On the Design of One-step Diffusion via Shortcutting Flow Paths
Authors: Haitao Lin
Peiyan Hu
Minsi Ren
Zhifeng Gao
Zhi-Ming Ma
Guolin Ke
Tailin Wu
Stan Z. Li
Venue: ICLR 2026 Poster (2026)
Keywords: Few-step Diffusion
Shortcut Model
Diffusion Model
Flow Matching
Primary area: generative models

Abstract: Recent advances in few-step diffusion models have demonstrated their efficiency and effectiveness by shortcutting the probabilistic paths of diffusion models, especially in training one-step diffusion models from scratch (\emph{a.k.a.} shortcut models). However, their theoretical derivation and practical implementation are often closely coupled, which obscures the design space.
To address this, we propose a common design framework for representative shortcut models. This framework provides theoretical justification for their validity and disentangles concrete component-level choices, thereby enabling systematic identification of improvements. With our proposed improvements, the resulting one-step model achieves a new state-of-the-art FID50k of 2.85 on ImageNet-256×256 under the classifier-free guidance setting with one step generation, and further reaches FID50k of 2.53 with 2× training steps. Remarkably, the model requires no pre-training, distillation, or curriculum learning.
We believe our work lowers the barrier to component-level innovation in shortcut models and facilitates principled exploration of their design space.

Published as a conference paper at ICLR 2026
ON THE DESIGN OF ONE-STEP DIFFUSION VIA SHORTCUTTING FLOW PATHS
Haitao Lin" Peiyan Hu 1,2 & Minsi Ren'
Zhifeng Gao gaozfēdp. tech
(linhaitao, hupeiyan, renminsi j@westlake. edu. cn
Zhi-Ming Ma²,
Guolin kel
Tailin Wa'n & Stan Z. Li (wutailin, stan. zq (.l..ijeewe @ewestllkee wew.stteeeeeeeeee.eee...
mazm@ant. ac cn
kegledp. tech
Department of Artificial Intelligence, School of Engineering, Westlake University;
2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences;
DP Technology, Beijing.
Code Repository: https: /ggi thub. wtw.://EEANNNNN/////ppppccccttt. git
Project page: httpp://edap/ii htt.hh..oooppplllsttooooooooooop
ABSTRACT
Recent advances in few-step diffusion models have demonstrated their efficiency and effectiveness by shortcutting the probabilistic paths of diffusion models, espe- cially in training one-step diffusion models from scratch (a.ka. shortcut models). However, their theoretical derivation and practical implementation are often closely coupled, which obscures the design space. To address this, we propose a common design framework for representative shortcut models. This framework provides theoretical justification for their validity and disentangles concrete component-lewe choices, thereby enabling systematic identification of improvements. With our pro- posed improvements, the resulting one-step model achieves a new state-of-the-aa FID50K of 2.85 on ImageNet-25662566 under the classifier-free guidance setting with one step generation, and further reaches FID50k of 2.53 with 2x training steps. Remarkably, the model requires no pre-training, distillation, or curriculum learning. We believe our work lowers the barrier to component-leved innovation in shortcut models and facilitates principled exploration of their design space.
INTRODUCTION
Diffusion-based models have become the dominant paradigm in deep generative modeling (Sohl- Dickstein al., 2015; Ho et al., 2020; Song al., 2020), progressively transforming samples from prior distribution toward the data distribution. However, dozens or even hundreds of neural function evaluations (NFEs) are typically required, resulting in slow inference and limited real-time use (Song& & Ermon, 2020; Salimans & Ho, 2022; Lu et al., 2025; Zheng et al., 2023). Consistency models (Song et al., 2023; Song & Dhariwal, 2023) are pioneering works that attempt to achieve one- step generation (Luo et al., 2023; Wang et al., 2023; Yin et al., 2024a;c; Salimans et al., 2024; Geng et al., 2023; 2025b), but a costly two-stage training process is required, i.e., first training a reliable diffusion model and then distilling velocity score from it. Despite the costly two-stage training, they offer fast generation, which motivates further research into improving training efficiency.
Recently, one-step diffusion models trained from seratch have emerged, such Consistency Train- ing (CT) (Song et al., 2023) as the training-frome from-scratch variant of consistency models, Inductive Moment Matching (IMM) (Zhou et al., 2025), and Shortcut Diffusion (SCD) (Frans al., 2025). These models aim to learn direct shortcut mappings between intermediate states along the probability flow trajectories of the probability flow, thus enabling one-step generation; refer such models as shortcut models. Building on this principle, continuous-tin Song. 2025) and MeanFlow (Geng al., 2025a) have been introduced, achieving ofate-of-the-an performance in one-step generation for image synthesis. Their efficiency and effectiveness in both training and generation have stimulated further exploration improving their sampling fidelity.
shortcut models such as sCT (Lu &
"Equal contribution. Corresponning authors.

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Published as a conference paper at ICLR 2026
Although these models share the same objective, the barrier to understanding the working mechanisms remains non-trivial. Specifically, the literature on them is dense on theory, derivations of method formulations and the corresponding learning objectives, well as technical details like time samplers and curriculum, and training tricks, etc., leading to a less intuitive design paradigm. As a result, it may inadvertently obscure the underlying design space, making each carefully crafted module appear indispensable, so that altering single component seems to threaten the integrity of the entire system.
Therefore, we first contribute to proposing common design framework for these shortcut models from a practical standpoint. We summarize that both discrete- and continuous-time variants share the principle of upproximating two-step flow map targets with one-step parameterized predictions. We also provide a general theoretical justification for the validity of this design paradigm. This framework allows us to disentangle the concrete modules within these models, offering clearer insights into how the components interact and what flexibility remains in shaping the overall method design.
Secondly, our contribution lies in elucidating the design space of shortcut models. We decompose each model into distinct modules aligned with their learning objectives, and then conduct in-depth empirical investigation and theoretical analysis of different module combinations. In summary, we demonstrate the advantages of linear paths in settings of shortcut model trained from scratch, discuss the scenarios where continuous- tituuouuu---eeeeeeo variants exhibit superior sampling fidelity over discrete-time ones, and figure out the impacts of time samplers on training convergence.
Further, the third set of contributions centers on improvements to the training of continuous-time shortcut models. Building on the previous analysis, we introduce three technical refinements for enhancing training stability: (i) the use of plug-in velocity and its correction under classifier-fre free- guidance training. (ii) gradual time sampler, and (iii) several established training techniques such as variational adaptive loss weighting. Our experiments demonstrate that these techniques consistently improve performance. Finally, we conduct a scaling-up up evaluation on ImageNet-2266 x256. By incorporating the proposed improvements into our modeling framework, we achieve an FID50k of 2.85 under one-step generation, setting a new state of the art among shortcut models trained from scratch. We believe that our work facilitates component-level innovation and thereby enables more systematic and targeted exploration of the design space of shortcut models.
EXPRESSING ONE-STEP DIFFUSION THROUGH SHORTCUT MODELS
2.1 SHORTCUTTING FLOWS WITH FLOW MAP SOLVERS
Diffusion models. Let Paata be the data distribution, and Pprior 2 be Gaussian distribution with zero mean and variance o². In the following, we write 1 by default for notational simplicity. According to stochastic interpolants (Albergo al., 2023), diffusion models establish probabilistic path between Po Pdatn and pl Pprior such that Ø, ar + O,E, where 3o ~ po. E ~ Pt, and aie, of 2 O; with boundary conditions ag = 0 = and a' = oo = Both the forward noising and inverse denoising processes are governed by the probability flow ODE (PF-ODE) as d; = vy (2))). where (x) is the marginal velocity v (x) = E( æ = x) + à, (E [x. = x).
Flow paths. Probabilistic paths satisfying the above are defined flow paths. For example, with the reformulation by Lu & Song (2025), the EDM preconditioner path (Karras et al., 2022) can be transformed cosine path (Ma et al., 2024) with a daat. a₁ = cos( ( and a. = sin( [ in Rectified Flow (Liu et al., 2022), o = 1, ag = 1 t and o; = t, leading to the linear path (Lipman et al., 2023; Tong et al., 2024). Since 1i (z) inaccessible, the conditional path is established for tractable training, where the corresponding conditional velocity is 1yjo that neural networks F (x,, t) are trained to approximate. In sampling, one can first sample 21 8 ~ Ppriors and then simulate trajectory of the flow through the PF-ODE as o; = F (xx, t).
U2 (, = óg + ôye
Flow maps. In order to shorteut established flow paths from time t (0 < S 1), we introduce the flow map notation (Boffi et al., 2025b; Liu, 2025; Boffi et al., 2025a) to express the design frame for simplicity. A flow map Xt,- is defined as the unique map such that Xx,r (ax) = 2r- for all rI, [o, 1J2, where æ, is the solution of PF-ODE, which corresponds to position in physics.
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sãcal Joeal prodiction
pratticl targt
X
practical prodiction
M
(b) ideal learning of DTSC
(d) practical learning of DTSC
ideal direction
Prjo
(r- ()dugr//tt
t)
(c) ideal learning of CTSC
(a) marginal velocity field
(e) practical learning of CTSC
Figure 1: The physical picture of ideal and practical learning of discrete- and continuous- timouusstttmeeee shortcut models (DTSC&CTSC) where u, denotes the target obtained by the two-step flow maps, and uf the models' prediction for one-step flow maps. (a) shows the marginal velocity field from N(0, 1) to Gaussian Mixture. (b) and (c) illustrate the ideal learning of DTSC and CTSC, where ær is sampled from the same trajectory of PF-ODE, and thus U serves the correct supervisory signal for training. (d) and (e) depict the practical learning of DTSC and CTSC, where the targets deviate from the trajectory, thus leading to models' prediction drifts away correspondingly.
Egg
According to the PF-ODE, one can easily derive the flow map solution through
2r = Xx,)( = x. + J," (x(x))dr
(1)
where J,¹ v, (z,)dr corresponds to displacement in physics.
Flow map solvers. With the definition of average velocity over time (Geng al., 2025a) as ur. we can rewrite Eq. I to express the flow map solution = (xx) through
(xt) = + (r (r-t) t) uher (2:e)
where (x,)) = --t J. Dr (z(2)) dr,
1
or to infer (x+) with the instantaneous velocity nt, through DDIM-solver (Song et al., 2021) as first-order approximation of DPM-solver (Lu al., 2022), which reads
(x,) = DDIM t,r) ā,,, + 3,,
(3)
where at,- = cos( (r-t)) and Br. 2 sin( 3 (r-t)) in cosine paths; and ã,,- and 3 r-
in linear paths. The general formulation and detailed derivation are given Appendix A.2.
With the solvers, if a model learns the solution to the flow maps from any to it can bypass the costly iterative procedure and achieve one-step generation by predicting X Xf,o(x)).
2.2 LEARNING TO SHORTCUT FLOW PATHS
Overall design frame. We claim that the previous methods shorteut the flow paths of diffusion model by regularizing a one-step flow map prediction against the two-step flow map target. In practice, they first sample time points 8, ( p(T) with I I, and then the consistency property (Liu, 2025; Boffi et al., 2025b) as detailed in Appendix A.1 to design a shortcut model trained from scratch, which reads
X (x,,)) (x.)
Specifically, these methods aim to construct a two-step flow map target from t to s, then to r, i.e., X o (x.), and then make the parameterized flow map X X², (2x,) approximate this target in
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single step. It allows the model to achieve one-step generation by predicting z! Xf,0(21) with 21 = E ~ p1- As a result, their learning objectives can be expressed as
one-step prediction
two-stop target
arg min Br Br..drpl)) a pppp w(r d( X2, (3:) (x))
sg (Xa, (xa((e)) )]
(5)
((
where is the weight term, X and X are flow maps obtained with the conditional velocity or the neural network F d(..-) is a loss metric function, such as the squared lg-distance, and sg(.) is the stop gradient operator in backpropagation. We call Xx (x,) two-step flow map targets and X {x, (2xx) one-step flow map predictions, and write the inner loss term of expectation as (at,, r, 8, 8,,,; 1;
Time sampler. To construct the training objective, time points (r, 8, () are sampled with S S t. We refer to this as the discrete-ting shortcut model (DTSC) when {r, s, tt are discrete time points. For example, in CTs, is fixed at 0, and and s sampled from non-uniform discretization curriculum that gradually changes from sparse to dense such that is always chosen to be one time step ahead of 8; SCD divides the time interval into equal segments based different powers of 2, and samples uniformly between adjacent grid points with spacing f, as denoted by (t, h) ~ Uniform log2(t, h). IMM samples time with r and uniformly from [0, 1], and (s, separated by fixed gap. As the gap between two time points becomes infinitesimal, the discrete-time shortcut model converges to continuous time form (CTSC). For example, sCTs and MeanFlows recover this by setting >
Network parameterization and flow map solution. We denote by of the neural network with parameters 0, whose architecture is instantiated as U-Net (Song et al., 2020) in the pixel space, or as DiT (Peebles & Xie, 2022) SiT (Ma et al., 2024) in the latent space. To obtain the flow map, Eq. Eq. 2, and Eq. 3 can all serve as solutions. Since the integral term J er (2, )dr in Eq. 1 is intractable in general, the DDIM solver with instantaneous velocity is adopted practically when estimating the flow map with of parameterized by Fd or the conditional velocity tyoo through Eq. 3. Alternatively, if F" parameterizes average velocity uf,-- the flow map be obtained directly through Eq. 2.
2.3 EXAMPLES: DISCRETE- AND CONTINUOUS-TIM SHORTCUT MODELS
Discrete-time shortcut models. CTs, SCDs and IMMs are representative DTSCs. If parameterizing velocity with neural networks as F (x, 1) = of (az). we can then adopt the DDIM as flow map solvers. Specifically, we first use the parameterized velocity uf (zx) to solve the flow map a xf,(x,) (æt), which serves as the one-step prediction. For the two-step target, we alternate between the conditional velocity Uyjo to obtain a. = (xx) and the parameterized velocity vº (ăt.) to obtain é, (..) In this way, can derive ((x,, 8, t; 0) in Eq. 5 as
let (ar, r, 8, 4,00 = LPIPS
(EPIP((DDIMO) (DDIM of (ax)),,, 4,),, sg (DDDMM(( (())),
(6)
where the loss metric is LPIPS (Zhang et al., 2018) applied directly in pixel space with w 1, and (er, t; O) coincides with its formulation in CTs with details shown in Appendix B. Alternatively, if we parameterize the average velocity as (x,) = F (x t, r), and estimate both the one-step prediction and the two steps in the target with neural networks Fo, we thus obtain the 1( r, 8, 4, 0) SCD through Eq. Due to equi-spacing time points as
s - h, it reads
lacd ( F,
8, r, 0) (2:) 2 ( + u.,
where the loss metric is set as squared ly-distance, a and a, x; + (s t)u",, (z))) with details shown in Appendix B.2. Moreover, for IMMs, conditional samples (1) e(0) j=z mini-batch of size are first partitioned into different groups, and s,t are drawn for each group. With similar flow map construction to CTs, the loss metric is implemented with a grouped kernel function as to minimize MMD (Gretton et al., 2012), applied to measure both inter- and intra-sample similarities between a:)) within the group, as further detailed in Appendix B.3.
e within
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Table I: Specific design choices employed by different shorteut models. "sg EMA decay' means that the parameters 0 in the stop- -apgrrddient targets are updated delayed manner with EMA.
