRegarding experiments, the paper lacks visual comparison results, which are crucial for method evaluation, especially when the FID quantitative metrics show minimal differences at the tiny scale discussed. I also recommend comparing with more methods and incorporating the proposed training speed improvement approach across more SD models. I believe the paper should present higher-resolution and larger-scale generation results. Given the computational benefits brought by this work's methodology, these tasks should be more computationally feasible.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)