Gaussian Mixture Flow Matching Models
Authors: Hansheng Chen
Kai Zhang
Hao Tan
Zexiang Xu
Fujun Luan
Leonidas Guibas
Gordon Wetzstein
Sai Bi
Venue: ICML 2025 poster (2025)
Keywords: GMFlow
diffusion models
flow matching
Gaussian mixture
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: GMFlow generalizes diffusion models by predicting Gaussian mixture denoising distributions, enabling precise few-step sampling and high-quality generation.

Abstract: Diffusion models approximate the denoising distribution as a Gaussian and predict its mean, whereas flow matching models reparameterize the Gaussian mean as flow velocity. However, they underperform in few-step sampling due to discretization error and tend to produce over-saturated colors under classifier-free guidance (CFG). To address these limitations, we propose a novel Gaussian mixture flow matching (GMFlow) model: instead of predicting the mean, GMFlow predicts dynamic Gaussian mixture (GM) parameters to capture a multi-modal flow velocity distribution, which can be learned with a KL divergence loss. We demonstrate that GMFlow generalizes previous diffusion and flow matching models where a single Gaussian is learned with an $L_2$ denoising loss. For inference, we derive GM-SDE/ODE solvers that leverage analytic denoising distributions and velocity fields for precise few-step sampling. Furthermore, we introduce a novel probabilistic guidance scheme that mitigates the over-saturation issues of CFG and improves image generation quality. Extensive experiments demonstrate that GMFlow consistently outperforms flow matching baselines in generation quality, achieving a Precision of 0.942 with only 6 sampling steps on ImageNet 256$\times$256.

Gaussian Mixture Flow Matching Models
Hansheng Chen
Kai Zhang? Hao Tan
Zexiang Xu² Fujun Luan
Leonidas Guibas' Gordon Wetzstein Sai Bi²
https://githii: wttp:tttw.ccmmmccccii////aaa
Abstract
Noise
(v (R:)
4a (A, kk,)
(C (k.)
4x (k,)
Diffusion models approximate the denoising dis- tribution as Gaussian and predict its mean, whereas flow matching models reparameterize the Gaussian mean as flow velocity. However, they underperform in few-step sampling due to discretization error and tend produce over- saturated colors under classifier-free guidance (CFG). To address these limitations, propose novel Gaussian mixture flow matching (GM- Flow) model: instead of predicting the mean, GM- Flow predicts dynamic Gaussian mixture (GM) parameters to capture a multi-modal flow veloe- ity distribution, which can be learned with a KI. divergence loss. We demonstrate that GMFlow generalizes previous diffusion and flow match- ing models where single Gaussian is learned with an L2 denoising loss. For inference, we derive GM-SDE/ODE solvers that leverage ana- lytic denoising distributions and velocity fields for precise few-step sampling. Furthermore, we intro- duce novel probabilistic guidance scheme that mitigates the over-saturation issues of CFG and improves image generation quality. Extensive ex- periments demonstrate that GMFlow consistently outperforms flow matching baselines in genera- tion quality, achieving Precision 0.942 with only 6 sampling steps on ImageNet 256x256.
Network
Network (ulk.,))
Network (mji,.))
Koswakk on(wix,,)
(u(ru
Ta (t.)
Flow DDPM steps (learned variance)
[x (tr)
ax(*,
(o (t..)
(a
network
au (mlan)
oo (wlt.,)
sofuls,))
GMFlow GM-SDE steps
Figure 1. Comparison between vanilla diffusion (with flow velocity parameterization) and GMFlow on 2D checkerboard distribution. (a) The vanilla diffusion model predicts mean velocity, modeling denoising distribution ge (æn (z;) as single Gaussian, and then samples from Gaussian transition distribution qe 9o( a2)2)) (b) GMFlow predicts a GM velocity and yields multi-modal GM denoising distribution, from which the transition distribution be analytically derived for precise next-step sampling. allowing more accurate few-step sampling (4 steps in this case).
1. Introduction
Diffusson probabilistic models (Sohl-Dickstein al., 2015:
Ho al., 2020), score-based models (Song & Ermon, 2019; Song al., 2021b), and flow matching models (Lipman et al., 2023; Liu et al., 2022) form family of generative
'Stanford University, CA 94305, USA Adobe Research, 95110, USA HHillbot, Correspondence to: Hansheng Chen <hanshengchen@ sdamf
models that share underlying theoretical framework and have made significant advances in image and video gener- ation (Yang et al., 2023; Po et al., 2024; Rombach et al.,
2022; Saharia et al., 2022b; Podell et al., 2024; Chen et al.,
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
2024; Esser et al., 2024; Blattmann et al., 2023; Hong et al.,
2023; HaCohen al., 2024; Kong et al., 2025). Standard