CT
SCD
IMM
SC
MeanFlow
Diffusion basis!
Flow path
Cosine
Linear
Linear
Cosine
Linear
Network
Architecture
U-Net
DiF
DTT
U--Nrr
DrT
Outpat
e²
x2"
Flow map construction
locks
aretuni
fir 200 I]
2 di
aaretan
r. (sigmoid( r.).
sigmuidtrs)
nh I
where
s.t. and with
arctani chai
yi 200001
(P Mia)
Time sampler
tel (old) all))")
Play
Mayta Af Pa))
where Uniform log, (r.A)
24/0....
Two-step larget
1st-step (8,)
DDIM Po DDIM (f.
an Rod (in))
DDIM Pgo DDIMEE. a.
DDMM Rgg 5,4)
DDIM Rog du (r- (on),
2nd-step (8x)
du (R.)
DD el
One-step prodiction (z/))
DDIM(a, a..).))
a: (ia))
DDIM(E, (x)
DDIM of G.))
as + (e- (ju.,
Training
Loss metric d'
LPIPS
Squared distaaeeee
Grouped kemnel
Squared by distance
Squared by distancce
sg EMA decay
!Demonstration of the configuration on ImageNet. *In CT, P. Tacxs oii are adopted from EDM, usually set as 7, 0.001 and 80. gradually increases from Kmn. (usually set as 2) to Kms (usually about 200); In CT's original paper, network output's are the score function and the reformulation is given in Appendix A.3. + is usually set as 12. 31n sCT and MeanFlow, since = 1 dt, which involves differentiaton west r, tems in loss metrics are normalized by dt. The expression is intuitive analogy, while the derivation given in Appendix B. A Although sCT originally initialized from tencher diffusion model, suppose that it attain comparable performance when trained from scratch, similar the behuvior observed in CT and MeanFlow.
Continuous-ting shorteut models. When the difference between two time points is infinitesi- mal, the resulting shortcut models are referred to CTSCs, by setting s dt and normalizing 1(2,, r, t; 0) by dt. For instance, MeanFlows are continuous time shortcut models in which 1 dr. They leverage linear paths with squared l2-distance as the loss metric and parame- terizes the average velocity ue (1:z) with neural networks F By writing 1(x,,, r, dl, 1;
w de (a:) - Xi-dt.r de and d de (xx) 0 U.ul,r ) + (V. (V.,),))xt)) Vt.
and applying Eq. 1 and 2 with approximation shown in Appendix B.4, correspondingly obtain
(at r,t 0) = w uu - sg Vgj + (r dul (xx))
(8)
dt
under squared l2-distance with adaptive weighting w, as detailed in Appendix B.4. Note that there predefined probability Preq such that f, which results in lnf w/w 1.12 during training. This training technique of instantaneous conditional velocity supervision is also employed SCDs.
sCTs, the continuous-time variants of CTs, use squared (2-distance instead of LPIPS. Under $ = -t dt, Appendix B.5 shows that the gradient of le (2 s,t: w.r.t. 6 can be approximated as Vol(w,, T, t; xe Vollof(x)) - sg(of + w()) di (x:)) By setting w(t) = cos(?t))
d x
(((())) sg (vf (2))) + w(l)
dDDIM(x; D1, of (xxe),
|
?
lace (x r, r, dl, 1))
(9)
di
Remark 2.1. sCT with linear paths is of the same form as MeanFlow, as proved in Appendix C.I.
Putting together. Table summarizes the deterministic variants reproduced from the discussed representative methods, including DTSCs and CTSCs, within our framework. The goal of this reframing is to disentangle the independent components that are often intertwined in prior work. Within our framework, these components combination of components will yield a functioning model. In practice, the relative effectiveness of different choices and combinations is the focus of our investigation in Sec. 3.
be explicitly separated, such that any reasonable
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2.4 DISCUSSION: SHORTCUTTING FLOW PATHS UNDER MARGINAL VELOCITY FIELDS
Q.1: Why share a common design frame?
We inherently aim to simulate the PF-ODE with the marginal velocity field, written as v,(x). Consequently, shortcut models essentially operate along the sampling trajectories of the flow governed by vi (x)) as shown Fig. ((a). Intuitively, the ideal construction of the learning target is to sample two distinct states In and H, along the same curved trajectory from the flow paths, so that the neural network can directly map 2y to 2, such that F" t, r) =e 2r, as illustrated in Fig. 1(b) and (c).
However, such pairs (æt, Lr cannot be obtained via simulation based sampling: once 2e is sam- pled, 2y remains inaccessible because both (x)) and its integral from t to r are intractable. To overcome this, common design paradigm is employed, which is to let the network's outputs, the conditional velocity alternatively, estimate Σ, in two steps: first producing an intermediate E., and then constructing an estimated target &,, as shown in Eq. 5 in Sec. 2.2. This makes training feasible. Although one may also construct multi-step (i.e, more than two) flow map targets for simulating the {æ,, pairs (Kim et al., 2023). the paradigms of two-step target construction approximated by one-step prediction are sufficiently general according to the following theoretical justification with detailed proof in Appendix C.2. Note that we classify the aforementioned methods' training objective into DTSC and CTSC. For example, lnnt and laa are instances of lawe-
Theorem 2.2 (Error bound of DTSC&CTSC (brief)). Under the mild assumptions with details given in Theorem C.I of (i) one-sided Lipschitz condition of marginal velocity and (ii) twice continuous differentiability with bounded second derivatives of X for any TI, E [0, 1]. Let po the density of vo. and ph the density of v/ = X2. (x1), under the squared lg-distance:
w2 (po. Po) S Ci (d) + C2(t s);;
W3 (po. Pb) S CzCenc(O),
where we write the training objective in Eq. 5 as (0) = En,p,tpppr) m (1,, 8, t; f) with E {dise, ctsc), W2( is the W distance, {C1, C23 are given in Theorem C.3, and C3 is given in Theorem C.4 and C.5 in Appendix C.2.
Q.2: What challenges in constructing flow map targets?
From this perspective, ideal learning for DTSC and CTSC shares a similar physical picture as shown from Fig. 1(b) and (c). However, the practical construction of the two-step flow map target inevitably causes the obtained é, and ž, deviate from V, and æ, on the sampling trajectory governed by marginal velocity fields as shown in Fig. (d) and (e), leading to bias and variance in estimating a, with क,. Introducing this deviation into the supervision of model training greatly affects the performance differences across various shortcut model designs justified in Prop. 3.1.
Q.3: Why distillation from pretrained velocity fields performs better?
From another perspective, this explains why distilling from pretrained diffusion model is often more effective than training from scratch (Song et al., 2023; Lu & Song, 2025). Unlike (s)CT, which trained from scratch, (s)CM benefits from distillation by learning from pretrained velocity field v (a). In practical training, the conditional velocity U:jo and network output of in sg(-) in Eq. and 9 are replaced with ve which closely approximates ny (ar.). This substantially reduces errors in estimating the two-step flow targets, providing more accurate supervision for network training.
3 ELUCIDATING THE DESIGN SPACE OF SHORTCUT MODELS
According to our design framework, we analyze existing shorteut models from several key perspec- tives, including the choice of flow path and design of time sampler, which primarily determine how the flow map is constructed. In the following, we aim to address several corresponding questions to empirically and theoretically elucidate the design space of one-step shortcut models.
Empirically, we evaluate the proposed formulation using unified codebase implementation with the same training iterations and batch sizes. For unconditional generation on CIFAR-IA 10, we employ U-Nets (Song et al., 2020) (~555M param.) the network architecture operating directly in the pixel space. For conditional generation in ImageNet-256x 256, with and without classifier- free guidance, 2556, use S/T-B/2 (Ma et al., 2024) architecture (~131M param.). operating in the latent space via

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100
300
203
Nearfiow
100
Hearfiow
140
ACT-mmeer
29.34
11
17.22
60.
5.05
400
320
320
(a) Uncond. CIFAR
(b) Cond. ImageNet
(c) CFG. ImageNet
Figure 2: Comparison of FIDSOK during training among different shortcut models described in
Table 1. (a) is the unconditional (Uncond.) generation CIFAR-I0; (b) is class-conditional (Cond.)
generation; and (c) is maassfiee-frre-guidde (CFG.) training on ImageNet-256x255
pretrained VQVAE (Rombach al., 2021). While sCT is originally initialized from the teacher diffusion model as stated in Lu & Song (2025), we train all the discussed models from scratch, for a fair comparison. Fig. summarizes the results of one-step generation on the two datasets, with additional setting of bassifier-guidancee details on settings are provided in Appendix D.I.
learning, as discussed in Geng et al. (2025a). Further
Q.1: Following linear or cosine paths?
I Linear paths are generally regarded as more analytically tractable and easier to employ for training and sampling tricks (e-g., classifier-free guidance), owing their simple formulation. By contrast, in pixel- space generative modeling, cosine paths are often considered more stable for training convergence, because they induce stochastic process with fixed variance. Exploration of these two flow paths in the context of shortcut models remains underexplored. Here we extend cosine- cosine-path- bath- models (CT and sCT) their linear-path counterparts. Fig. 2 shows that shortcut models with linear paths are more competitive. We attribute this to the fact that the marginal velocity fields generated by linear paths as conditional paths are at lower convex transport cost (Liu et al., 2022), implying lower curvature of the velocity-fiell targets are less likely to deviate from the ideal. Furthermore, while cosine paths are optimal in the setting of diffusion and flow matching (Santos & Lin, 2023) under Fisher information metrics, theoretically justify that linear paths in the setting of shortcut models are optimal conditioned on data samples in Appendix C.3. Based this, subsequent analysis will focus the linear path.
trajectories. Consequently, the simulated two-step flow map
Q.2: Shortcutting flow paths discretely or continuously?
Under the same training setup and within a unified codebase, continuous- time shortcut models clearly outperform their discrete-time counterparts. As shown in Fig. 2(a), both sCT and MeanFlow achieve lower FIDSOK scores on CIFAR-10 compared CT and SCD, similar conclusion can be drawn on ImageNet-256x from Fig. &(b)) Below, we analyze the inference error of the discussed methods with linear paths Prop. 3.1, and characterize the regimes which each objective preferable. We denote sCT and MeanFlow with linear paths by subscripts ctsc, thanks to their same formulations according to Remark 2.1, and discrete-time models by dtsc as well. In addition, we write the parameterized of in sCT uf,0 under the linear path according to Appendix C.1.
Proposition 3.1 (Inference enror analysis). Under regularity conditions shown in Appendix C.4.1. the Wasserstein-2 distance of shortcut models with one-step generation is bounded as:
BVase +Var d dt u. (x:)) + 8o
( p3) s
(10)
r=0,1=11
W p3) s BV + 882 Var (xx)] + 8(1 + E282) oan 1
(11)
where BV. = Bias + Bias + 2Vr ui (x1)) with E {clse, disc), and Bias and Bias? 2
Josss are defined in Prop. C.8 51 = 62 r; l is the local Lipschitz constant of u²: o is the variance of the conditional velocity, defined by oz = Var(v, (xe (2o)); o = o
for CT's two-step flow map targets, or 2 = Var uh (2x)) when using SCD's 's flow map targets.

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From Theorem 2.2, for CT and CTSC, we conclude that if Vara u&, (xx)] and Var d ui, (x:) are of the same order, the right-hand side of Eq. 11 contains an additional term 68309 d dise compared with Eq. 10, which is likely to result in higher inference error and instability in training, as the proof in Appendix C.8 shows the inference error already subsumes the training error bound. Further, when + t. and o dominates both 02 Var ui,,- (a:)) and Varl d u (z))( the training convergence di
and sampling fidelity of CTSC and CT are both closely tied to the variance of the conditional velocity used for supervision. Therefore, being able to provide a low-variance velocity supervision during training. such as one obtained from pretrained neural network, helps to improve shortcut models.
Q.3: Fixing the terminal time or not?
Since sCT-linear is a special case of MeanFlows where the terminal time is fixed at O. the empirical results on CIFAR-IO in Fig, 2(a) and on ImageNet-256x 256 in Fig. 2(b)&2(c) demonstrate that, in general, random sampling of is beneficial in capturing the overall shorteut patterns. However, in the early stage of training (approximately before 20-40k epochs), sCT-linear exhibits faster convergence in terms of FID50K for one-step generation. We conjecture that in the early stages, continually adding supervision of Zo. akin to a denoising task, provides a simpler learning task that accelerates convergence toward favorable local optima. Yet, without intermediate flow path targets Fy where r > 0, the model may remain stuck in these sub-optima during the later training stage.
IMPROVEMENTS TO TRAINING
Building on the above analysis, all subsequent techniques and developments will be carried out under the continuous
shorteut model with linear paths, so we choose MeanFlow with SiT-B/2 architecture as our baseline implementation with its default hyperparameters shown in Appendix D.2. Table 2 presents an ablation study that shows the effectiveness of our improvement techniques, where ESC as explicit&gasier shortgut model is the CTSC with all the proposed techniques as follows.
Plug-in velocity instead of conditional one. Since the marginal velocity is intractable, training relies on the conditional velocity, obtained by sampling 3i from the finite training set ( j==1- Based on it, we derive of (wee (y ) as the marginal velocity under the empirical data distribution, which we refer to as the ideal velocity in the following:
Proposition 4.1 (Marginal velocity of empirical distribution and bias- variance comparison). Assume the data distribution is the empirical distribution, as po(y) = EN 1 1v, (3). the marginal velocity reads
N 0 (2 (ONNN = Σ
Nax;; ony 0
dz (x. (xy ))) σ1
(a.g()
(12)
Σj Nime; o of
where x, a4Io + OiE. Specifically, under mild assumptions in Prop. C.13 in Appendix C.5.2. substituting v, in Leve with it significantly decreases Eq. 10's last term Teyo which reduces the variance by O(1 1/N) while increasing the bias by O11NN).
E B|uerjo
vell ?
Replacing the conditional velocity eojo in Eq. 10 with the ideal velocity obtained from the full training set the variance of the velocity term to O(1/N), As result, since N is usually large number, ac- cording to Prop. 2.2, employing the ideal velocity field can therefore provide more stable supervision during training and lower error in inference. However, its computation requires summing over the en- tire data set, which is infeasible for large-scale data such as ImageNet (N = 1, 281, 167). To address this limitation, we adopt the plug-in velocity during training instead, which reads 1ū (ax (+) ) .)-
The above computation is restricted to a mini-batch {y ()) 73
Fi=1 with pseudocode implementation pro- vided in Algorithm 1. This can be viewed as mixture of conditional velocities from the
samples, reducing the level of variance vego in Eq. 10 to (1/B), at the minor cost of increased bias. Theoreticallyy, we give further details on the validity of the training objective employing plug-in velocity in Prop. 15 in Appendix C.6.