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Gaussian Mixture Flow Matching Models
diffusion models approximate the denoising distribution as
The main contributions of this paper are follows:
Gaussian and train neural networks to predict its mean,
We propose GMFlow, generalized formulation of diffu- sion models based on GM denoising distributions.
and optionally its variance (Nichol & Dhariwal, 2021; Bao et al., 2022b;a). Flow matching models reparameterize the Gaussian mean as flow velocity, formulating an ordinary differential equation (ODE) that maps noise to data. These formulations remain dominant in image generation as per user studies (Artificial Analysis, 2025; Jiang et al., 2024).
We introduce a GM- sampling framework consisting of novel SDE/ODE solvers and probabilistic guidance.
We empirically validate that GMFlow outperforms flow matching baselines in both few- and many-step settings.
However, vanilla diffusion and flow models require tens of sampling steps for high-quality generation, since Gaussian approximations hold only for small step sizes, and numeric ODE integration introduces discretization errors. In addi- tion, high quality generation requires a higher classifier-fre guidance (CFG) scale (Ho & Salimans, 2021), yet stronger CFG often leads to over-saturated colors (Saharia et al., 2022a; Kynkäänniemi et al., 2024) due to (OOD) extrapolation (Bradley & Nakkiran, 2024), thus lim- iting overall image quality even with hundreds of steps.
2. Diffusion and Flow Matching Models
In this section, we provide background on diffusion and flow matching models as the basis for GMFlow. Note that we introduce flow matching as special diffusion parame- terization since they largely overlap in practice (Albergo & Vanden-Eijndenn 2023; Gao et al., 2024).
Forward diffusion process. Let æ E R. be a data point sampled from distribution with its PDF denoted by p(ar). A typical diffusion model defines time-dependent interpo- lation between the data point and a random Gaussian noise e ~ (N I), yielding the noisy data D; = a;æ + dzE, where t E [0, T] denotes the diffusion time, and a2 at are the predefined time-dependent monotonic coefficients (noise schedule) that satisfy the boundary condition 2o = æ. ØT e. AApparently, the marginal PDF of the noisy data p(æ; can be written as the data distribution p(æx) convolved by a Gaussian kernel p( az)) = Nax;;; 0,40o o I), i.e., p(æ:) fRz p(æ, 22 (eo)p(zz) (zo) dæ Alternatively, p(æ;) can be expressed as a previous PDF p( Ar) convolved by a transition Gaussian kernel p(z; [x, Ad). given by:
To address these limitations, we deviate from previous single- Saagle-Gassiin assumption and introduce Gaussian mix- ture flow matching (GMFlow). Unlike vanilla flow models, which predict the mean of flow velocity u, GMFlow predicts the parameters of Gaussian mixture (GM) distribution, rep- resenting the probability density function (PDF) of u. This provides two key benefits: (a) our GM formulation captures more intricate denoising distributions, enabling more accu- rate transition estimates at larger step sizes and thus requir- ing fewer steps for high-quality generation (Fig. 1): (b) CFG can be reformulated by reweighting the GM probabilities rather than extrapolation, thus bounding the samples within the conditional distribution and avoiding over-saturation, thereby improving overall image quality.
p(z,|
B. af af- A
=N Dii
Ot
i-ss )1) (1)
We train GMFlow by minimizing the KL divergence be- tween the predicted velocity distribution and the ground truth distribution, which we show is generalization of previous diffusion and flow matching models (S 3.1). For inference, introduce novel SDE and ODE solvers that analytically derive the reverse transition distribution and flow velocity field from the predicted GM, enabling fast and precise few-step sampling (8 3.3). Meanwhile, we develop a probabilistic guidance approach for conditional generation, which reweights the GM PDF using a Gaussian mask to enhance condition alignment ($ 3.2).
Ft Abs
o
a1-2t
where the transition variance is denoted by Br, for brevity. A series of convolutions from 0 to T constructs a diffusion process over time. With infinitesimal step size At, this process can be continuously modeled by the forward-time SDE (Song et al., 2021b).
Reverse denoising process. Diffusion models generate samples by first sampling the noise DT +- and then revers- ing the diffusion process to obtain æo. This can be achieved by recursively sampling from the reverse transition distribu- tion p(æ; = pla;-ai) pr)p( (m)- DDPM (Ho et al., 2020) approximate p(æ;-11 (z;) with a Gaussian and repa- rameterize its mean in terms of residual noise, which is then learned by a neural network. Its sampling process is equiv- alent to a first-order solver of reverse-time SDE. Song et al. (2021b) reveal that the time evolution of the marginal distribution p(x,) described by the SDE is the same as that described by flow ODE, which maps the noise IT to data mo deterministically. In particular, flow matching models
For evaluation, we compare GMFlow against vanilla flow matching baselines on both 2D toy dataset and ImageNet (Deng et al., 2009). Extensive experiments reveal that GM- Flow consistently outperforms baselines equipped with ad- vanced solvers (Lu et al., 2022: 2023; Zhao et al., 2023; Karras et al., 2022). Notably, on ImageNet 256x256, GM- Flow excels in both Precision and FID metrics with fewer than 8 sampling steps; with 32 sampling steps, GMFlow achieves a state-of-the-art Precision of 0.950 (Fig. 6).
p(a.)
2

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Gaussian Mixture Flow Matching Models
train a neural network to directly predict the ODE veloc- ity field dæ: They typically adopt linear noise schedule, de
Table I. Comparison between vanilla flow models and GMFlow.
Vanilla diffusion flow
GMFlow
which defines T == 1, a == 1 - o, := t, thus yielding a simplified velocity formulation dæ: = Em Bmoo-(mooja.) [u]), with the random flow velocity u defined as:
Transition
small p(at ar lax)
Derive p(z (ax) from the predicted GM
de
assumption
is approximately a Gaussian
Network
Mean of Hlw velocity pa(
GM params in (o(M(as) > AuN ps. s'
output
u
Ft 30
Training
C flve [1(
an [ log So (o(
Tt
loss
Sampling methods
ost-ord: Fuler, DDPM... 2nd-ord: DPM++, UniPC...
Ist-ard: GM SDF/DDD 2nd-ord: GM-SDEVODE 2
This reveals that the velocity field dæs represents the mean of the random velocity u over the denoising distribution p(aoo/rx)) p(mo)p((m, imo)
di
Mean extrapolation (itt( c) - (1 ())
GM reweighting
Guidance
2 no (o c)
p(x,)
Flow matching loss. Flow models stochastically regress dm, using randomly paired samples of zy and Ty+- Let dt (o(()) denote flow velocity neural network with learnable parameters O, the L2 flow matching loss is given by:
mixture (GM):
K
go (++) = Σ k=1
AuN(u: Ex);
(4)
1 2 u (a(
where {As, Mk: } are dynamic GM parameters predicted by network with parameters 0, and K is hyperparameter specifying the number of mixture components. To enforce >n A = with A₁ 2 O, we employ a softmax activation expax where {ax E R} are pre-activation logits.
C = E ar m
In practice, additional condition signals (e.g.. class label or text prompt) can be fed to the network ( c), making it a conditioned diffusion model that learns p( [)).
A = Ee expax"
We train GMFlow by matching the predicted distri- bution (u((a)))) with the ground truth velocity distri- bution p(u Specifically, we minimize the Kull- back Leibler (KL) divergence (Kullback & Leibler, 1951) Eu-ppuule:) le.,) [log p(ul(2;,- ( log to (ul uu]ae))]. where log p(ul (a,,) does not affect backpropagation and can be omitted. Dur- ing training, we sample
Sampling errors in flow models can arise from two sources: (a) discrctization errors in SDE/ODE solvers, and (b) inaccurate flow prediction μ (zt, c) due to underfitting (Karras al., 2024). While discretization errors can be reduced by reducing step size, it increases the number of function evaluations (NFE), causing significant computational overhead. To mitigate prediction inaccura- cies, CFG (Ho& & Salimans, 2021) performs an extrapola- tion of conditional and unconditional predictions, given by wpo (I,, c) + (1 - w) (zzz)., where w E [1, +o0) is the guidance scale. Such a method improves image quality and condition alignment at the expense of diversity. However, higher may lead to OOD samples and cause image over- saturation, which is often mitigated with heuristics such as thresholding (Saharia et al., 2022a; Sadat et al., 2025).
Limitations.
(z,) by drawing pair of Do It, and then calculate the velocity using Eq. (2). The resulting loss function is therefore reformulated as:
L = E2 m m [- log go (u [xx)]-
(5)
In Eq. (6), we present an expanded form of this loss function.
Why choosing Gaussian mixture? While there are a large family of parameterized distributions, we choose Gaussian mixture for its following desired properties:
Mean alignment. The mean of the ground truth distribu- tion p( u x2) is crucial since it decides the velocity term in the flow ODE and the drift term in the reverse-time SDE ($ C.I). When the loss in Eq. (5) is minimized (assuming sufficient network capacity). the GM distribu- tion go go (ar)) guarantees that its mean aligns with that of p( u v..).
3. Gaussian Mixture Flow Matching Models
In this section, we introduce our GMFlow models, covering its parameterization and loss function ($ 3.1), probabilistic guidance mechanism ($ 3.2), GM-SDE/ODE solvers ($ 3.3), and other practical designs for image generation ($ 3.4). Table I summarizes the key differences between GMFlow and vanilla flow models. Algorithms and 2 present the outlines of training and sampling schemes, respectively.
Theorem 3.1. Given any distribution p(u) and any sym- metric positive definite matrices {Ex), J. if {ar nT)) are the optimal GM parameters w.r.t. with the objective min man Em--p(;) (- -log Σx Au, p Ex)], then the GM mean satisfies Σx Ai = Eu-p(u) [w].
3.1. Parameterization and Loss Function
Different from vamilla flow models that regress mean veloc- ity, we model the velocity distribution p( u (u) as a Gaussian
We provide the proof of the theorem in $ C.2. It's worth pointing out that not all distributions satisfy this property
3

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Gaussian Mixture Flow Matching Models
(e.g. the mean of Laplace distribution aligns the median instead of the mean).
defined conditional distribution (o(u e). This allows us to formulate probabilistic guidance, a principled approach that reweights the predicted distribution while preserving its intrinsic bounds and structure.
Analytic calculation. GM enables necessary calculations to be approached analytically (e.g., for deriving the mean and the transition distribution), as detailed in $ D.
To reweight the GM PDF qo (u) c) analytically, we mul- tiply by a Gaussian mask w()):
Expressiveness. GMs can approximate intricate mul- timodal distributions. With sufficient number of com- ponents, GMs are theoretically universal approximators (Huix et al., 2024).
w(u) Tu ( c) = 7 (ul(x,), c),
where Z is a normalization factor. The reweighted Iu (ula;, e) remains a GM with analytically derived pa- rameters (see D.2 for derivation), and is treated as the model output for sampling.
Simplified covariances.
The expansion of Eq. (5) involves the inverse covariance matrix Σ which can lead to train- ing instability. To mitigate this, we simplify each covariance to a scaled identity matrix, i.e., Σn = I, where & E I8+, is the predicted standard deviation shared by all mixture components, It's worth pointing out that this simplification does not limit the GM's expressiveness, which is mainly dominated by K. Moreover, Theorem 3.1 implies that the structure of the covariance matrix is irrelevant to the accu- racy of the mean velocity, mitigating the need for intricate covariances. Under this simplification, the expanded GM KL loss is reduced to:
Then, our goal is to design w(u) so that it enhances con- dition alignment without inducing OOD samples. To this end, we approximate the conditional and unconditional GM predictions qa (u Dee c) and qo (z)) as isotropic Gaussian surrogates N ((u: plc, seI) and N(u: Pu: s by matching the mean and total variance of the GM (see $ A. for details). Using these approximations, Gaussian mask as:
define the unnormalized
K
w(u) = N( Me + Ws..pm (1 w s
)
(8)
C = Er,
log
exp
N(u; Pe; (T)
k=1
}
where v E [0, 1) is the probabilistic guidance scale, and A pn == M4 is the normalized mean difference.
252 pull
D log + log As
(6)
Intuitively, the numerator in Eq. (8) shifts the conditional mean by the bias TssA,,, enhance conditioning, while re- ducing the conditional variance according to virinvar decomposition. Notably, for any w, a sample from the nu- merator Gaussian satisfies Ex [[|eu /el / = x². This ensures that the original bounds of the conditional distribu- tion are preserved.
which can be interpreted as a hybrid of regression loss to the centroids and classification loss to the components.
Special cases of GMFlow. GMFlow generalizes several formulations of previous diffusion and flow models. In special case where K = s, 1. Eq. (6) is identical to the L2 loss in Eq. (3). Therefore, GMFlow is generaliza- tion of vanilla diffusion and flow models. In another case where } are velocities towards predefined tokens and s x O, then {As} represent token probabilities, making Eq. (6) analogous to categorical diffusion objectives (Gu et al., 2022; Dieleman et al., 2022; Camphell et al., 2022).
Meanwhile, the denominator in Eq. (8) cancels out the orig- inal mean and variance of the conditional GM, so that the reweighted GM gw (ular, e) approximately inherits the ad- justed mean and variance of the numerator. Since the "mask- ing" operation retains the original GM components, the fine-grained structure of the conditional GM is preserved.
u-to-20 reparameterization. While the neural net- work directly outputs the u distribution. we can flexi- bly reparameterize it into an To distribution by substi- tuting u = #.-0a into Eq. (4), yielding go (0(2))z)) =
With preserved bounds and structure, probabilistic guidance reduces the risk of OOD samples compared to vanilla CFG, effectively preventing over-saturation and enhancing overall image quality.
Σx AwNi Mxg; I), with the new parameters fxx =
It- JzH4k and 8x = o,s. The velocity KL loss is thus equivalent to an xo likelihood loss L E, -- go [z).
3.3. GM-SDE and GM-ODE Solvers
In this subsection, we show that GMFlow enables unique SDE and ODE solvers that greatly reduce discretization errors by analytically deriving the reverse transition distri- bution and flow velocity field.
3.2. Probabilistic Guidance via GM Reweighting
Vanilla CFG suffers from over-saturation due to unbounded extrapolation, which overshoots samples beyond the valid data distribution. In contrast, GMFlow can provide well-
GM-SDE solver. Given the predicted based GM
4