Plug-in velocity under guidance training. From the comparison between Fig. 2(b) and Fig. 2(c), is evident that elassifier-free guidance (CFG) is crucial for high-quality image generation (Geng et al.,
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Table 2:
Evaluation of training improvements under one-step generation with SiT-B/2 F
Algorithm 1 Calculation of Plug-in Velocity.
x: training batch (B,D)
Training configuration
FID50k
# t: sampled time
= randn_like(M
MeanFlow under CFG. (Baseline)
6.09
+A1
Plug-in in velocity (Pplug-in 1.0)
6.01
xt (1- t)
t
x_ex, xt_ex x[: None, ;], xt[None,
+A2
Plug-in velocity (Pplug-in 0.5)
5.98
Plug-in velocity (Polug-in = 1.0)
eps (xt_ex (1- t) x_ex) /
+B1
6.08
coas-consistent batching
Plug-in velocity (Polug-in 0.5) & consiconnsittnt batching
1ogp_fn Normal 1).log_prob logp sum weight sof softmax dim=0)
+B2
5.96
dim=2)
+C
Gradual time sampler
5.99
+D
sCM training techniques
5.95
v_end = eps x_ex v_plugin = matmul (weight.T, v_cnd)
ESC (Baseline + B2 D)
5.77
2025a). With CFG, the class-conditional velocity or (2x, aoo c) leverages instance-level supervision from the label c. In contrast, oz drawn mini-batches, which is likely to dilute erase the class-specific signal. To this end, we employ a plug-in probability Pplug-in that substitutes the conditional velocity with the plug-in velocity, as trade-off between lowering variance during training and retaining class guidance. The other trick is consissnntttnnt mini- batchig: When applying CFG during training, we ensure that each bati- is sampled within the same class. In multi-GPP training, the class labels of mini-batches across different processes are independent of each other.
) ==)) ); is computed by averaging over randomly
Gradual time sampler from SCT to MeanFlow. As discussed in Q.3 from Sec. 3, we design a tome-sampling schedule that gradually evolves with training iterations. During the first Kixo iterations, the sampler selects r = 0 with probability Pixo. and with probability 1 1- paxo follows the MeanFlow sampler shown in Table 1. The value of Paxa decays from 1.0 to 0 under cosine schedule the beginning of the training, so that after Kio iterations the sampler fully adopts the MeanFlow's strategy, where Kixo is usually set 20k in practice.
Adoption of training techniques. Moreover, since sCT can be regarded variant of CTSC, several training strategies have already been explored in its original work, such as variational adaptive loss weighting (Karras et al., 2024) and tangent warmup (Lu & Song, 2025). These techniques also applicable to CTSC and bring performance improvements in the cases given in Appendix D.3.
5 SCALING-UP EVALUATION
Setting. In this part, evaluate the proposed ESC an improved variant of CTSCs to illustrate its effectiveness at scale. We conduct a scaling-up experiment on 2mageNet-256x2555 in latent space, and employ SiT-XI/22 (~~76M param.) as the backbone model. We follow the training setting of McanFlow with CFG, where the model is trained from scratch with 240 epochs (~1.2M 2M iterations). Furthermore, ESC+ is trained with 480 epochs (~^..4 iterations). In addition, for CIFAR-10 (Krizhevsky, 2009), all the shortcut models use the same U-Net (Ronneberger et al., 2015) architecture from Song et al. (2020) (~-55M param.). ). The code repository is provided for reproducibility!. For further details on setting, please refer to Appendix D.3.
Benchmark comparison. In Table 3, we compare our results with previous methods by bench- marking the FID50k under one-step generation (1-NFE), the context of single-step generation, the proposed techniques bring more improvements with the large-scale network architecture (SiT-XL/2) than with the basic one (SiT-B/2), as ESC achieves state-of-the-art performance of an FID50k of 2.85 with 240 epochs and 2.53 with 480 epochs. This represents improvement of 16.9% and 26.2% compared to the prior one-step result of 3.43 obtained by MeanFlow, respectively, and even better than the two-step generative fidelity of MeanFlow (FID50k 2.93). For visualization of images generated by ESC with different network architectures, please refer to Appendix D.4. Moreover,

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Published as a conference paper at ICLR 2026
Table 4 gives unconditional generation results CIFAR-10 10, showing that improved models achieve competitive performance with prior approaches. For a full comparison including other families of methods, please refer to Appendix D.6. Notably, find that the performance gains from ESC with SiT-X1/2 over MeanFlow baseline are much more significant than with SiT-B//2 which discuss in Appendix E.2.
Table Evaluation on ImageNet-256xx
Values with underline
denote the best except shortcut models, values in bold is the best shortcut diffusion model under one-step generation.
ESClw/ dass- consist. J-IFFE
ESCle/o casss-conistt))-- -NFFE
Family
Method
Param.
NFE
FIDS0K
BigGAN (Brock al., 2019)
112M
6.95
GAN
GigaGAN (Kang et al., 2023)
569M
3.45
StyleGAN-XI (Karras et al., 2019)
166M
2.30
AR/Mask
AR w/ VQGAN (Esser al., 2021)
227M
1024
26.52
MaskGIT (Chang al., 2022)
227M
8
6.18
2.52
VAR-d30 (Tian et al., 2024)
2B
10x2
1.92
MAR-H (Li al., 2024)
943M
256x2
1.55
ADM (Karras al., 2024)
554M
250x2
10.94
1000 600
800
1200
Difii Flow
LDM-4-G (Rombach al., 2021)
400M
250x
3.60
iterations fk]
SimDiff (Hoogeboom et al., 2023)
2B
512x2
2.77
DIT-XL/2 (Peebles & Xie, 2022)
675M
250x2
2.27
Figure 3: Convergence of FID50k.
SiT-X1.2 (Ma et al., 2024)
675M
250x2
2.06
SiT-XL/2+REPA (Yu ct al., 2025)
675M
250x2
1.42
Table Uncond. CIFAR-10.
iCT (Song & Dhariwal, 2023)
675M
34.24
SCD (Frans et al., 2025)
675M
10.60
method
NFE
FID
Shorteut
IMM (Zhou et al., 2025)
675M
Ix2
7.77
iCT
1 2.83
3.43
ECT
1 3.60
McanFlow (Geng al., 2025a)
676M
2
2.93
SCT
I 2.97
ESC (w/o-class-oomiist))
676M
2.92
IMM
3.20
ESC (w/-class-consist.)
676M
2.85
MeanFlow
2.92
ESC+ (w/-class-consist)
676M
2.53
ESC I 2.83
The time cost of plug-in velocity is minimal. Computing plug-in velocity involves an O(B²) weighted operation within each mini-batch. but with DDP training, per-device batch size is small (B = 16 in experiments). As a result, the extra overhead is negligible because profiling over IM iterations shows 554 ms/iter vs. 558 ms/iter for conditional vs. plug-in velocity (= 0.7% increase). Despite small batch size introducing larger estimation variance and bias relative to the ideal velocity, compared to the conditional velocity, it stabilizes training by theoretically reducing variance by 1/B) at almost no additional computational cost and minor increase in estimation bias.
Class-consistent mini-batching brings faster convergence. While the final reported results show comparable performance with and without class-consistent mini-batching, we observe from Fig. that the convergence of FID50k during training is substantially faster with the technique, where Appendix D.7 gives full details. This suggests that the training technique is advantageous in scenarios requiring finetuning with limited training iterations. Exploring its broader applications will be direction for future work.
CONCLUSION
We focus one-step shortcut models trained from scratch and propose general design framework with theoretical justification of its validity. Building on this, we elucidate the design space of shorteut models through theoretical analysis and empirical evidence, and further propose improvements for continuous-time shortcut model training. Our improved model achieves state-of-the-a performance in image synthesis. More broadly, our work lowers the barrier to innovation in one-step diffusion and enables more systematic exploration their design, with limitations discussed in Appendix F.
ACKNOWLEDGEMENTS
This work was supported by the National Science and Technology Major Project of China (No.2021 YFA1301603), the Project (No. WU2025B006) from the SOE Dean Special Project
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Fund (SOE-DSPF) Program of Westlake University, National Science and Technology Major Project (No. 00222D0115101) National Natural Science Foundation of China Project (No. 624B21 No. U2IA20427), Project (No. WU2022A009) from the Center of Synthetic Biology and Integrated Bioengineering of Westlake University, Project (No. WU2023C019) from the Westlake University Industries of the Future Research Funding and Hangzhou Postdoctoral Daily Funding Program (Grant No. 10314002582250,, 2025). In addition, we gratefully acknowledge the continuous sup- port from DP Technology, Beijing AI for Science Institute, and Westlake University Center for High-performance Computing, for their sustained provision of computational resources for this project.
ETHICS STATEMENT
This work investigates one-step diffusion for generative modeling at the methodological level. The datasets used in this study are publicly available benchmark datasets and do not contain sensitive or personally identifiable information (e.g., ImageNet, CIFAR-10).
Potential risks include the possibility of misuse, such as generating misleading or harmful content, or propagating societal biases present in the training data. Our method itself does not explicitly address these issues, but we highlight that appropriate safeguards should be adopted in downstream applications, including content filtering, bias auditing, and domain-specific restrictions.
Overall, we believe the contributions of this work pose minimal ethical risks and can positively impact the community by advancing the efficiency and effectiveness of one-step generative modeling.
REPRODUCIBILITY STATEMENT
We have made every effort ensure the reproducibility our results. All datasets used this work are publicly available (e.g., ImageNet, CIFAR-10). The preprocessing steps, model architectures, training hyperparameters, and evaluation protocols are described in detail in Sections and 5.
To further facilitate reproducibility, we release source code through https::/ //gi thub. com/ EDAPINENUT DEEINENUT/Explict ici 1ShortCut/, and will release trained model checkpoints and experiment scripts upon publication. This will allow researchers to reproduce all reported results and extend our approach in future work.
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Appendix
Table of Contents
Background Diffusion Models
Stochastic Interpolants Flow Map
A.2 Flow Map Solver
18
A.3 Derived Flow Path from preconditioner of EDM
19
B Derivation of Flow Map Construction and Loss
20
B.I
Consistency Training
20
B.2 Shortcut Diffussoon..
21
Inductive Moment Matching.
................................
22
MeanFlow
23
B. s-Consistency Training
23
C Proof of Theorems and Propositions
24
C.I Proof of Equivariance of MeanFlow and sCT-lincar (Remark. 2.1)
24
C.2 Proof of Error Bound (Theorem 2.2)
24
C,3 Optimal Path Shortcut Model (Q.1. in Sec. 3)
29
Proof Inference Error Analysis (Prop. 3.1)
31
C.5 Proof of Ideal Velocity and its Bias Variance Analysis (Prop. 4.1)
36
C.6 The Convergence of CTSC Loss Employing Plug-in Velocity (Sec.
38
D Experimental Details
40
D.1 Details for Empirical Analysis of Fig.
40
D.2 Details for Empirical Analysis of Table 2
42
Details for Scaling-up Evaluation Sec. 5
42
D.4 Visualization Examples for ESC
42
D.5 Algorithm for Plug-in Velocity Calculation.
43
D.6 Full Comparison of ESC other SOTA benchmarks
44
D.7 Details of ESC-XL/2 convergence with and without class-consistent mini-batching
45
E Further Analysis
45
Plug-in Velocity Stahilizes Training
45
Large Models Gain More Performance Low Variance Training
45
F Limitations and Future Works
46
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Figure 4: Images generated by ESC with SiT-B/2 trained on ImageNet-256x256, 256x256, with FID50K 5.77.
H.
Figure 5: Images generated by ESC with SiT-XL trained on ImageNet 256.xx26622 with FID50k 2.85.
A BACKGROUND OF DIFFUSION MODELS
A. STOCHASTIC INTERPOLANTS AND FLOW MAP
Here give formal definition of stochastic interplants and flow map:
Definition A.1 (Stochastic Interpolants (Albergo et al., 2023)). The stochastic interpolant I, between probability densities q and p1 -N I) is the stochastic process given by
2t ax3o + oi
(13)
where at, or E CC+((O, 1)) satisfy on σ = 1 and 01 = ao = 0. denote the distribution of xr as Pt-
Proposition A.2 (Probability Flow), For all t [0, 1]. the probability density of xz is the same as the probability density of the solution
i;
(x()),, Do pm(x).
(14)
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where [0,1] x R Rd is the time-dependent velocity field (or drift) given by
Ua (x)) = Rg a0. x²).
(15)
More specifically,
(x)) dr (aro 24 2:) [2(2 Dy x) Definition A.3 (Flow Map (Boffi et al., 2025b; Liu, 2025)). The flow map X R + R2 for Eq. 14 is the unique map such that
(16)
xx,)(x,)= 2t, for all (s.t) E [0.1]3
(17)
where (2s))c)jo1j is any solution to the ODE Eq. 14.
Proposition A.4 (Consistency Property (Boffi et al., 2025b)). The flow map X xx) satisfies the
Consistency Property
(Xe, (x)) = (x)
(18)
for all (t,,, E 10, 1]3 x Rd. In particular. X,, (X),((),)))) = for all (s,t, E [0, 1|3 x Rd.
A.2
FLOW MAP SOLVER
A.2.1 EULER SOLVER
With probability velocity field (z) which can be derived from a pre-defined probability path or approximated by of (æ) with neural network, an Euler Solver can predict the Flow Map from to with
x = (xx( = xz + J: v. (x))T.
(19)
Besides, if the integral of (x(x) is given utr r- Je d, parameterized with u the flow map can be easily obtained with
It, = (x,) = + (r t))(t,r).
(20)
A.2.2 DDIM SOLVER
Let the forward process be defined by
Er = a(r) + a(r)e,
e~,(
(21)
so that at any I we have
xt = a() a ot)),
v: å(t) ö(t)e.
(22)
Conditional formulation. Conditioned on æt the posterior distribution of (æo. e) is Gaussian, and hence both H, and v, can written as linear functions of æo, E. Taking conditional expectations yields
[o
de
E
E
(23)
Vt
E
and by inversion, we obtain
T6p
-ot
E
(24)
a,o, o.:o
-
at
1i
DDIM update.
Substituting back into the expression for æ, gives
Biz, xy] = a, E(zu
x o, Ele x
(25)
att, r) x B(t,r) ve.
(26)
where the coefficients
a(t, r) a,0; 0, a;;; o,:0"
-0,0; + 0,0 ar², - ai
B(trr)
(27)
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Cosine Path. In the cosine path, the schedule of at a(t) and de (t)) reads
π
a(l) cos
((l) = sin 2
a(1) 2
sin
t(1) 2 (1
Then:
a ( (r- B(t,r)
π (r 1) sin
a(t, r) = cos
(28)
2
Linear Schedule.
In the linear path, the schedule a, = a(t) and o, (t) reads
a =1
1 1-
=()====(()==--- a(t)= =
Then:
a(t,r) (1 (1-r)
(1 (1-1)())-((1)
B(t,r)= -(1 -(1-r)t+r(111
Therefore,
rtt,r)=1, B(t,r) r
A.3 DERIVED FLOW PATH FROM PRECONDITIONER OF EDM
The original establishment of EDM based on the score- score-based diffusion model, while in this part, we aim to demonstrate that although in EDM, and d2 do not satisfy
0g = 1, 01 0; og 0, 01 1,
the preconditioner of EDM is equivalent to the cosine path in paper, namely TrigFlow in sCT, by using the change-of--varalle This part is mostly based on Appendix of TrigFlow proposed by more unified view from stochastic interpolants (Albergo et al.,
Lu & Song (2025), while 2023) and SiT (Ma et al., 2024).
A.3.1 SCORE- BASED VIEW OF EDM.
EDM forward diffusion.
We draw Ty ~ Pdata and NN(0, d I) and define ar 04 37p
0e E,
(29)
where > 0 and > 0 are schedule functions determined by noise scale ot)):
data
o(t)
Volua + (1)
Oe = V
V Jama (1)2
(30)
Here ddata denotes the data standard deviation, and the EDM noise schedule is
(()) lia + 2/P min
a 1/0 1/2))² )
LE [0,
(31)
with typical choices Tmin R3 2 x 10TH 3 Omax R3 80.0, and p = 7.
A.3.2 FROM EDM PRECONDITIONER COSINE PATH.
Score parameterization. score
We may interpret EDM as a score-based model. Specifically, define the
(x.)) = Vn, logpe(xe))
(32)
and approximate by a neural network
60 (x0,1)) 1) (x)))
(33)
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Since the Gaussian corruption satisfies - (o())s((z)) the EDM predictor is written D (mi, 1) = Cskip()) (1) Dt + Cout (t) (cin())) Cnoise
(34)
since Thp (ro + (()e) Cakip (1)a,, and according to Eq. 29 and Eq. 30, with scaling coeffi t
cients ensuring unit-normalized training targets:
Cin (1)
Cakip (1) = Ot::
Cout (1) = data Oz-
(35)
and the denoiser reduces to
DO (2 1) = Ctt dx Br. daata (it/ ddata, Croise (1))
(36)
Cosine reparameterizattion. Since az 38 = 1, can introduce cosine time variable t' E [0, 1] such that
a cos {1)):
3 = sin (C) t' 2 arctan A 2
arctan a(t))
(37)
On this cosine path, we may equivalently define
all') cos _
(l')) sin +1((
(38)
which again satisfies (x))) + (t') = 1. Moreover, since EDM one typically samples t ~ AogA((P [eman,, Pe Pata). under the change variables /' = arctan((), the resulting sampling matches the time parameterization used in CT and sCT (Table 1).
A.3.3 FROM SCORE PARAMETERIZATION
VELOCITY
In general, we can denote /' E [0, 1] as 1, with Eq. 38, which leads the denoising predictor of Eq. 36
to
(a t) = cos( t
2 at sin( x
2 ddata ( Croise (t)),
(39)
Since D (xx, t) aims to approximate wo. and if we write
v (xx,,) t) data (x Choise (t)).
it reads
(Et, (x( 1) = cos( π 21) + sin( Lin(-11)' (xte,))
the coefficient t)) and 2 sin( 7) coincides to a(t, 0) and B(t, 0) in Eq. 28, the parameter- ization of of (xx)) = d F (xx, 1) is equivalent to denoise the path of preconditioner in EDM schedule, The same evidence is also provided with Eq. Lu & Song (2025).
B DERIVATION OF FLOW MAP CONSTRUCTION AND LOSS
B. CONSISTENCY TRAINING
In CT's and CMs (Song et al., 2023), the original paper uses the EDM preconditioner as the components of the basic diffusion model. By using a neural network oe to approximate the score function 8, (z), its target is map any noised samples in to 0 which in the flow map notation, reads
x (z:) = f (xz) = Cakap (f)xz, + dould (xx))
such that
lam d(s (xx,, (+t))),
(40)
where
111 +
T
(0 10 σ L/pp]
K
lo max +
7
(0 - σ ]/10))
(41)
K
0 1,..,.... K],
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and é, is the same conditional flow path that generates zy. By adopting the equivalence of EDM preconditioner to Cosine path, as shown in Appendix A.3, we can write F as the approximator v under the change-of- variabbl,,, which reads
E~ e~~(0 (0011)
po,
to + sin π
= cos(
2 t)e
2
oijo sin --1 1)
π
π 1)000 + cos(
()e
(42)
2
2
f, cos( aa sin 2
2
Then, following the derivation of Eq. 28, by substituting the coefficients ât,s and 3r we can equivalently express
(x:) ã. (D Mejo, 8) (x,) = 2, @ (tt)) (x,()) = x = DDIM( F (xt)),
(43)
Finally, by replacing d Eq. 40, reads
la( r, t: = LPIPS( (ax)), sg( (iz))), (XP, sg( & (i)
(44)
LPIPS (DDIM( ( (xt), sg (D(DII(( (0.),,,))
where a. (D,, Urjo, s), which coincides the description of CTs Sec. 2.3, Further, under the change-of-variable the time sampler Eq. 41 will be of the form as described by Table 1.
B.2 SHORTCUT DIFFUSION
In the original paper of SCD (Frans et al., 2025), it parameterizes the velocity with the neural network as
Fd (æx, L, r) = v 1, r),
while we claim that the o (x,, 1, r)) is not the instantaneous one, because it requires the entries of both as the start point, and r as the end point. Instead, we regard it as the average velocity, leading to our parameterization of
F (" 1,r) (2x)) In this way, is first sampled from a linear path, as
Io ~ po: 8~ N(0
2; t)zo +oo++e te
(45)
2 = Dog +
Then, the flow map is constructed via
X (2x)) G, Dt hug (2:z)
(x,( = G, I2 - hu (5x)
(46)
X2, al. at 2hu (x1)
Finally, according to the consistency property of flow map shown in Prop. A.4, and by setting w = 1/2 the loss term for the regularization of velocity can be rewritten as
ladd (Z,, s,
||
2h (xx) - sg (zz huf (xx) - huu (x: - (xff(()z)))
4h2
||,.
-2hu (It) - sg (xt l - hug (x.))
4h2
u (xx)
5gg(ui, (x:)) + u (8.))
(47)
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where s,t t equi-spaced, such that t Specifically, in the time sampler, it defines the total step K with = logg K.. For each sample, draws 2h E {2 , 2-1 ...... 2 (T 1-"T-1) 1) and e ~ 2( 1), then computes
r--- h, which coincides Eq. 7 in Sec. 2.3.
r=t 2h, = h, as the time sampler for fr, s, t). We denote the time sampler as (t, h) ~ Uniform log, (t, h), and h = 24.
t= K 2hK
(K 2hK + 1)
(48)
B.3 INDUCTIVE MOMENT MATCHINNG
Given a mini-batch of size B, IMM first draws ((x)) (x) size M. Within each group, triplet (r, s, t) is sampled, where (r, t) are drawn uniformly from [0, 1] with separated by a fixed difference, as
) and partition them into groups of
~ 2(00 1]
n2 1
1
t 2
(49)
+1
[[, t],
where y usually set as according to code implementation. Each group thus defines M correlated particle samples that share the same flow interpolation times. For each group of M particles, IMM constructs an x M kernel matrix based on a positive definite kernel function as metrics d(,;) .) (e.g., RBF). The objective is
Limm (0) = Em, Em. ma..a. w(r t) (keT(Zt.r: 2 +krr( Ze.rs z'a.) ker (zi, z'a,r) -or)(z/rr zs,r))
(50)
where
trr DDIM(E, (Zx),
-,, sg(DDIM( of (x)),,
and w(r, t) is prior weighting. Intuitively, samples are repelled by intra-group pairs (e.g. žt, vs. z(,-)) while attracted towards inter-group matches (zt,r vs. 2 This ensures both intra-sample diversity and inter-sample alignment.
In practice, a batch size B divided into B/M groups, and the IMM loss computed as an average over these groups. For kernels, RBF and negative pseudo-Huber kernels are common choices for ker(-,), -), which guarantee moment matching up to all orders.
Further, we bridge the IMM loss with the common flow map learning objective in the following. In Eq. 50, gives the group kernel function, according to Appendix. C.3 Zhou et al. (2025), we can write the loss here as Limm (0)
-m (rr t) (ker(zur) ) + ker(Ze,n xls,) (eer(Et.r 2',r)) ker(z',r Beirl)]
-Er,,t (rrr (Ex,, or (kor(ze,r; :), ker (z), ") + (xcr( "), kor(z :)
- (ker(Zzr, :), ker(>' (2)r)> )) (ker(z),r ( ker(zary -)3
=Brait w(r, t) (o(r.))(Be (e.. (xxr(x,,rrrr (x:),,) ker(sg (x.)),
Ex; [ker(X]] (x (x(), ker(sg(X (a())) (1)
=Er,st wo(r 1) Fan, [ker( X (20e).) ker(sg( (x9 (x (2x.))
(51)
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where T. DDDIM(#, Dejo. t, s) is estimated with conditional velocity, and E [ (Xf, (ax),.)
ker(ss((X (s.g s,r (# -)] | the Maximum Mean Discrepancy commonly defined on Reproducing H
Kernel Hilbert Space (RKHS) H. with positive definite kernel in IMM. Then, according to Jensen's inequality,
Er,,, wor,,)) Eee, (x)) ker(ss(X" (i+)),
(52)
Slra w(r, ()) (2x), ker(sg(X (x. (1:))) (x., -)
In this way, we define d(x y) RKHS discrepancy | :) kerly, .) 1 7 Limm (0) S E (erpe merpp [w(r, () d( ((X,,, (at)) sg( &ar (x:(())))))
it reads
(53)
Therefore, minimizing Eq. 5 is equivalent to upper-bounding the IMM loss.
B.4 MEANFLOW
As MeanFlow takes the Linear flow path, we can easily obtain
DDIM( Ue, t) + (s t)ur,
(54)
which means the DDIM solver and Euler solver are the same. With the flow map construction, we write the corresponding terms into 1|2 of d(;;:) .) follows:
(xx + (r t)uf, (xx)) (xt + (s t) + (r- (r-s)ee,, (Ix))
(55)
=(r- - l)u (xx)) (s 1)0e (r s)ue. (2rs).
By substituting s = t - dt and normalized by dt, we get
((r - t) (2ta) + dl uz (r - dt) de,r (xDe ))
d (r ()(e,((0))) dt
=dt Ui +
/et
(56)
d
=0, - uh, (t - dt
(xx.)
However, the marginal velocity v. is inaccessible in training, so it can replaced by the conditional velocity tajo- From Eq. 6 in Geng et al. (2025a), by adding the adaptive loss term w, this loss coincides with the training objective MeanFlow Eq. 8, as
du., (are)
1(1 t dt, 0) w- (x) - sg Urjo + (r
(57)
dt
Further, McanFlow adopts an adaptively weighted squared L2 loss. Given the regression error
A Ug Ueg, where Urgr = sg Vejo + (r t) dul (#x)) training, MeanFlow reweights || 11 with
the squared L2 loss is || l/ To stabilize dt
w
(A c
(58)
where avoids division by zero and p controls the weighting (p 0.5 recovers "Reudo-Huber style loss). The final loss is defined as sg(w) L, where sg(.) denotes the stop-gradient operator.
B.5 S-CONSISTENCY TRAINING
We here simplify the derivation with datta = 1. According to the Eq. 44, we can write the correspond-
ing terms with squared la-distance with t Al and 1(x,, 8, 0) normalized by At, as the
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following:
w(I) limm [DDIM (4,4,)) - sg(DDIM (D, vi s,r) )11
==( At
At-30
=w(t) lim
s=t-A [DDIM tf, -sg(D))((2 s,rl1
A1-+0
(t lim
(DDIM (xe, ft,,,) (arr)--s(DDDDD))) sg(DDIM(2E, sr)))
(59)
At
A4-+0
DDIM (1,, 0l ttrr sg( (gDDDMM( E.., s,r))
At
(1)DDDD (zr, (,, - (DDM (.,,, dDDIM (Ie, of, tr
dt
By fixing r = 0, from Eq. 28, can be obtain that the gradient of Eq. 59 w.r.t. is
dDDIM (Et, Di t,r)
w()V 8
(DDIM (2t, 4,,,
sg(DDIM C.,, (,r)
dDDIM (I,, tt,,,r)
=w(t)V
DDIM (Ie, trr sg
(cos(( ()ar, sinn( 1)mo T
dDDIM (or,, oi r)
=w(()Vo
ag
dt
=Vo (v,)
-sin( T (t)(t)
dDDIM (ate, ui 1,,
sg
dt
dDDIM (It, of t, 1
=Vollu sg
i wt)
dt
(60) where w(t) tan Frj and w'(t) = - sin( (t) cos (tt), prove that this flow map construction corresponds to the original loss with the specific time sampler, and the derived loss is in the same form as Eq. 9 where rewrite w' by
C PROOF OF THEOREMS AND PROPOSITIONS
C.I PROOF OF EQUIVARIANCE OF MEANFLOW AND SCT-LINEAR (REMARK. 2.1)
Sketch of proof. First note that under linear paths, (xx)) (w:) DDIM(# of (xx,,, w; tof (z:). As for the training objective, with w(t) and linear path. Eq. be easily written as I(,,,,, dt, t: 0) 0=
Since in SCT, is fixed to 0, | (x.) sg(u, + (r t) 4 (z.)) P=
parameterization of the neural network can be invariant to r, leading to of (ax)) F (a,, 0). Thus, in sampling, by replacing u,,0 and with Fe in Eq. 2 and 3, respectively, the sampling processes are the same when following linear paths.
C.2 PROOF OF ERROR BOUND (THEOREM 2.2)
In this section, we aim to prove the error bound of DTSC&CTSC. Specifically, the theorem is stated follows.
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Theorem C.1 (Error bound of DTSC&CTSC)) Assume the marginal velocity of the flow path satisfies the one-sided Lipschitz condition, where
3C E [] (v.(x))
-c yll; for all (1, 2. (, y) E [0, 1] Rd XR.
Assume XL,s are twice continuously differentiable with bounded second derivatives, the weighting
function w(r, 8, 1) is non-negative and bounded. For DTSC, also assume p(r 0) > 0, 1st-step
satisfies (x,) It. and 3t .... iN s.t p(0, tay tn+1) > O, w(0, wn, tn-))++)= 0. Under d(x, y) a y1l.. given æ1 Pr, let Po the density of To, and pó the density of 2g X1 (z)0(01) that is estimated by neural network with parameter 0 then
W (po. p8) 2 C. Edisc + C2 ()-)),.
(2 pb) S Cate
where write the training objective in Eq. 5 as L. Br.,,toplr) mi (2,, s, 8)], with E {ctsc, disc), and W2(+.) is the Wasserstein-2 distance.
We note that MeanFlow loss and sCT loss are all Lase (f), and CT loss and SCD loss are all Lase IMM's loss is calculated across different conditional paths, finally bounded by the Lasee (0) as shown in Appendix B.3. The mentioned previous methods all satisfy the assumptions about w(r, 8, t), p(r,s, s,t), f), and (x,) It. As for the assumption of d(x; y) = | y l2 it holds for all the mentioned methods except CT. which takes LPIPS as the metric function. The convergence of La has already been proved by Song et al. (2023).
We prove the theorem in three steps: (i) establish the error bound for DTSC; (ii) derive the start point differential CTSC bound; and (iii) further derive the end point differential CTSC bound.
C.2.1 ERROR BOUND OF DTSC
Lemma C.2. Assume d and XL, are both twice continuously differentiable with bounded second derivatives, the weighting function w(r, 8,1) is non-negative and bounded, and Ist-step satisfies Xe,(2x)) = Ez. We define a loss L1 follows:
(0) = R [ 1) x [x:) (xx))))
(61)
Then,
Laise (0) L( + Oct -
where Lâue is the discrete-time shortcut models' loss.