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Gaussian Mixture Flow Matching Models
(0 Ex (mn Mxks s the reverse transi- tion distribution qo (at (m)) can be analytically derived (see& $ C.3 for derivation);
Eq. (10), yielding (( In an ideal scenario where both GMs are accurate, [24) perfectly matches go (ar (mo/ (x,);; otherwise, their discrepancy is extrapolated following the scheme. To perform GM extrapolation, we adopt a GM reweighting scheme similar to that in $ 3.2.
4o (z) (2)-))(
K
More details are provided in $ A.2.
+ c2flx:: (os + e)st) (9)
3.4. Practical Designs
c.s of au-ae 02
with the coefficients Ci
Ou-Ai At+ of
Pixel-wise factorization. For high-dimensional data such as images, Gaussian mixture models often suffer from mode collapse. To address this, we treat each pixel (in the latent grid for latent diffusion (Rombach et al., 2022)) as an inde- pendent low-dimensional GM, making the training loss the sum of per-pixel GM KL terms.
C3 = 0r,4: of oi- By recursively sampling from to ar) we obtain GM approximation to the reverse-time -teese SDE solution. Alternatively, we can imple- ment the solver by first sampling d: ~ (2)l and then sampling p( Lt, 20) = =N(x;-Ari C1 + czăo, ezf) (Song et al., 2021a) (see $ C.3 for details). Theoretically, if ( (as)) is accurate, the GM-SDE solution incurs no error even in a single step. In practice, a smaller step size At increases reliance on the mean (see $ C.4 for details) over the shape of the distribution, which can be more accurate as per Theorem 3.1.
Spectral sampling. Due to the factorization, image gener- ation under GM-SDE solvers performs the sampling step do ~ qo (3 (0 (z)) independently for each pixel, neglecting spatial correlations. To address this, we adopt spectral sampling technique that imposes correlations through the frequency domain. Specifically, correlated Gaussian random field from a learnable power spectrum. The per-pixel Gaussian samples are then mapped to GM samples via Knothe-Rosenblatt transport (Knothe, 1957; Rosenblatt, 1952). More details are provided in $ A.3.
generate spatially
GM-ODE solver. While GMFlow supports standard ODE integration by converting its GM prediction into the current mean velocity, it also enables unique sampling scheme with reduced discretization errors by analytically deriving the flow velocity field for any time 7 < I, thereby facilitat- ing sub-step integration without additional neural network evaluations. Specifically, given the Zy based GM qo (zo ax)) at Zt, we show in $ C.5 that the denoising distribution at æt can be derived as:
Transition loss. Empirically, we observe that the GM KI. loss may still induce minor gradient spikes when s be- comes small. To stabilize training, we adopt a modified loss based on the transition distributions (Eq. (9)), formu- lated as Lirms E,,.. - I- log go (z)))= where we define Al == AL with the transition ratio hyperparam- eter À E (0, 1]. When À = 1, the loss is equivalent to the original; when À < 1, the transition distribution has a lower bound of variance cz, which improves training stability. The modified training scheme is described in Algorithm 1.
p(x, (20) ( x1) Z p(xx xx))
g0 (2 ze).
(10)
where Z is normalization factor, and q(z)) (x,) is also GM with analytically derived parameters. This allows us to instantly estimate GMFlow's 's next-step prediction at æ, from its current prediction at æt, without re-evaluating the
4. Experiments
neural network. The velocity can therefore be derived by reparameterizing ((zg) x in terms of and computing its mean. This enables the GM-ODE solver, which integrates curved trajectory along the analytic velocity field via multi- ple Buler sub-steps between and t - At (Algorithm 2 line 18 18-2)). Theoretically, if go (x;) is accurate, the GM- ODE solution also incurs no error. In practice, the velocity field is only locally accurate as т + t and x, - xt. thus multiple network evaluations are still required.
To evaluate the effectiveness of GMFlow, we compare it against vanilla flow matching baselines on two datasets: (a) simple 2D checkerboard distribution, which facilitates vi- sualization of sample histograms and analysis of underlying mechanisms; (b) the class-conditioned ImageNet dataset (Deng et al., 2009), a challenging benchmark that demon- strates the practical advantages of GMFlow for large-scale image generation.
Second-order multistep GM solvers. Vanilla diffusion models often use Bashs---shfortth-ikk codder mul- tistep solvers (Lu et al., 2023), which extrapolate the mean To predictions of the last two steps to estimate the next midpoint, and then apply an Euler update. Analogously, we extend this approach to GMs. Given GM predictions at times & and + At, we first convert the latter to time via
4.1. Sampling a 2D Checkerboard Distribution
In this subsection, we compare GMFlow with the vanilla flow model on a simple 2D checkerboard distribution, fol- lowing the experimental settings of Lipman et al. (2023). All configurations adopt the same 5-layer MLP architecture for denoising 2D coordinates, differing only their output
5

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Gaussian Mixture Flow Matching Models
NFE
NFE
NFE
8 NPE 16 WFE =222
NEE 2 NFE
NPE
NEE:
WFF
NPE
(a) SDE method comparison
variance
K
(DDIM
DPM+
GM-SDE
GM-SDE
==1
Euler (DDIM)
(b) ODE mothod comparison
GM-ODE
GM-ODE 2
UniPC
(a) GMFlow K and solver comparison
NPF
GM-ODE =
K
GM-SDE 2
EM-SDE 2 wio (wr,(( conversion
Figure 2. Comparison among vanilla flow models with different solvers and GMFlow. For both SDE and ODE, our method achieves higher quality in few-step sampling.
(b) Ablation study on the (xolx,, conversion
Figure 3. (a) Comparison of first- and second order GM-SDE and GM-ODE solvers with varying NFE and GM components Increasing K improves few-step sampling results. Second-order solvers produce sharper histograms with fewer outliers than first- order solvers. (b) Ablation study on the ((<o)(x;) conversion in second order solvers. Removing this conversion causes samples to overshoot and concentrate at the edges.
channels. For GMFlow, we train multiple models with dif- ferent numbers of GM components K (with À = 0.9); for GM-ODE sampling, we use n [128/NFE] sub-steps.
Comparison against flow model baseline. Fig. 2, we compare the 2D sample histograms of GMFlow us- ing second-order GM solvers (GM-SDE/ODE 2) against the vanilla flow matching bascline using established SDE and ODE solvers (Lu et al., 2023; Zhao et al., 2023; Ho et al., 2020; Song et al., 2021a). Notably, vanilla flow mod- els require approximately 8 steps to achieve a reasonable histogram and 16-32 steps for high-quality sampling. contrast, GMFlow (K 64) can approximate the checker- board in single step and achieve high-quality sampling steps. Moreover, for vanilla flow models, samples generated by first-order solvers (DDPM, Euler) tend to concentrate toward the center, whereas those from second-order solvers (DPM++, UniPC) concentrate near the outer edges. In con- trast, GMFlow samples are highly uniform. This validates GMFlow's advantage few-step sampling and multi-modal distribution modeling.
Xs C.lll x
Pixel-wise (Me) X x C x
Pixel-wise (Au)
x xI x
DIT
Soft
c,
Embedding
2-Layer MLP
grad
Figure 4. Architecture of GMFlow-DiN The original DiT (Peebles & Xie, 2023) is shown in blue, and the modified output layers are shown in purple.
the expressiveness of GMFlow.
Second-order G. solvers. Fig. 3 (a) also compares our first- and second-order GM solvers across various NFEs and GM component counts. Comparing the left (first-order) and right (second-order) columns, we observe that second- order solvers produce sharper histograms and better sup- press outliers, particularly when NFE and K are small. With K 8, the GM probabilities are sufficiently accu-
Impact of the GM component count K. Fig. 3 (a) demon- strates that increasing K significantly improves few-step sampling results, leading to sharper histograms. Notably, GM-SDE sampling with = 1 is theoretically equivalent to DDPM with learned variance (Nichol & Dhariwal, 2021), which produces blurrier histogram. This further highlights
6

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Gaussian Mixture Flow Matching Models
GM-ODE 2
UniPC
Euler
GM-ODE
UniPC
Euter
GM-ODE 2
UniPC
Euler
GM-ODE 2
UniPC
Euler
NFE
NFE
NFE
NRE
Figure 5. Qualitative comparisons (at best Precision) among GMFlow (GM-ODE 2) and vanilla flow model baselines (UniPC and Euler),
GMFlow produces consistent results across various NFEs, whereas baselines struggle few-step sampling, exhibiting distorted structures.
GM SDE 2 GM -DE
DDPM small DPM 2M SDE
GM ODE
D 2M
rate that the second-order solvers do not make difference, aligning with theoretical analysis.
GM: ODE
Euler UniPC
DDPN
SA-Solver
DEIS
SDE solvers
ODE solvers
Ablation studies. To validate the importance of the ((2ro) (æx) conversion in second-order GM solvers, we con- duct an ablation study by removing the conversion and di- rectly extrapolating the 2y distributions of the last two steps, similar to DPM++ (Lu et al., 2023). As shown in Fig. (b), removing this conversion introduces an overshooting bias, similar to DPM++ 2M SDE (NFE- 8), underscoring its importance in maintaining sample uniformity.
10
Dest FID
3.00
Bost Best Praciision
0.55
0.65
0.85
4.2. ImageNet Generation
10
22
125
8
10
32
126
For image generation evaluation, we benchmark GMFlow against vanilla flow baselines on class-conditioned Ima- geNet 256x256. We train a Diffusion Transformer (DiT- XL/2) (Peebles Xie, 2023; Vaswani et al., 2017) using the flow matching objective (Eq. (3)) as the baseline, and then adapt it into GMFlow-Dii by expanding its output channels and training it with the transition loss (A- 0.5). As illus- trated in Fig. 4, GMFlow-DiT produces K weight maps and mean maps as the pixel-wise parameters {Ak; p J. Mean- while, a tiny two-layer MLP separately predicts using only the time l and condition e as inputs. Empirically, we find this design to be more stable than predicting 8 using the main DIT. The adaptation results in only 0.2% increase in network parameters for K 8. During inference, we apply the orthogonal projection technique by Sadat al. (2025) to both the baseline and GMFlow since it universally im- proves Precision. Additional training and inference details are provided in A.4.
(a) Best FID and Precision vs. NFE
SDE solvers
ODE solvers
1.0
Prectaion
0.9
0.9
0.8
0.8
0.7
0.7
Recall
Recall
(b) Preciiionn-eecaaee curves N
=8
Figure 6. (a) Comparison the best Precision and best FID among GMFlow and vanilla flow model baselines using different solvers across varying NFEs on ImageNet. For best FID, GMFlow signif- icantly outperforms the baselines in few-step sampling: for hest Precision, GMFlow consistently excels across different NFEs. (b) Precision Recall curves of different methods at NFE 8. Points corresponding to the best FID and best Precision are marked on the curves. GMFlow achieves superior Precision and Recall.
For quantitative evaluation, we adopt the standard metrics used in ADM (Dhariwal & Nichol, 2021), including Fréchet Inception Distance (FID) (Heusel et al., 2017), Inception Score (IS), and Rrecil (Kynkäänniemi et al., 2019). These metrics are highly sen-
Evaluation protocol.
sitive to the classifier-free guidance (CFG) scale: while smaller CFG scales (w x 1.4) often yield the best FID by balancing diversity and quality, human users generally favor
7

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Gaussian Mixture Flow Matching Models
higher CFG scales (w > 3.0) for the best perceptual quality, which also leads to the best Precision. The best Recall and IS values typically occur outside these CFG ranges and are less representative of typical usage. Therefore, for a fair and complete evaluation, we sweep over the CFG scale wo or the probabilistic guidance scale w for each model, and report the best FID and best Precision. We also present Precision-Recce curves, illustrating the qiveity-diversity trade-off more comprehensively. Additionally, following Sadat et al. (2025), we report the Saturation metric at the best Precision setting to assess over-saturation.
Table 2. ImageNet evaluation results at best Precision (NFE 32). The reported Saturation values (Sadat et al., 2025) are relative to the real data statisties (Saturation=0.118)
Method
Orthogonal projection
Guidance Precisiont Saturation
DDPM small DPM++ 2M SDE
w 3.3
0.944
+0.032
w 2.9
0.938
+0.032
GM-SDE
A 0.47
0.950
-0.019
GM-SDE 2
ab 0.39
0.950
-0.019
Euler
w = 3.3
0.934
+0.052
UniPC
10 3.3
0.931
+0.048
GM-ODE
Φ 0.47
0.947
-0.024
GM-ODE
1 0.47
0.947
-0.024
Comparison against flow model baselines. For baselines, we test the vanilla flow model using various first-order solvers (DDPM (Ho et al., 2020), Fuler (DDIM) (Song al., 202la)) and advanced second-order solvers (DPM++ (Lu et al., 2023), DEIS (Zhang & Chen, 2023), UniPC (Zhao et al., 2023), SA-Solver (Xue al., 2023)). Details on adapt- ing these solvers for flow matching are presented in $ A.5. In Fig. 6 (a), we compare these baselines with our GMFlow (K = 8) model equipped with second-order GM solvers (GM-SDE/ODE: 2). GMFlow consistently achieves superior Precision in both SDE and ODE sampling across various NFEs. Notably, GM-ODE reaches a Precision of 0.942 in just 6 steps, while GM-SDE 2 attains a ofate-of-thh-ar Pre- cision of 0.950 in 32 steps. For FID, GMFlow outperforms baselines in few-step settings (NFE < 8) and remains competitive in many-step settings (NFE 2 32). Fig. 6 (b) further illustrates that the PeecisionRReall curves of first- and second-orde GM solvers consistently outperform those of their baseline counterparts. Qualitative comparisons are presented in Fig. 5.
DDPM small
3.3
0.939
+0.056
DPM++ 2M SDE
2.9
0.933
+0.054
GM-SDE
@
0.27
0.943
+0.023
Fuler
3.3
0.931
+0.064
UniPC
3.3
0.929
+0.060
GM-ODE
1 0.27
0.933
+0.016
GM SDE
GN-CDE2 NFE-6
GM-SDE 1FE-32
GM-20E2
NFE =8
MFF -2
0.007
Beat Beat Precieion
Dest
4-
0.557
0,965
0.935
0.93
Figure 7. Best FIDs and best Precisions of GMFlow models with varying numbers of GM components K.
Table 3. Validation NLL of ImageNet training images (latents) under different GMFlow configurations.
K = 16
Method
K 1
K
K
16
+spectral
Saturation assessment. Table 2 compares the Saturation metrics of different methods at their best Precision. GM- Flow (K = 8) effectively reduces over-saturation, achieving Saturation levels closest to real data. Visual comparisons in Fig. 5 further support this finding. Additionally, ablating the orthogonal projection technique results in the same trend: GMFlow consistently achieves the best Saturation, while baseline methods perform even worse without orthogonal projection.
NLL (bits/dim)
0.346
0,263
0.242
0.224
0.173
posterior sampling applications.
Ablation studies. Table 4 presents ablation study results evaluating the impact of various design choices in our method. For GM-SDE, removing spectrum sampling (A2) leads to noticeable degradation in FID. Replacing our probabilistic guidance with vanilla CFG (A3) naively shifting the mean of the predicted GM-results in severe quality degradation (lower Precision) and over-saturation, which is also evident in Fig. 8. Replacing GM-SDE 2 with the second-order DPM++ SDE solver (A4) worsens both FID and Precision. Reducing the transition loss to the orig- inal KL loss (A5) degrades FID. Finally, for GM-ODE 2. directly applying ODE integration without taking sub-steps (B) 1) leads to significantly worse FID.
Impact of the GM component count K. In Fig. 7, we analyze the impact of the number of GM components on the metrics. Few-step sampling (NFE = 8) benefits the most from increasing GM components, particularly with the GM- SDE 2 solver. The metrics generally saturate at K = 8. Be- yond this point, the GM-SDE 2 Precisions decline because spectral sampling introduces larger numerical errors when more Gaussian components are used. Additionally, Table 3 presents the time-averaged negative log-likelihood (NL.)) values of data under the predicted distribution qo (2 c), showing that increasing the number of Gaussians signifi- cantly reduces NLL, suggesting its potential benefits for
Inference time. GMFlow only alters the output layer and uses solvers based on simple arithmetic operations. As a result, it adds only 0.005 sec of overhead per step (batch size 125, A100 GPU) compared to its flow-mater wow-matching counterpart,
8