Proof. As
2; = x. (t s)U, O(t s).
and here define the 0- the parameters in the model which stop-grad operates, for notational simplicity. By using Taylor expansion, we can get that
L1 (0) E (r, 8, X (ax)), x (ax))
E
w((r s, X (xx.) + 0,X (xx) - + (x, - s)v, + oft - s)
x (xx)
E w(r, s, t) X (x.) 0,x00 (x.) 5) + sss-ss,s x (a:)))
= E w 8, (d(x)) (x,), x (x.)) 0id(X² (x:s), x (z.,) (z, X. (xx) - 8)
+ O(t -).
As for (x,), there are three ways to calculate it as stated in Sec. 2. Ways in Eq. and Eq. 3 are numerical solvers, have (2:)) x. O(t s). Eq. 2 also satisfies this equation since
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have the assumption that (x)) æt and Xt,s is twice continuously differentiable with bounded second derivative.
With similar derivation, also have
dase (0) E w(r, s, d(x,, (x1)) X2, (2i))
E ( dl (x (x.)) + 0, (x.) - +0 - - -s)]
E s, [d(x syr (xx), X (xx)) 0,d(' (xx), (x, (x,)) (o,,),,xx (x,) - =])
+ +
By subtracting the two equations, we obtain
Laise (0) (0) O(t s).
Theorem C.3. Assume XL, is twice continuously differentiable with bounded second derivatives,
the weighting function w(r, 8, 1) is non-negative and bounded. Also, assume p(r 0) > 0, Ist-step
satisfies (xx) Et. and Etg I ..... tN s.I p(0, Eas tn+1) > O, w(0, t, bn+1) 0. Under
d(ær y) 1 y1
W2 pi) S C₁ d + Ct(t-s)
(62)
where C1 = En= M0,,aof (0. 11) and W2(+s)) ) is the Wasserstein-2 distance.
Since we have proved that the difference between Lusee and L1 is O(s t), only need to prove
Proof.
W (p (O)
Because for r- = 0, 80, to s.t p(0, 80, 40) > O, w00 90, 40) > 0.
(0) E F[w(r
X (x:))
2 p(0, 50. 40 m 40 40) (2 X (ztt))
we have
E
dd(x 10.0 (Ze.) Xx [Lo S
p(0, so, to w so. to) (0).
(63)
We define
c,D = Xx,o(xx)) - (x.))
Then,
et,.0 Xx,,o(xt..) X X((o(
= -X - buti 104t,0000 (1 + Xp of (().,)
-ta-1.0 + (x1 (Btnr)) - X. (30,0
Consequently,
x ( - x² -x (l)))
C1, C0,0
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where 60.0 0. Using Eq. 63, can get
W (po.pr p6) s olle
E|| Bl|x in (0 o 211
[dx 1m11. (ow. X o(
(1)
p(0, buy 1 (0 Lus bnt1)
n=1
Li (1).
p(O, An ln+ (O, li bnts)
With Eq. 61, we finally have the Theorem
C.2.2 ERROR BOUND OF START POINT DIFFERENTIAL CTSC
Next, we prove the error bound of the start point differential CTSC. The derivation is adopted from Boffi et al. (2025b).
Theorem C.4. When + I, the CTSC loss can be written as
Lesereo E X (xt) (x.) + ve VX (x:)11
Under d(xx, y) || yll. then
W (po. po) S C3Loso-so000000 (0); where C3 = c, and W2(,)) ) is the Wasserstein-2 distance.
Proof. Firstly, from the chain rule,
d x (xx) (x:) + VX
di
we can simply use uτ,r as the model output, and write the term into the expectation of Laso-stto as
(0 (xx) + (x)(())))
=1 dt s,, (ax))12
-11e, + (r - -t (x:)11
dt
= -Ilur,- v. (r dut, (xt) dl
which coincides to the McanFlow loss Lmt in Eq. 8. While in Remark. 2.1, sCT loss is equivalent to MeanFlow loss in linear paths, the CTSC loss when - r also of the same form as claimed. The cosine path version of sCT loss is variant, so we did not include in this stage, and will consider it as our future work. Then, we first define that
E = IB| (x,) X2, (x::)..
After differentiation, we can get
dF,r
dx (xx))
E
xx)))))) 2 (X,,r (xx)) xt, (2+))
dt
dt
dx (xx)
E (x), -x,
dt
[2 (Xe,r (xx) - xx,((x)) (xxx,((x)) + + (x:))
S F,, + a x (xte) + V (2:)
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So,
e' 0c Bs, - Err I <''Ello, (x) 1a (a:))
-,e Eir S e' (2)) 1 VX (x:))l
With E,. have
Fiy S ( e" eT (2:)) + (r:
(xx)) ||a
S e! [ Bloxxt (xi) + (1)
S eLessssoor
By setting C3 e, the theorem is proved.
C.2.3 ERROR BOUND OF END POINT DIFFERENTIAL CTSC
Finally, we provide the proof of the error bound of the endpoint differential CTSC. Theorem C.5. Assume the marginal velocity of the flow path satisfies the one-sided Lipschitz condition, where
3G, E I. [0, (0(())),((1)) (x (x- v) 2 -C -- yll? for all (1, 21,
(,,yy) y) E [0, 1] x Rd x Rd.
When s r, the CTSC loss can be written as
Lane (x (at)) O X (2:))12
Under ((x v) =1|x 212 then
W (po.pi) Po) S (0))
where C3 el 2 IC, and (2( :) is the Wasserstein-2 distance.
Proof. Using the one-sided Lipschitz condition, we can get
- (x), Xx, (v)) (x)) (x,((x,()))) S (Crl| - (2)
We then define
E,, = Exe Exol -x,, (x:)
With differentiation, we have
dEEr
(Xer
dx (20e)
2
(xxr X², (x:)))
dr
dr
dr
-2 (x)) X², (v(Xe, 0, (xx))
= -EE (xx, (x:) (xi)) - 0,X(((
(xx))
2T [(x,, (m)) (2:)) (xx,((a))))
S E. (x,, (x E. (æ:) -0,X% (x:)
2E (Xtr (1x)) x (vx)
< Es, + Ez 10 (X (a:)) 0, 0. (x:) 20, Er,r
= (1 - E,r + E. l (xx)) - 0, 111
So,
0, 0 r- 25 C. (.r) < 2--255 C. R Emll - (xxx(()))))
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With E O, we have
Bir S S. e e- Bal (x (xi)) 0, x, (2xx) 1B27
S 1 1C, [ Eall Exllo(X) (xx)) - 0.X (x; 11ddr.
Therefore, when & = 1, = O, we have
E1,0 S e 22 Cildt [ Ex (x)) 0, (2x))
Finally, due
(2 p0) - (1,x1, Xf. xf,((=))1
and
Letsc-sto-r00) = EEl (x,)) - 0 x (2:))12
S J..1P w(1, r)) 11 X (ax)) 0 x (2:))
we obtain
(po. - ( -x
14122 1o [Cilds Letsc
By setting C3 cl+2,6 [Cildt, the theorem is proved.
C.3 OPTIMAL PATH SHORTCUT MODEL (Q.1. IN SEC. 3)
Previous works have claimed that the cosine path is optimal for diffusion models from the perspective of Fisher information metric (Santos & Lin, 2023). Here, we provide the analysis of the optimal path for one-step models under the Fisher information metric.
We first briefly introduce the Fisher information metric. Treating probability distributions p(v) as smooth manifold, the Fisher information metric defines Riemannian geometry that enables the computation of distances between them. Specifically, the definition is
a
{()) = Ex Ex
log (X) 07g
log p. (X)
0%
When the distribution family is exponential as
p(#ln)) h(x)) exp(n()) - (())),,
the Fisher information metric becomes (Karczewski al., 2025)
I, (om(1))
[0u(7)
Oy
where
p(7)= E((x)) 1n1 1-1=
T(x)p(r 1ddr
the expectation parameter. Then we can naturally define the optimal schedule the geodesic between two distributions, which leads to the theorem below.
Theorem C.6. (Zhang, 2025) The optimal schedule under the metric 1(7) is generated by yp of the form
(²(>)= AA7)),,, where A(s) S VT(rjdr.
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With these preparations, now turn to the one-step diffusion. We point out that, for the one-step diffusion, since our goal becomes modeling the average velocity, we no longer consider the manifold of p(zz), but rather that of p( (o (zo)) p(æ, zn). Then, claim that the linear path is the optimal conditional schedule as follows.
Theorem C.7. For Vao, the linear schedule is the optimal schedule considering {p(uo. æo. t)}. ie,
A '(M7), where = t).
Proof.
Et 20
puuu tu.t) = p(
æu t)
04
20 oze
pl
Fo, t)
0,400 - 3o of
Nu
T)
t exp (-
[|
2a, - 2 T 1
+ ((0 -
(2m)4/20( 207
t
So uo,t 20,t) t) is exponential, and we have
22
20,
-
1) 1) =
Wo: 1)
20,
t(a
1)
2
20,
of
20
and
((uo,)) = =(uo0tt 1/ 11
Then
u( 1)
E ((uo.)))) (oo 1]
T(uo,))(2oo. æo. d)
/ (uo,.. (uo.el1))NNu
0,:0
2o
of
1 duuo
t
04 To 0440
xo
t2
So
t(at - 1)
(a: - 1 + )a
2t(as 1)810t
Ory(zoo 1) t(zo,t)
o7
of
4top
40,0,
40
Cg 1
6,t - + 1
0pl(zo
Eo
t
t2
a(zo. t)
Or
dit 04 +
1
2a 2to
2x0
2
Eo
f2
4
a; +
12
2(
2
2(art
0q + 1)
2aya 2to
13
11
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Substituting the linear schedule at 1 and de we obtain
11 ((-1)
-2 f2 21 1)t + 21
0m(2v, t) (xv. 1)
200
12
1 8-13
t
-213
21 + 21 + 21
t
and
1-t-1
- 1-1+
0p( xo. t)
0(vo. ()
1
21
26
2300
00
2
Based on these two equations, we get
- 0)
T
T( 1)
0
2a0
which means the metric under the linear path is uniform. So the probability p(uo, (Z, t) travels at constant rate, leading to the optimum.
C.4 PROOF OF INFERENCE ERROR ANALYSIS (PRoP. 3.1)
We first give a detailed version of Prop. 3.1 as following.
Proposition C.8 (Inference error analysis). Under mild regularity conditions shown in Appendix C.4.1, the Wasserstein-2 distance of shortcut model one-step generation bounded as:
(po. pO) < 2 BV + 8Var d - Wirr (x:)) + 80
(64)
dt
W2 20) BV 882 Var "ar[u",r (w)) 8(1 + e (8 d 1-000-11
(65)
where BV. = Bias
Biasz.m 2Varfu (x) r]] with (ctsc, dtsc),, and Binsk and Bins"
are defined as
Bias El| u., (xx)) Y. [|
(66)
Bias" t Ell. (az,, r; 0]]
Y. is the two flow map target when You in each model, i.e.
d u (x.,ttr)
Yose di
uejo(zz; (2o))
Yase 5n U2 o (mr)) + 82 (x: + 81 (en (ma)) use CT loss
Vasc 61 vn (mr) + 82 u u + 6g (
if Juse SCD loss.
1, (wx,, r. 8, 8, 0) is term the expectation of different training objectives as given Sec. 2.3; 8, = 8. 8y = & is local Lipschitz constant of u²; o is the variance the conditional velocity, defined by 2 (ar(ve)(( 2 o² when using CT's flow map targets, or 2 Varf uf when
using SCD's s targets.
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C.4.1 ASSUMPTIONS FOR PROP. 3.1
We state the regularity conditions required for the theorem.
Assumption C.9 (Velocity variance). The conditional velocity oejo approximates the ground-truth velocity vt., such that we can write
(x) vz(xx,) 775-
where we assume the discrepancy Th has variance o Because E(n, [oo] = 0 (take expectation on both sides), so Var(Pejo) o Assumption C.10 (Lipschitz continuity). There exist constants E such that for any h,
||u + h,, u (2,,,, r)sl S [
with independent of (a,, r,
C.4.2 LEMMA USED FOR PROOF
Lemma C.11 (Bias-variance Saaariannce-ooaaianneeBVCCC decompostion) For random vectors A, Rd with finite second moments,
B|A B| |IEA EBI + tr tr Cov 2r Cov [o,[[A, , B).
(67)
Proof. Denote A and I3' expectations by μ4 IEAA and μg TEB. We start by expanding
E A - B =
(A- B)] = E| All + B| EEAATA B).
Each term can be decomposed as follows:
||| = (r(B[AAT) (r(Cov[A) + MAHA)
= Ir Cov(A] + ||A||?
E BII tr(E[BB) 1) tr tr (B] - (BUB
Cov [B] + ||MB||
[[ A B] = tr(EEAAB
tr (rCCov [A, B] (APT)
= tr Cov [A, B] + MARB.
Substituting these expressions back, we obtain
B 2
||A 2Bll + tr Cov[A] + Cov[B] - tr Cov[A,
This establishes the bias-variance-covaaiinn identity.
Lemma C.12 (Variance lower bound under local bi-Lipschitz). Let f(x) = u (x, r, and assume local bi-Lipschirz: there exists c > 0 such that for all sufficiently small h.
[ + h) f(x)]l cl/tall.
Fix æ. and let W be random vector. Define
Z f(x, + 8, f(x.).
Then, conditioning on the o-field that renders æ, deterministic,
tr Cov(Z)z.) 2 c² 87 tr Cov(W)xz).
Consequently.
Cov 2 2 2 B Cov(WW
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Proof. Set W == [ and write
Z fa + 81 W) f(x; + k, + f(a + & - f(x)))
2'
constant given
Adding a constant does not change variance, hence tr Cov(Z| = tr Cov(Z'(zs)) By the bi- Lipschitz lower bound,
||z|| || (a: + 51 (W - w)) - f(x))) 2 cδi I - -WI-
Squaring and taking the conditional expectation.
[| 2 e² 8f BlIW - - = 88 tr (o:(W
Since ir Cov( ( v e² of ir Cov(W s4). Taking expectation a. and using the law of total variance gives the second claim.
| we obtain ir Cov (Z)z)) ir Cov ( (x)))
>
C.4.3 PROOF FOR THEOREM 3.1
Write 6 i, 62 and note that in inequalities below we only use 81, 82.
Step 1. Upper bound for CTSC (MeanFlow and sCT with linear path). Here we consider the sampling error from i as
Eir- = 1e (x:)
It can be written
[[[[ ( (2:) (r- (r )12]
[I]((r (r - t) (r - t) (xte (r - t)y + (r - Y 112)
<2 + 2 2r (2+) - Yessl & + 2(0 + )2)) (xt - Yess 1
where Yasse = (61 + 62) d uuf (xxt) - U:jo( x00)
First, consider the first term take
Uh, (xt),
Yetsc = (6) + 62) u u (2+)) m)
Applying Eq. 67, 2(6, + E I|A B12 2(,, +22) (J BETA-BBII-4 + + Var[B]
Cov (A, B)).
Bias?
According to -2Cov(A, B) I Var + Var[B], we can get
E [||A - B
(68)
< Biasaac-ut + 2 2 [ + 2 Var[[Yuu)]
(69)
-Bass + 2Va (asc),
(70)
because Var[ur.r] = 0.