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Gaussian Mixture Flow Matching Models
which is minimal compared the total inference time of 0.39 sec per step- most of which is spent DiT.
Table 4. Ablation studies on GMFlow (K 8, NFE 8) using ImageNet evaluation.
ID Method
Best FID)
Best Precision
Saturation
5. Related Work
AO Full model (GM-SDE 2)
3.43
0,939
-0.003
Prior works (Nichol & Dhariwal, 2021; Bao et al., 2022b;a) extend standard diffusion models by learning the variance of denoising distributions, which is effectively a special case of GMFlow with K = 1 and learnable s. GMS (Guo al., 2023) further extends this approach to third-order moments and fits bimodal GM to the moments during inference. In $ B.3, we present a comparison between GMFlow (K 2) and GMS. Conversely, Xiao et al. (2022) employ a genera- tive adversarial network (Goodfellow et al., 2014) to capture more expressive distribution, though adversarial diffusion typically relies on additional objectives for better quality and stability (Jolicoeur- Volicoeur-Martineau et al., 2021; Kim et al., 2024). However, none of these methods address deterministic ODE sampling.
AI AO GM-SDE
6.96 (+3.53)
0,938 (-0.001)
+0.009
A2 Al w/o spee. sampling
8.98 (+2.02)
0,940 (+0.002)
-0.022
A3 A2 Vanilla CFG
9.02 (40.04)
0,917 (-0.023)
+0.049
A4 AO + DPM++ 2M SDE
4.59 (+1.16) 0,912 (-0.077
-0.062
A5 AO + = 1.0
4.49 (+1.06)
0.941 (+0.002)
-0.019
BO Full model (GM-ODE 2)
2.77
0,946
-0.028
B1 BO w/o sub-steps
7.47 (+4.70) 0.947 (+0.001)
-0,031
A2: Prob.
guidance 0.55
A3: Vanilla CFG
A growing line of work trains networks to directly predict ODE integrals or denoised samples, enabling extremely few- step sampling (Salimans & Ho, 2022: Song et al., 2023; Yin et al., 2024b; Sauer et al., 2024). These approaches fall under the category of distillation methods, whose quality typically remains bounded by the many-step performance of standard diffusion models, unless supplemented by ad- ditional adversarial training (Kim et al., 2024; Yin et al., 2024a).
Figure 8, Ablation study on probabilistic guidance, comparing samples from Table 4 A2 and A3.
over-saturation. Extensive experiments demonstrated that GMFlow significantly improves few-step generation while enhancing overall sample quality. This framework lays the foundation for potential future research in both theoreti- cal and practical directions, including applications such as posterior sampling with GMFlow priors.
Efforts to refine CFG beyond thresholding (Saharia al., 2022a) and orthogonal projection (Sadat et al., 2025) in- clude disabling CFG in early steps (Kynkäänniemi et al., 2024), which compromises Precision, and adding Langevin corrections (Bradley & Nakkiran, 2024), which reduces efficiency.
Limitations
To apply Gaussian mixture to high-dimensional data, we adopted pixel-wise factorization for image generation, which may not fully exploit the potential of GMFlow, leav- ing rooms for further development.
Beyond diffusion models, Gaussian mixtures have also been employed in other generative models. DeLiGAN (Guru- murthy al., 2017) and GM-GAN (Ben-Yosef & Weinshall, 2018) enhance the latent expressiveness of GANs through the introduction of mixture priors. GIVT (Tschannen et al., 2024) models the output of an autoregressive Transformer as Gaussian mixture distribution, enabling continuous data sampling and outperforming its quantization- counter- part.
A Acknowledgments
We thank Liwen Wu, Lvmin Zhang, and Guandao Yang for discussions and feedback. Part of this work was done while Hansheng Chen was an intern at Adobe Research. This project was partially supported by Qualcomm Innovation Fellowship, ARL grant IN and Vannevar Bush Faculty Fellowship.
6. Conclusion
In this work, we introduced GMFlow, a generalization of diffusion and flow models that represents flow velocity as a Gaussian mixture, offering greater expressiveness for cap- turing complex multi-moda. structures. We derived prin- cipled SDE/ODE solvers unique to this approach and pro- posed novel probabilistic guidance technique to eliminate
Impact Statement
This paper presents work whose goal is to advance the field of generation models. There are many potential societal consequences our work, none which we feel must be specifically highlighted here.