Then, by Assumption C.9 (Var[ueo] = 2220))
Var[(O] +52) d
Var[Yaue]
dt ui.
[ajo]
< 2 Var[(61 + 52) d
u
8
+ Var
dt
d
= 2(8) + Var
0
20
[
db
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Therefore,
B ( Yeieell
Bias ass--gt + 4(61 + 82) Var d [ + 40 vajo
Secondly, take
A = uf, (x+),
= Yescc = (ó1 + 82) dt
ui (2t) (11o(2,)206)
and it coincides that IB] ue (2+)) - Y Yinocl2 = El(caee Applying Eq. 67, we have
E [esec]
|EA EBI1 + Varl Var(AA) Var[B] 2 Cov (A, B).
Bins
It can also be easily to obtain
E [ctsc]
1
Bias
+ 2aa
ue, (xx)) + + + +82)
dt ui, (xx)) + 40 40
cisc: loss
In summary,
Eir
d <2(0) + +2)² BBissueege + Biias + 2+a Varrlll, (x,, + 8(66 + )2) -
(2x1) + 80 di
Specifically, when 1,, =0 O,
E1.0 2 Binssnauigt + Bias lass + Va (x11) + Var d dt u (x:)] + 80 |
Step 2. Upper bound for CT. Then, let's still consider
Eer- = E(l (x;))
by setting Ya:
5,4622 ( 81 (21 + 82 u 2.,((1z; (x, + 51 (x which equals to
Ei - t) (xx)
(r - (r (2,) (x:)
Blll(r (m,)
(r - t)u,, (#x) (r l) + (r-1)
2 + 52)) (2xt) Yallă + 2(51 + 52) B (xx)) Y: 12
Firstly, set
1 (81 (1)((,) + 82 u + 6n (z, {=)o(=
(xx))
B Yet
61 + 82
By Cauchy-Schwarc -2 Cov(A, B) is again absorbed into the X notation. So we have
E | - B1]
(71)
< Biaser + 2ar [utr] [Vr
(72)
2Bi + 2V Yal,
(73)
For Var[Ya], expand the second term as
( (x, + 51 vejo) (x,) + (u (x. + 61 vojo) u (.:))
Apply Lemma C.12 with f(x) u², and W u: (2xe Conditioning on æ; and using the
Lipschitz upper constant f > O, we obtain
Var u (x,, + 8, vzlo) u (2x) xe;) S 8² of Var(vsj((222200) xx).
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Taking expectation in æe yields
Var u (2) + ò1 vH1o) (2)((x)) S 6 o %0100
Therefore,
20 o2 o Var[Ya] 5 (ò) + 82)2
+ 20 V (x:)] + 21 87 02 )
Hence, we get
4
B (
Yarl22 < aig
03
+ (1+ (8)68
(81 + 82)3
Secondly, set
A ul,r (x)e,
B Yet
ô1 + 82 (ò1 0 + 82 tareh, u.. + 61 (
Then B|] 1E (2:e) Yall [B(aa). Easily following the above derivation, we can obtain
Blfec-loss] I Biase + Var u (xx:)
+ (61 82)2
4
88 Varlu. (xx)) + (1 683)08
To sum up, we have
Esr $ 2(01 52)² Biaser.gr + Bias + + (Var
03 Var Varfug, + (1 (1++1- (113833007 )e.1)) (61 02)2
8
+
When 0, 0. L, the inequality becomes
E1,0 2 2Biis + Bias + Var u (3x))
88% Var Var (a:)] +8(1 + l 282) o m
Step Upper bound for SCD. this case, consider
Ei, = E[[| (x,r (x, (2)
by setting Yoed
6,+o2 ( ui (x+) + 82 ug,((00 (xc + 51 (x:))) ) which equals to [[ (2x)) - (r t)ue, (2t)
= [Ill t)uer (20) - (r t)ug, (xt) - ((--))Xe t) + (r - Ynd
<2(6; + 52) 2 ( - Yood 1 + 2( + +02) " (x Yood
We can find that the only difference between SCD and CT is the second term in Ysed- So we only need analyze this term:
8 (x; 51 (xx)) (xs, + (ul, + 61 u., (z:)) u (x:)). By Assumption C.10 (Lipschitz continuity), this term contributes a variance of order e² of Varlu (x:)). Hence,
Var[Ysna] S of Varfu (2:)) + 02 Varfua, + l 28003 V
Similar to the derivation of CT, can get
Ei, < 2(0; + 6 Biiss
+ Bias loss
2ar (2.
02
(1 of (a::)))
(8) + -2)2
When 0,r 1,, the inequality becomes
E1,0 s Biassang + Bias?
2Va (x))
802 Var (xx)) + +8(1 + e (2x))) -=0,t=1
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Step 4. Conclusion
Since
W2 S E ( x1 = E1,0
The proposition is proved.
C.5 PROOF OF IDEAL VELOCITY AND ITS BIAS- VARIANCE ANALYSIS (PROP. 4.1)
C.5.1 THE FORM OF IDEAL VELOCITY
Proof. By definition,
Pe 200) vi I De (o(20)120))
po(
pe (x()
We aim to rewrite
(x(x))) = f (1 mo)
pr( (o) po(x))) (x(e
darg Ey (2.) los 2o))
(74)
The forward (noising) process is linear Gaussian:
3t 04 oiE,
E ~~N(0, I),
(75)
so that
Pa (x xo)) Nari a, of1)
(76)
Assume the conditional velocity has the form
er (xe xo) = T, + O,E.
(77)
Since E = (xx - -a,xt)// we can eliminate the noise and write
we (x(
xo)
de + (ār
diau
x0
(78)
de
de
ax3; + 6,30.
(79)
Suppose the prior Po is empirical:
N Σ 1 (v). 1=1
Po(v) = N
(80)
Then the marginal and posterior are finite mixtures:
N N (xss ar3is of1).
px(xx)) = 1 N
(81)
i=1
p(as y: xx) We (mre)
wr(xx) 2 Naas; a.y. of1).
(82)
Σj= wo (x;))
Taking the expectation of the linear form yields
(x1) = a, + by E|wa xz).
(83)
From Bayes' rule,
w. wx,
Eizo xt) =
Σ m. (ax:) yis
mxx))
(84)
Σ (xx)
In conclusion, under the empirical prior, (x,))) is obtained as westeri average of the conditional velocities associated with each training sample y::
0 (xi)
(= y: xx) v. y:).
(85)
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C.5.2 THE BIAS AND VARIANCE OF PLUG-IN VELOCITY
Under mild assumptions and with the Bias- Decomposition, we can analyze Ell - v, which consists of the bias term and variance term. The proposition below shows that although there an increase in bias of order (1/N), the variance is significantly reduced by O - 1/N).
Proposition C.13 (Bias- Variance Decomposition of Ideal Velocity), Assume there are the empirical distribution Pemp on any (N and the data distribution po has the finite normalization constant
(Pemp
(u)] 2 2 20 O.
Suppose 3M1, M2
0,8 2, s.t. (e [o))lee- I M1, and (x S M2. Then we have
(
MP +-M
Var (x,
Blio vol1 S C
N
N
Proof. According to Eq. 67
A B| Il /Bll Var[A] Var[B] Cov [A, B),
and the Cauchy-Schward incquality
-2 -2 Cov(AA, < Var[A] + Var[B].
have
E B| 11 < (EEv,- - E0,))² Varlu;] + Var[u:]
So analyze the bias and variance of e.r below. Bias of plug-in velocity.
E|v, - 1 |E Pemp )pxx x xo) - po(xx)p(zz) 20)00(<0 xo)1
E[v"]
E
20
s M, B Bilpeemp pollcy]
where
||l|c == sup {
du S (9)), 1/l s $
The previous work (Kloeckner, 2018) has proven that
C
E Pemp Polley
VN
so we finally get Variance of plug-in velocity. Let
CM2
N
7 7(x, ( == p(zz: 2o Do) (xo))d.
1
Σ p(:x ())
N
Then, under the empirical distribution, we can write
0i NZ N t)
vr (0
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which leads to the variance of oi as
N
Var[e;] -
Var
2
i=1
N
- -(1 (r (2 12
N
E
N2
C²
De (2r, 3 (0)
E
C²
N
1e (2(22))/(0)) + (B [er(acile 1)))
Var
2
N
Var (1((ar))((()) [0] + ==(a)))))
N
C'
(Var (2te + +M)
N
C.6
THE CONVERGENCE OF CTSC LOSS EMPLOYING PLUG-IN VELOCITY (SEC. 4)
Lemma C.14. Define the normalization constant
Z(q) = / p(20z (g)(()dy
For the weight function
wy(y) g(y)p((z y)
Z(q)
if there two distribution q and with finite normalization constant
Z(g), Z(r) zu > 0,
the following inequalities holds
(1) - wr(3)) S (21)d/22
+ (2T)d/2 of 40
Wi(gr))
L
(w)(2)) w. (1))1 (2T)do dzg
+ (2T) d
Wi(g,r).
Proof. First, since p( () Na y, of There exist constants L 0 such that
0 < p(x; y) S (2 of la(x, y) p(ax, v')l S L y [|-
Denote p(ace v) as (z))). We decompose
(3v) r(y) K2(z) (9()) (((3))-(3)) r(v)
Z(r) Z(a)) -(0))
wy( - w, (v)
r(y)
7(g)
2(r)
Σ(g)
Z(g)2(r)
Then, by Cantorovich-Ruubinsten duality,
[Z(g) Z(r) = 1/ Kr d(g -n) LWi(gr)).
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Also, using Z(q), 2 20 and Ki(y) S (2x)4/207 we have
(w)(y) --(()l= =
(2x (2x)4/220 la(y) r(y)l + (2 d/2 2/ 00
[44) - 2(r)
L
- (2T)d/20 +
22T W r).
Next, since 0 S ug(y)= S
1 /20, we bound the squared difference:
v2n0.
[wg(y)² (r) = (w)(v) w. ())())(())())) + wx)
I.
I 22/20/20 (27)d/220f2 + (2
Wi(g.r)
L
(27) (2m)do7da) Wi(g,r).
Proposition C.15. Denote the empirical distribution as Pemp, then the difference between the training loss employing plug-in in velocity and marginal velocity can be bounded by the Wasserstein distance between Pemp and po as
Cplus-in (0) Lmarginal (0) CW CW Po).
Proof. Substituting the form of plug-in velocity in Eq. 12, we have
d
-ptug-in in (0) E || |1,2,, (r - ui (xt)
(2
dt
E ((s)))(0s)) (r 1) d
d (t))
Ma (1) 10
(au (8) + àt (x. - ozy 11)
Σn a. 11
N
N(m ( 00,1
E
N(z; (
(1:)) (r- 1) d
uu (xx)) (árg + ō (a. (aa-a)))))) (1)))))
dt
az
We denote
N(oriay,,of)
wrx y, ty 1=)1))
Σg a0 of1)
(r -r) d (x)) (ayy + o. - dt
1(x1, y, 0,,,,
ag))
at
then
Cplug-in (0) = E Σ
w (x (+)) ()),),,)) (0)) tt,,,,, (t,,,,,)) b)]
Recall that Pemp (y)
1 N EN 1. (z), then we can obtain
w(x y, (8) N .) Wpmm (Iee, 3)
And the training loss with marginal velocity is
mmarginal (0) = E
Σ 1=1
Ww (w, 12) 2,,,, ,,)
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Finally, by the boundedness of and Lemma C.14, get
plug-inn (0) marginal (0) E
s O sup w - w
I CW1 (Pemp)
Remark C.16. We point out that the Wasserstein-1 distance between the empirical distribution and the true distribution decreases as the number of data samples increases, which has been established in previous literature (Fournier & Guillin, 2015).
D EXPERIMENTAL DETAILS
D.1 DETAILS FOR EMPIRICAL ANALYSIS OF FIG.
We conduct experiments on unconditional generation on CIFAR-10, 10, conditional generation on ImageNet-256x255 2266 with without the classifier-free-guidae setting. We here give more details of each method's setting of implementation.
Uncond. CIFAR-10. We use a unified setting with batch size as 512 and iteration number as 800k (~8000 epochs). For stability, adopt exponential moving average (EMA) update the model parameters, with decay ratio set to either 0.99995 or 0.9999. We find that 0.9999 ema decay usually performs better under 800k iteration with batchsize 512. We report results using the best-performing EMA setting. For all the experimental trials, we trained them with Nvidia-A A100x8.. The detailed hyperparameter configurations for each model are as follows:
CT and CT-linear. Both variants adopt LPIPS the loss metric, where the difference lies in the choice of time path: the former uses a cosine path while the latter employs linear in training 2e-4. Following the official JAX implementation,
path. We set the learning we adopt progressive time sampler such that the scale Kmin is initialized at 2 and gradually increased maximum of Kmax 150. This implies that the interval [omin maxx] is partitioned according to
h 1/0 K lo 2122)1) K S
{ (omas +
with Omin 0.002, maxx 80.0. After the change-of-variable, 2 arctan() is operated to scale the time from [0.002, 80] to [0.1] in cosine path, while in linear path, the sampled time is normalized by ट [0, 1). In addition, curriculum learning strategy is introduced to regulate the evolution of K with respect to the training iterations. When updating the model inside sg(-) via EMA, decay rate Temm is employed to further stabilize training. In detail, at training step [1......] J} with total steps 800k, the progressive scale K(j) and the corresponding EMA decay rate Tema() (j) are computed as
[V (Kmaxx + 1)2 K2iin) K2 K2in 1 +1,
K(J)
- Mog((eem-minn Kmia
Toma (j) exp
K KJJ)
Here K(j) is lower bounded by 1, and rema()) smoothly interpolates between r'eaa-in 0.9 and as training progresses.
SCD. For SCD, since the official release does not include the configuration for training
CIFAR, the same hyperparameter settings those used for ImageNet in the
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official release. K defined as the total number of steps that we divide the time interval into is set as 128, and the Pug = 0.25 as the probability of training the average velocity with instantaneous conditional velocity supervision described in Sec. 2.3. We set the learning rate in training as le-4.
IMM. Unlike the summary in Table 1, IMM here employs a cosine path with an EDM preconditioner. M as the group size is set as 4 and 7 12 for calculating the difference be- tween 8 and t, as its default configuration. For the grouped kernel function, is implemented by the RBF kernel. We set the learning rate in training to le-4.
sCT and sCT-linear. In time sampler, P memn -1 and Pad 1.4. Tangent warmup iteration for gradient ratio set as 10000. We set the learning rate in training to le- 4. Besides, the variational adaptive weighting techniques are not employed for better understanding the modularized contribution of each models, while the tangent normalization is employed in the sCT for stabler training, but not implemented in sCT-linear.
MeanFlow. In time sampler, Piman 220 and Paw 2.0. The Pleq 0.25 as the proba- bility of training the average velocity with instantaneous conditional velocity supervision. We set the learning rate in training to 6e-4. The power for adaptive weighting is 0.75.
Moreover, for CIFAR-10 10, to enable a fairer comparison, we keep the models identical except for the time sampler. Specifically, disable adaptive loss in MeanFlow, variational adaptive weighting in sCT, and tangent warmup, and instead use a squared l loss with a learning rate of 2c-4. Under this setting, with Peqq 0.25, we obtain FID50K of 4.64 for McanFlow and 4.81 for sCT-linear on CIFAR-10, which also validates our conclusion in the Sec. 3.