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Gaussian Mixture Flow Matching Models
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A. Additional Technical Details
Algorithm 1: Complete GMFlow training scheme.
A.1. Details on Gaussian Surrogates
Input: Data distribution p(æo. c), transition ratio À
Output: Network params 0
For a GM distribution of the form Σi=1 A we approximate it with an isotropic Gaussian surrogate N N(p', 2 r) by matching the first moments. The surro- gate mean computed
Initialize network params O
for sample (æo, cj ~ p( c) do
I)
3
use logit-normal then
4
L Sample ~ LogitNormalOO, (0,
else
6
L. Sample U(0
"
Aspu
(11)
At AL
Sample 0i-ai p(at: (To)
Sample æ; ~ p( (0;4((;=111
To determine the surrogate variance, we equate the trace of the GM's covariance matrix (i.e., the GM's total variance) with the trace of the surrogate covariance, which yields:
10
Predict GM params in go (z (z c)
// E.Eq. (9)
11
Lives - log qo (1:- -
Predict magnitude spectrum
13
Lypa <- - logooo (vec(=)) O, diag(s:)
1/ Eg. Eq. (20)
Backpropagate and update 0
K
g²2
Ap Ll +1112
(12)
Algorithm Complete GMFlow sampling scheme.
Input: Steps NFE, sub-steps n, guidance scale wi,
A.2. Details on Second-Order GM Solvers
condition network params O
For GM extrapolation, we follow the method in $ 3.2 and fit the isotropic Gaussian surrogates (xxo p4+, s2.11 : qo (20 (zx),. (woi , I) s (( æ.). We then define a Gaussian mask:
Output: Zo0 At NPE #1
(O, I), Cache 0
while t > 0 do
Predict GM params in go (aro (a:,, c) if Ø 0 then
5
Predict GM params in qo (a (ass) Compute qu qa [22 c)
N ((zu M+ Ap,, (s2 1|4 )r)
6
I/ Eq. (7)
p(xg)
(13)
go (2ooo c) parm
qu Vuloool/e e)
N( ph: $2.
use 2nd-order then if Cache * {} then Compute @( (22, e) from Cache // E/ (10)
where Ap ال so that μ + Ap represents the ex- trapolated mean the next midpoint. The final extrapolated GM is obtained via the reweighting formulation:
Cache +- {we, (2oolae [,), Compute (Fo c)
// E/. (14)
13
go (o: (oul c) laa (ma at,, c)
Text (exr(200|at (20) p(Zo)
qo (2)o)a)).
(14)
else
L Cache + {mr, go (ano))) ( c)}
We then feed gear (2) (e)) to the first-order GM-SDE or ODE solver as a substitution for qe q0(zo (#)(
16
if use GM-SDE then
if use spectral sampling then Predict magnitude spectrum sy Sample % (mr ar c)
Empirically, we observe that second-ordes order GM solvers using the above formulation slightly underperform with higher guidance scales, which is common issue with multistep solvers (Lu et al., 2023; Zhao et al., 2023). To address this, we rescale the mean difference Ap by an empirical
// // (q.((19)
20
clse
Sample zu ( (22, c)
Sample Bi-Ai (LI-AGG + c200,
23
else if use GM-ODE then
factor y max (o.)4 (with the hyperparameter 0.005), so that high t practically disables multistep extrapolation.
T
At/n
while At do Compute q(20 (c.,c)
#1 Eq. (10)
Tr Z+
AIE ATE (xgla)))))))
22
0r
h
A.3. Details on Spectral Sampling
29
t f At
Due to pixel-wise factorization, image generation un- der GM-SDE solvers performs the sampling step to ~ go (2 (x)) independently for each pixel, neglecting spatial correlations. Spectral sampling addresses this by estab- lishing an invertible mapping between a frequency-space
distribution (power spectrum) and the pixel-space GM distri- bution. During training, the power spectrum optimized via
13

---

Gaussian Mixture Flow Matching Models
Pixel-wise (Mag] Kxx cx #
Compute
mean Ee. As ArMag Cx XW
Var of plxal means
Pixel-wise [Ag]
2-Layer MLP
Stop grad
Pixel-wise var Ee Av 11llltt
Ex AaMeael
Meen of vexee
Softmax
x
Sx based output
Viewed
Convert IFFT
Reshope
Sample from (0 diiag(s))))
KR transport
C
rendom field
Figure The optional spectral sampling pipeline used during inference. The spectrum MLP takes the statistics of pixel-wise GMs input, and predicts the power spectrum s²². Alternatively, if probabilistic guidance second-ordel GM solvers are employed, one can re-use the mean and variance from the numerator the Gaussian mask to compute the input statistics, which serves good approximation.
likelihood loss, while during inference, frequency-space samples transformed back into pixel space to introduce spatial correlation.
which can be expressed as:
(dvec(z)) N(voo(z;); 0, diag (s:)))
26 (2oglaz) (2xe))
det
axp
We establish an invertible mapping using two building blocks: the fast Fourier transform (FFT) and Knothe- Rosenblatt (KR) transport (Knothe, 1957; Rosenblatt, 1952). The FFT bridges a frequency-space Gaussian distribution and a pixel-space Gaussian distribution, while KR trans- port bridges each per-pixel Gaussian to its corresponding Gaussian mixture (GM) distribution.
(17)
where the absolute determinant can be easily derived using the properties of FFT and KR transport:
det (dvec(z)) )
Exc = det @vec(z)) )
det (Ovec(S)
@vec(S)
Exo
qo (2D
(ive(()) 0,
During training, given a real image vo E RCxHx and the factorized denoising distribution (0))))); = I E Az.ij.N(Zo.ij (Rrss. s KR transport defines an invertible mapping for each pixel:
qo [Zo (x:)
(18)
N(vec( 0)
Substituting Eq. (18) into Eq. (17), we obtain:
Tij To.sj + Sij:
(15)
NV (vve(z;;;; 0, diag(s;) N N(vec(z.);;
4g (10g ((00)]2x)) qo (m la:))
(19)
such that each Sij be viewed as standard Gaussian ple. We then assemble all Sij into tensor IR Cxxxx and apply forward 2D FFT with orthogonal normaliza- tion, yielding the complex frequency representation z = FFT(() CC Since is Hermitian symmetric, we can derive a real-valued representation z, while preserving invertibility:
With the derived PDF, the entire model can be trained by minimizing the negative log- likelloodd (NLL) of data sam- ples under the PDF, which inherently includes the GM KI. loss (Eq. (5) and (6)) as one of its terms. Therefore, the total loss consists the GM KL loss (replaced with the transition loss in practice) and an additional spectral loss, defined as:
2, Re[z] + Im| lm|z].
(16)
.V (Vvo((z,); diag(s;)2)
Lypec = Er,
log
NV(ve((z)) 0,
Finally, we impose a zero-mean Gaussian prior 2,, given by N ((vvo((z,) (2)); 0, diaag(s;)3) where Sp E R represents the magnitude spectrum.
= Et,
2 (diag((:))
veo((2))) 211
dynamically model 8p E RR, we use tiny two-layer MLP, which takes the mean of per- pixel GM variances and the variance of per-pixel GM means input, and outputs the power spectrum sf² with softmax activation, where (.)02 stands for element-wise square. With this invertible mapping and the spectral Gaussian prior, we can derive the model's spectrum spectrum-enhancee enhanced denoising PDF using the change variables technique in a similar way to normalizing flow models (Rezende & Mohamed, 2015),
+ log det det(diag((ss)
(20)
In practice, we stop the gradient flow through KR trans- port to prevent spectral learning from influencing the main GMFlow model.
During inference, we employ the spectral sampling pipeline illustrated in Fig. 9.
14