Cond. ImageNet. In this setting, we include the class label as part of the network input for conditional training. Since CTs require LPIPS their loss metric, replacing it with squared l2 loss on latents causes training to diverge. For all the experimental trials, we trained them with Nvidia-A x00 Therefore, we do not report CT's results in the latent space. For the other models, the settings are as follows:
SCD. The configuration is identical to that used for CIFAR-10.
IMM. It is implemented with linear path in latent space. M as the group size is set 4 and Y = 12. We observed that IMM fails to converge (FID does not decrease to a reasonable range) on SiT-B/2 when the B E {512, 1024]. Convergence appears only when we increase batch size B to 2048, at which point the model begins to generate valid images. This phenomenon is consistent with IMM's grouped loss: with group size M 4, each mini- batch provides only B B/M independent group-level supervision signals for backpropagation. 2048 yields 2048/4 = 512 effective signals, which seems to be a practical threshold for stable training in our setup. Therefore, in Fig. 2(b), we report IMM with bsz = 2048; the corresponding training epochs are scaled by the grouping factor, i..., 4 240 960 epochs, to match the effective number of parameter updates. Others are the same as the setting for CIFAR-10.
Consequently,
sCT and SCT-linear. We use the same hyperparameter setting as for CIFAR-10, since the original paper uses U-Net in the pixel space, we cannot use the provided official configuration.
MeanFlow. In time sampler, P Pmeen -.4 and Pas 1.0. The Pigy 0.75. We set the learning rate in training to le-4. The power for adaptive weighting is 1.0.
CFG. ImageNet. For one-step generation, our training setting with CFG follows MeanFlow, as introduces mixing scale k and defines the velocity under CFG
of (Z c) wo ( c) + ku (xe c) + (1 (1-w- w- f (x (xx))
(86)
This satisfies the original CFG formulation with an effective guidance scale k. As it is proposed to bridge the instantaneous velocity and average velocity under classifier-free guidance, applying this technique directly to sCT, which models the instantaneous velocity, is not entirely straightforward. However, for sin-rr, since we have shown its near equivalence to MeanFlow, the CFG training technique can be directly adopted. In addition, for IMM, applying CFG requires two NFEs during
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inference compute U. As our focus is one-step generation, i.e., 1-NFE, we therefore do not include IMM the comparisons.
In addition, we adopt the best hyperparameter configuration recommended in the official McanFlow implementation with DiT-B/2 while our network is changed into SiT-B/2. i.e., w = 1.0, K 0.5, class-dropout= 0.1 and CFG triggered if in [0, 1], while keeping all other settings identical to those used for Cond. ImageNet. As an improved variant of SiT over DiT, it leads the FID50k to 6.09, better than 6.17 reported the original paper.
D.2 DETAILS FOR EMPIRICAL ANALYSIS OF TABLE 2
Here, we adopt the exact same parameter setting as MeanFlow with SiT-B/2.
sCM training techniques. In addition, in ESC, we, following sCM (Lu & Song, 2025) and EDMv2 (Karras et al., 2024), introduce variational weighting output head, where the output of the time embedder is passed through linear layer one-dimensional scalar the adaptive weighting function, which reads w (t, r) and is then used reweight the original loss in Eq. We keep the the SiT architecture blocks untouched, while architectural improvements are orthogonal and possible. ier
Moreover, ratio n'gaad min Kand" }} for tangent warmup is implemented for mitigating some gradient spikes during training, where K grad is set 10k, the same sCT training.
(2.) (2.) sg Vij0 + r'gnal (r - -) dul, (2ze)
lesc (24, r,t dt, t;
w
D
dt
w adpi (1,r),
In this way, we gives the full hyper-parameter setting for Table 2, conclude in left column of Table 5. For all the experimental trials with network architecture SiT-B/2, we trained them with Nv0dia-A100x8.
D.3 DETAILS FOR SCALING-UP EVALUATION IN SEC. 5
We conduct caassuunconiitioall generation experiments on CIFAR-I0. Following the official MeanFlow setting, adopt the Adam optimizer with learning rate of 6 10-4, batch size 1024, and momentum parameters (31, 32) = (0.9, 0.999). We use a dropout rate of 0.2, no weight decay, and EMA decay factor of 0.99995. Training is performed for 800k iterations, including 10k warm-up phase. For time sampling, we draw (r, I) ~ LogNorm(-222 -200, 2.0), with probability 75% that o The adaptive weighting exponent is set to p 0.75. Data augmentation follows the protocol of Karras et al. (2022), except that vertical flipping and rotation are disabled.
CIFAR-10.
Regarding our proposed improvements, we observed that variational adaptive weighting from EDM2 did not yield further gains and was therefore not adopted. Instead, we found that setting the plug-in probability Pplug-in E [0.2, 0.5) improved training stability, although a performance gap remained. Moreover, we set Kix 20k and K Kgrat 10k. All CIFAR- 10 experiments with U-Net architectures were conducted on Nvidia 100 GPUs
ImageNet-256xx555 For large-scale evaluation, we adopt SiT-XL/22 as the backbone of our im- proved CTSC variant, denoted as ESC. The hyperparameters follow the default configuration recom- mended by MeanFlow under the CFG setting, with details provided in the right column of Table 5. In practice, we find that the tangent normalization technique does not further brings performance improvements in the continuous-time shortcut model with linear path in training SiT-XL/2. All ImageNet experiments with SiT-XL/2 were trained on 16 Nvidia A100 GPUs.
D.4 VISUALIZATION EXAMPLES FOR ESC
We provide visualization results of ESC-generated images ImageNet-256x256 with different network architectures: SiT-B/2 (Figure 4) and SiT-XL/2 (Figure 5). All samples are generated in single step using the same noise initialization from the latent prior and identical class labels for classifier guidance. Additional CIFAR- 10 examples generated by ESC at different training epochs shown in Figure 10.
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Table Configurations on ImageNet 256 256 for Table 2, (w/-cc) means 'with-clas--consistent
and (w/o-cc) means 'iiboout-clss-ccosissten"
Experiment
Sec. 4
Sec. 5
Configs
McanFlow
A1
A2
B1
B2
ESC
FSC(w/-ce)
ESC(W/-cc)
Architecture
B/2
XL/2
purams (M)
131
676
FL.PPs (G)
23.1
119.0
depth
12
28
hidden
heads
12
patch size
2x2
2x2
Training and optimization
epochs
240
240
batch size
512
256
dropout
0.0
0.0
optimizer
Adam (Kingma Ba, 2017)
Adam
schedule
constant
constant
Ir
0.0001
0,0001
Adam (81, (2))
(0.9. 0.95)
(0,9, 0.95)
weight decay
0.0
0.0
emn decay
0.9999
0,9999
Time sampler
Phog
0.75 logmorm(-0.4, 1.0)
0.75 lognorm(-0.4, 1.0)
(r. sampler
power for adaptive weight
1.0
1.0
CFG settings
Eq.
1,0
in Eq. 86
0.5
0.92
cls-cond drop
0.1
0,1
triggered if in
[0.0,100]
[0.0,0.75]
ESC improvments
Dglug-in
0.0
1.0
0.5
1.
0.5
0.0
0.0
0.5
0.2
0.2
Kgnd
10k
10k
00k
0Ok
Kaso
1
20k
20k
20k
20k
cluss-consistent batching
No
No
No
Yes
Yes
No
No
Yes
No
Yes
variational adaptive weighting
No
No
No
No
No
No
Yes
Yes
Yes
Yes
80
Meanflow-8/211M IMFE
ESC-B/2-INEE
ESC
McanFlow B/2
40
3.5
3.18
8:38
4
4
2.5
Derations 00
iterationn (K)
kerations (k]
(a) MeanFlow-B/2
(b) ESC-B/2
(c) ESC-XL//2
Figure 6: Comparison of FID5Ok under 1-NFE and under 2-NFE during training among different methods.
D.5 ALGORITHM FOR PLUG-IN VELOCITY CALCULATION.
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Algorithm 2 Calculation of Plug-in Velocity
Require: Training batch a E RBxDD, sampled time
1: Sample noise ~ (0, /)
2: Compute noised samples: 22 = (1 - t)æ + te
3: For all sample pairs (i,j) )) in the batch, compute
Dt.j (1 t)
4: Evaluate log-pro pe-probbbbllitis:
log Pi.j
Σ
log.N (Ergja 0,
d=1
5: Compute normalized weights along index i:
exp(log
wij Σn Pr
6: Compute conditional velocity:
Vend,i.j
Ei.j x2
7: Aggregate obtain plug-in velocity:
Σ Wij Vend.i.
pplg-iin.j
Ensure:
Uplug-in
(Ulug-inn.
D.6 FULL COMPARISON OF ESC VS. OTHER SOTA BENCHMARKS
We further include comparisons with the current state-of-the-art diffusion and autoregressive models for completeness, as shown Table for ImageNet 256x256, and Table 7 for CIFARIO.
Table 6: Evaluation of ESC and other benchmarks under one/few-step generation on ImageNet-
256x256. Underline means overall the best, while bold means the best in shortcut models.
Family
Method
Param.
NFE
FID50k
BigGAN (Brock et al., 2019)
112M
6.95
AR/Mask GAN
GigaGAN (Kang al., 2023)
569M
3.45
StyleGAN-XL (Karras et al., 2019)
166M
2.30
AR w/ VQGAN (Esser et al., 2021)
227M
1024
26.52
MaskGIT (Chang et al., 2022)
227M
8
6.18
VAR-d30 (Tian al., 2024)
2B
10x2
1.92
MAR-H (Li et al., 2024)
943M
256x2
1.55
ADM (Karras ct al., 2024)
554M
250x2
Diff/ Flow
LDM-4-G (Rombach et al., 2021)
400M
250x2
3.60
SimDiff (Hoogeboom el al., 2023)
2B
512x2
2.77
DiT-XL/2 (Peebles & Xie, 2022)
675M
250x2
2.27
SiT-XL/2 (Ma et al., 2024)
675M
250x2
2.06
SiT-XL/2+REPA (Yu et al., 2025)
675M
250x2
1.42
iCT (Song & Dhariwal, 2023)
675M
34.24
SCD (Frans al., 2025)
675M
10.60
Shortcut
IMM (Zhou al., 2025)
675M
1x2
7.77
3.43
McanFlow (Geng et al., 2025a)
676M
2.93
ESC (w/osclasscconsist.)
676M
2.92
ESC (w/-class-conssist)
676M
2.85
ESC+ (w/-class-consist.)
676M
2.53
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Table 7: Full comparison on unconditional generation on. CIFAR-10.
Family
method
NFE
FID
Diff-Instruct (Luo et al., 2024)
1
4.53
Distill
DMD (Yin et al., 2024b)
2.66
SID (Zhou al., 2024)
1.92
iCT (Song Dhariwal, 2023)
2.83
ECT (Geng et al., 2025b)
3.60
Shortcut
sCT (Lu & Song, 2025)
2.97
IMM (Zhou et al., 2025)
3.20
MeanFlow (Geng al., 2025a)
2.92
ESC
2.83
1-NFE FID
92
2.85
2.5
200
400
600
800
1000
1200
iterations (K)
Figure 7: Convergence of FID with ESC-XL with without class-consistent.
D.7 DETAILS OF ESC-XL/2 CONVERGENCE WITH AND WITHOUT CLASS-CONSITTER MINI-BATCHING
Here we give the convergence of FID with ESC-XL. with or without class-consistent in the complete training process, as shown in Figure 7.
E FURTHER ANALYSIS
E. PLUG-IN VELOCITY STABILIZES THE TRAINING
To figure out whether the plug-in velocity helps to stabilize the training of shortcut models, here give the training loss vs. iteration steps for MeanFlow and McanFlow with Plug-in Velocity. We show the comparison of the first 200k iteration in Figure 8, where all the training setting the same in our paper with batch size set as 512. further illustrates that incorporating the plug-in velocity significantly stabilizes the training of MeanFlow.
E.2 LARGE MODELS GAIN MORE PERFORMANCE FROM Low VARIANCE TRAINING
As shown, performance improvement for SiT-XL/2 over the MeanFlow is 16.9% while it is 5.3% for S/T-B/2 architecture. We attribute the performance gap to two key factors:
Optimization Dynamics. In larger networks (e.g., XL/2), the representational capacity increases substantially, amplifying the impact of optimization stability. As shown in Figure 8, MeanFlow exhibits higher variance and less stable loss behavior during training, whereas ESC maintains stable optimization and is therefore more likely to converge to a better solu- tion. In smaller models (e.g., B/2), the representational capacity is nearly saturated, leaving
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40
30
20
50c
120k
150k
200k
40k
B0K
1.2k
1904
200c
Training Rerations
Training
(a) MeanFlow without Plug-in Velocity (MeanFlow)
McanFlow with Plug-in Velocity (ESC)
Figure 8: Stable loss fluctuation with plug-in velocity.
80
40
MeanFlow-B/2- ESC-B/2-1NFE
MeanFlow-XU/--INE ESCXU/2-INFE
40
F1
20
10
6.
3.40
85
4-
100
200
300
400
500
600
200
400
600
800
1000
1200
Iterations
iterations
(a) McanFlow ESC with SiT-B/2
(b) McanFlow ESC with SiT-XL/2
Figure 9: Convergence of FID with different model architectures.
limited room for further improvement. In contrast, for larger models, ESC's improved stability enables it to better exploit the additional capacity, resulting in more noticcable performance gains.
Statistical Generalization. As the parameter space dimensionality increases, gradient noise also grows, making variance-reduction mechanisms- as EMA, momentum, gradient clipping, or the proposed plug-in velocity- more beneficial. This observation aligns with the theoretical intuition in Kaplan et al. (2020), where the generalization gap (or overfitting) is linked to the variance term scaling. Within scaling law framework, bias dominates in smaller models, while variance becomes the main factor as the model scales up. To illustrate this, we compare the FID convergence curves ESC-B/2 vs. MeanFlow-BB (trained for 600k iterations) and ESC-XL/2 in Figure 9. Empirically, in the smaller B/2 setting, both methods converge rapidly to similar FID values. However, in the larger XL/2 model, MeanFlow's FID curve plateaus in the later training stages, while ESC continues to improve and reaches 2.85. This suggests that in large-scale models, variance dominates generalization behavior, and the variance reduction introduced by plug-in velocity significantly enhances final performance.
McanFlow-XLL (trained for 1.2M iterations), as shown
F LIMITATIONS AND FUTURE WORKS
Slow convergence in few-step generation. An interesting phenomenon we observed is that, under the proposed improvements of ESC, employing two-step generation, i.e.,
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Zy X Xf.005(01) (z1)). led to slower FID convergence compared to one-step generation. This effect is particularly evident under the SiT-XL/2 architecture, whereas for B/2, the two-step scheme still achieves better performance, shown in Fig. 6. Although MeanFlow also exhibits relatively slow convergence with two-step generation, it still outperforms one-step (2.93 vs. 3.43). One possible explanation is that, in variational adaptive weighting, predictions from 0 to 1 are inherently more difficult. With the stronger expressivity of the XL/2 architecture, the training naturally allocates higher weights to u while the simpler sub-task u0.5,0 receives less weight. In contrast, for the more sapacity-limiteee B/2 architecture, fitting the easier task like 40.5.0 proves beneficial for the overall convergence. We leave a deeper investigation of this phenomenon as future work.
Inflexibility in training with CFG. We observe that introducing CFG leads a relative improvement of (33.05 6.09)/33 /33.03 = 81.5%, indicating that training with CFG is essential. However, the current approach follows Eq. 86, which inevitably introduces two additional hyperparameters, w and k. As shown in Table 5 and in Table 4 of the original work, the optimal values of these parameters, as well as the triggered intervals, vary significantly across architectures. This greatly complicates hyperparameter tuning, and for large models such as XL/2, results in substantial computational overhead. Therefore, we argue that alternative approaches, such as representation aligument (Yu et al., 2025), representation entanglement (Wu et al., 2025), or RL-guided generation (Zheng et al., 2025), may offer promising replacements by injecting class class-relate related semantic information into training or enabling CFG-free diffusion generation. We leave the exploration of these directions for future work.
Approximation for fast JVP. Since computing JVP is required, techniques such as FlashAt- tention cannot be directly applied in architectures like SiT. Although this does not incur significant time overhead, it leads to substantial memory consumption. Moreover, the computation of JVP itself is relatively expensive and introduces additional memory usage. In future work, plan explore numerical approximations of JVP to reduce reliance explicit differential operators.
Generalization to downstream tasks and more models. Our current work focuses purely generative modeling. An important future direction is to extend the proposed framework to downstream tasks where generation is conditioned on additional modalities, such as text- to to-image synthesis, image editing, molecule design. Incorporating cross modal alignment mechanisms and scalable conditioning strategies would allow the model to generalize beyond unconditional settings, making applicable to wider range of real-world scenarios. In particular, extending the framework to text-to- text-to-image generation represents a natural and promising step, enabling richer semantic control and practical applications. Furthermore. the proposed techniques like plug-in velocity, should be regarded as general training technique. Since our paper includes extensive modular decomposition and performance comparisons across wide range of methods, it is difficult to perform with/without plug-in velocity evaluations for all models under limited computational resources. We will consider extending the proposed techniques for evaluation to a broader set of models part of our future work.
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(a) 4000 epochs, FIDD50==3.39
(b) 8000 epochs, FID50k=2.98
(c) 12000 epochs, FID50k=2.93
(d) 16000 epochs, FID50k=2.95
Figure 10: Images generated by ESC trained with CIFAR-10 of different epochs
48


Reviews:
[{"review_id": "VgeOy8B7D9", "reviewer": "Reviewer_jD45", "summary": "This paper first summarizes and presents the design space of shortcut flow paths in diffusion models for one-step generation, such as consistency model, shortcut diffusion, MeanFlow, etc. The authors further propose plug-in velocity along with multiple pratical techniques to improve training of continuous-time shortcut models (SC), and enhance generation performance of shortchut models in their experiments.", "questions": "1. Regarding the multi-GPU training implementation, are the batches gathered across all devices to compute a plug-in velocity, or is the plug-in velocity computed locally on each device's batch?\n\n2. Do the authors investigate the impacts of the number of samples used for calculating the plug-in velocity? For instance, setting computational efficiency aside, does increasing the sample size lead to further performance gains?\n\n3. Table 3 indicates a significant performance gap between the MeanFlow-based approaches and the other methods benchmarked. Could the authors offer an explanation or formalize the reasons for this observation?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "+ The paper presents a formalization of shortcut models within a unified framework. This can provide a valuable foundation for subsequent work.\n+ The paper elucidates the design space of SC models. Both the mathematical formulation and the overall writing quality are presented with clarity.\n+ The paper makes several theoretical contributions that will likely benefit future research. These include: i) A Wasserstein distance bound for the objectives of discrete-time (DT-SC) and continuous-time (CT-SC) models. ii) An inference error bound for both CT-SC and DT-SC in terms of the variance of the average velocity. iii) The optimality of linear paths for SC models under Fisher information metrics.", "weaknesses": "- The empirical results indicate that the proposed plug-in velocity yields marginal performance gains, suggesting that its practical benefits over existing methods may be limited.\n\n- The experimental comparison would be strengthened by the inclusion of other state-of-the-art baselines, such as rectified flow and reflow, for a more comprehensive evaluation.\n\n- The improvement techniques presented appear to be specific to the MeanFlow architecture. The paper's impact could be broadened by exploring the generalizability of these techniques to other SC models. For example, the authors might include a discussion or experimental analysis on the selection of loss metrics for different SC model variants.", "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": "CIrWYUwPuv", "reviewer": "Reviewer_enqR", "summary": "This paper proposes a unified framework for various shortcut model methods and provides both theoretical and empirical analyses of the advantages and disadvantages of different designs. Based on these analyses, several improvements are introduced, such as plug-in velocity, gradual time samplers, and class-consistent mini-batching, collectively referred to as the explicit&easier shortcut modeling method, which achieves state-of-the-art performance in one-step ImageNet-256×256 generation.", "questions": "- The results in Tables 2 and 3 suggest that the proposed method brings more improvements with the large-scale network architecture than with the basic one. What causes this difference?\n- In Figure 3(b), several comparison curves included in Figure 3(a) are missing. What is the reason for this omission? The same question applies to Figure 3(c).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This paper provides a comprehensive framework for a series of shortcut model methods, offering effective tools for analyzing this family of approaches.\n- The analysis of each component is detailed and well-structured. By disentangling the individual components, the paper makes the design space considerably more transparent. Theoretical analyses are extensive and appear to be well-structured.  \n- The proposed method achieves SOTA results on one-step ImageNet-256×256 generation, demonstrating the effectiveness of the improvements.\n- This paper releases detailed code, ensuring reproducibility.", "weaknesses": "- According to Figure 3, it is difficult to claim that “the convergence of FID50k during training is substantially faster with the class-consistent mini-batching technique.” More evidence is needed to support this statement.\n- As shown in Figure 6, at the XL scale, the proposed method achieves worse FID under 2-NFE compared to 1-NFE. This might indicate a potential scalability issue of the proposed approach.\n- There are a few minor typo errors. For example, in line 198, $X^\\theta_{t,r}(xt)$ should be $X^\\theta_{t,r}(x_t)$; in line 263, $l_{scm}$ seems to refer to $l_{sct}$ mentioned earlier.", "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": "wmve2Tg3y3", "reviewer": "Reviewer_YqvP", "summary": "This paper proposes a unified framework for ODE based one-step diffusion models trained from scratch. The authors systematically analyze existing methods (CT, SCD, IMM, sCT, MeanFlow) by decomposing them into modular components and studying their design choices. \nBuilding on this analysis, they introduce ESC (Explicit & Easier Shortcut model), which incorporates several technical improvements including plug-in velocity, gradual time sampling, and adaptive loss weighting. ESC achieves improved FID50k of 2.85 on ImageNet-256×256 with one-step generation, without requiring pre-training, distillation, or curriculum learning.", "questions": "- how to interpret IMM as a special case in the proposed framework? e.g. build connection between eq(5) and eq(50)\n- In fig.1, it shows that practical prediction matches practical target, which is unrealistic. In addition, in fig 1 (c)(e), we have vector addition $v_{t|0}+u_{t|r}^\\theta=(r-t) d u_{t|r}^\\theta / dt$, which is inconsistent with eq(8).\n- In sec 2.4 Q3, the authors claim that \"(s)CM benefits from distillation by learning from a pretrained velocity field\", however, in sCM, \"We always initialize the CM from the EMA parameters of the teacher diffusion model. For sCD, we always use the $F_{pretrain}$ of the teacher diffusion model with its EMA parameters during distillation.\" That's to say, for sCT the teacher model is used as initialization for the one-step generator, which is critical.\n    - Given that the authors admit that at least a pre-trained model is important for better few-step performance as answer to this Q3, and that it's impractical to train large-scale text-to-image model from scratch, the authors failed to justify why in this work they insist working on training from scratch.\n\n-------- below could be biased (does not affect the rating) -------\n- focus of the paper. while aims to \"disentangles concrete component-level choices, thereby enabling systematic identification of improvements\", the empirical results are mainly focus on improved training techniques for MeanFlow baseline.\n- the authors show in Fig 1 visually and with prop 3.1 that the challenge of constructing flow map targets and the consequence during inference, however, this challenge is not directly addressed in the paper. \nThe training technique/design are proposed to enhance training stability, while no empirical results can validate the training instability of the baseline.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper is clearly written and easy to follow.\n- While the work does not introduce a fundamentally new method and offers only moderate novelty, it contributes a valuable unifying perspective on ODE-based one-step diffusion models.\n- The theoretical contributions are strong: Theorem 2.2 establishes error bounds for both DTSC and CTSC; Proposition 3.1 presents an insightful bias–variance analysis clarifying when CTSC outperforms DTSC; and Theorem C.7 theoretically justifies why linear paths are optimal for shortcut models under Fisher information metrics.\n- It provides comprehensive ablations and analyses that thoroughly examine the effectiveness of the proposed techniques, demonstrating overall high quality.", "weaknesses": "- Limited Novelty of Core Framework: While the unified view is valuable for understanding, this is already well established in previous work Flow Map Matching[1,2], which further hurt the contribution of \"propose a common design framework for representative shortcut models\".\n    -from discrete to continuous, one can either set $s= t−dt$ to get backward formula such as MeanFlow and sCM; or set $s=r+dt$ to obtain the forward formula such as AlighYourFlow [3].\n- Empirical Gaps:\n    - Slow convergence in 2-NFE generation (Fig. 6, Section E) suggests the improvements may be overfitted to 1-NFE\n    - Limited comparison with recent distillation-based methods that achieve better FID scores [4,5,6]\n    - The improvement compared to the baseline is nevertheless incremental.\n- Technical Concerns: The time sampler design (gradual transition from sCT to MeanFlow) and the choice of $p_{plugin}$ appears heuristic without principled justification\n\n[1] Nicholas Matthew Boffi, Michael Samuel Albergo, and Eric Vanden-Eijnden. Flow map matching with stochastic interpolants: A mathematical framework for consistency models. Transactions on Machine Learning Research, 2025. \n\n[2] Qiang Liu. Icml tutorial on the blessing of flow. International conference on machine learning,\n2025.\n\n[3] Amirmojtaba Sabour, Sanja Fidler, and Karsten Kreis. Align your flow: Scaling continuous-time flow map distillation. arXiv preprint arXiv:2506.14603, 2025.\n\n[4] Tianwei Yin, Micha¨ el Gharbi, Richard Zhang, Eli Shechtman, Fredo Durand, William T Freeman, and Taesung 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.\n\n[5] Mingyuan Zhou, Huangjie Zheng, 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 on Machine Learning, 2024.\n\n[6] 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. Advances in Neural Information Processing Systems, 36:76525–76546, 2023.", "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": "v4WhLYylR5", "reviewer": "Reviewer_rWuT", "summary": "This submission summarizes and organizing algorithmic design space of one-step diffusion models including consistency training, shortcut diffusion models, and mean flows.\nAlthough differences in derivation and wordings of the methods, they share computational framework of training by aligning one-step predictions toward targets that are acquired by two-step computation, which can be viewed as learning \"shortcut\".\nThe manuscript includes theoretical analyses claim that 1) both two-step DTSC and CTSC have bounded errors w.r.t. Lipschitz constants of the velocity field, and 2) inference error of  shortcut models measured Wasserstein-2 distance are bounded using bias and variance defined using targets and losses.\nFollowing the insights,  improved training method Explicit&easier Shortcut Model (ESC) is proposed. It uses techniques from the summarized literatures and new plug-in velocity, which aggregates in-minibatch velocity.\nExperiments using CIFAR and ImageNet shows that ESC can make improvements over MeanFlow slightly (-0.1 -- -0.3 in FID).", "questions": "- In algorithm 1, some lines are too much pythonic to interrupt non-programmer readers. for example, x[:,None,:] and logp_fn = Normal(0, 1).log_prob (function treated as an object) may be replaced if possible.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- One-step diffusion models are a hot topic in 2025 after Shortcut diffusion and Mean flow. This submission is a timely and nice follow-up to help understanding.\n- Theoretical analyses are deep and well motivated for understanding shortcut behavior of continuous and discrete models. (although I could not check all of the proofs in the near-30-page appendix.)\n- In the method part, plug-in velocity (Algorithm 1) is novel and shown to be useful in the experiments.\n- The overall paper are well organized.", "weaknesses": "- Practical impact: surely the SOTA results in FID are achieved but the FID gains provided by ESC may be marginal. I think that the differences of FID ranging within 0.1 -- -0.3 hardly impacts on human opinions on the quality of images.\n\n- Experimental/theoretical supports of design choices are relatively weak: Table 2 shows the design choice picked from the design space, but I think these selections looks empirical and I could not grasp how the theoretical results are exploited. Please point if I missed some parts.", "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": "XzeJZeLnSl", "reviewer": "Reviewer_k1TZ", "summary": "This paper addresses the problem of designing one-step diffusion models, often called shortcut models, which can be trained efficiently from scratch. The authors argue that existing works are complex, with theoretical derivations and practical implementations being \"closely coupled.\" This makes it difficult to understand the core design principles, compare methods, or innovate on individual components. To solve this, this paper proposes a unifying framework that reframes existing shortcut models under a single principle: approximating a two-step flow map target. They empirically and theoretically demonstrate that continuous-time shortcut models (CTSC) using a linear flow path consistently outperform discrete-time (DTSC) or cosine-path-based models. Building on this analysis, the authors propose an improved CTSC model called ESC with three key designs: plug-in velocity, class-consistent batching, and a gradual time sampler. Using these improvements, the authors train a scaled-up (SiT-XL/2) ESC model from scratch, achieving a new state-of-the-art FID of 2.85 on ImageNet-256x256 with a single function evaluation (1-NFE), without requiring any pre-training, distillation, or curriculum learning.", "questions": "1. Does the trained model actually satisfy the flow consistency property $X_{0.5,0}^{\\theta}(X_{1,0.5}^{\\theta}(x_1)) \\approx X_{1,0}^{\\theta}(x_1)$? The 2-NFE failure suggests it may not (Figure 6). Could this imply the model has simply memorized a single shortcut rather than learning the underlying flow field?\n2. Following the previous question, I'd like to know if ESC can be used in a multi-step fashion. If yes, are there any results or visualizations? \n2. Is ESC applicable to video DiT models? Are there any results?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper's primary strength is its common design framework. The \"one-step prediction vs. two-step target\" (Eq. 5) is a powerful and intuitive abstraction that successfully unifies a complex and dense field of research. Table 1, which decomposes prominent models into their constituent parts, is an excellent contribution to the community.\n2. The motivations are clearly stated and verified. The analysis in Section 3, which ablates flow paths (linear vs. cosine) and time sampling (discrete vs. continuous), is thorough and convincing, backed by both empirical results (Fig. 2) and theoretical analysis.\n3. The final result, an FID of 2.85 (1-NFE) on ImageNet-256x256, is a significant achievement, especially given that the model is trained entirely from scratch. This result challenges the long-held belief that one-step models require costly distillation from a pre-trained, multi-step teacher.", "weaknesses": "This is a strong paper without major technical weaknesses. Please see my questions below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)