---

Gaussian Mixture Flow Matching Models
A.4. ImageNet Experiment Details.
w/ sub-steps
w/o sub-steps
w/ sub-steps
w/o sub-steps
We train both the baseline and GMFlow-DiT on ImageNet 256x256 with batch size of 4096 images across 16 A100 GPUs, using total training schedule of 200K iterations. We adopt the 8-bit AdamW (Dettmers et al., 2022; Loshchilov & Hutter, 2019) optimizer with fixed learning rate of 0.0002. Following Stable Diffusion (Esser et al., 2024), both mod- els sample t from a logit-normal distribution during training (Algorithm 1), which accelerates convergence.
NFE
NFE
While set the transition ratio to À 0.5 in the main experiments, the results in Fig. 7 and Table 3 are based an earlier design iteration that uses randomly sampled tran- sition ratios À ~ LogitNormall (0, 1). This setting slightly increases Precision and decreases Recall, though the overall differences remain minor.
NFE = 8
NFE 32
We densely evaluate the models across different guidance scales to identify the optimal FID and Precision. For vanilla CFG, we use guidance scales from the set (1.2. 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.1, 2.3, 2.6, 2.9, 3.3, 3.7, 4.3, 4.9, 5.7, 6.5); for probabilistic guidance, we use probabilistic guidance scales w from the set {0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.11, 0.13, 0.16, 0.19, 0.23, 0.27, 0.33, 0.39, 0.47, 0.55, 0.65, 0.75). In practice (Algorithm 2), we implement probabilistic guidance Tu bu-based GMs, which is equivalent to guidance on u.
Figure 10. Qualitative comparison (at best Precision) from the ab- lation study on GM-ODE sub-steps. Without sub-steps (forcing r = 1), few-step sampling tends to produce over-smoothed tex- tures and poor detail.
For DPM++ (Lu al., 2023), DEIS (Zhang & Chen, 2023), UniPC (Zhao et al., 2023), and SA-Solver (Xue et al., 2023), we use their Diffusers implementations (von Platen et al., 2022) and rescale their noise schedules to match the flow matching noise schedule. This approach is similar to how the EDM Euler solver (Karras et al., 2022) rescales a srriance-preserving diffusion model into a variance-exploding one at test time.
For inference with GM-ODE solvers, we generally set the number of sub-steps to n = max( INFE 2). which performs
128
well when NFE 2 8. For the exception when NFE = 4 or yields better performance.
6, we observe that redueing
In Table 3, the time-averaged NLL values are computed 50K samples from the training dataset using the following equation:
1 Er 20 20. .Ec [- logz 9o (2 (2 xx)),
B. Additional Experiment Results
NLL D
(21)
B.I. Ablation Study on GM-ODE Sub-Steps
where f U(0,1) 1) and D
H x W. This is equiv- alent to the original training loss (A- 1) with a uniform time sampling scheme. When spectral prior is enabled, we replace ge (e(zolz;) (z:) with (2 (x:) (Eq. (19)).
In addition to the results in Table 4 (BO and B1). Fig. 10 presents additional qualitative results from the ablation study on GM-ODE sub-steps, using K = 8 and the GM-ODE solver, As shown in the figure, sub-steps are essential for producing detailed textures when NFE- < 8.
A.5. Adapting Diffusion Solvers for Flow Matching
For DDPM solvers (Ho et al., 2020), implement them as special cases of GM-SDE solver with K = 1. The original DDPM solvers include large variance variant (3) and small variance variant (3). These are equivalent to setting and = 0, respectively.
B.2. Additional Qualitative Comparison
Fig. 11 presents a comparison among uncurated random samples (conditioned on random class labels) from GM- Flow and vanilla flow matching baselines, showcasing their respective best results under the many-step setting. Overall, the images generated by GMFlow exhibit more natural color saturation and, in some cases, improved structural coher- ence. These observations are consistent with the quantitative results shown in Fig. 6 and Table 2.
s
?
For DDIM solvers (Song et al., 2021a), its stochastic variant (1) = 1) is equivalent to DDPM with small variance, whereas its deterministic variant (17 = 0) is equivalent to Euler solver in flow matching models.
15

---

Gaussian Mixture Flow Matching Models
Table 5. FIDs on CIFAR-10 10 unconditional image generation. Com-
positive definite matrices. We aim to show that if
petitor results are sourced from Guo et al. (2023).
K Hkx Exs)
NFE
25
50
100
{ 14 = arg min B pump(u) {au-1ec)
- log
AuN
DDPM large
205.31
84.71
37.35
14.81
k=
DDPM small
34.76
16.18
11.11
8.38
(25)
SN-DDPM
16.33
6.05
4.19
3.83
with as R. Mr IR, An alignment property holds:
exp
then the mean
EF ki exp
GMS
13.80
5,48
4.00
3.46
GM-SDE
9.11
4.16
3.79
3.76
Σ Apu Eunp(u)(e)
(26)
B.3. Comparison with Moment Matching Methods
Table 5 presents a quantitative comparison among GM- Flow (K = 2), GMS (Guo et al., 2023), SN-DDPM (Bao al., 2022a), and DDPM (Ho et al., 2020) for CIFAR-10 (Krizhevsky al., 2009) unconditional image generation using SDE sampling. The GMFlow model is trained from scratch using the same U-Net architecture (Ronneberger et al., 2015; Ho et al., 2020) as its competitors, but with mod- ified output layers as in GM-DiT. The results demonstrate that GMFlow significantly outperforms its competitors in few-step sampling.
with 4
exp
_ exp
For brevity, define
qu(u) = ALN(u; uig Ex),
(27)
K
q(u)= = Σ (n()..
(28)
1=
Since {ak m minimizes the objective, the first-order opti- mality conditions imply
C. Additional Theoretical Analysis
3w u(u)p 1 log [
da
C.1. Details on the SDE and ODE Background
log q(u)
For notation references, this subsection briefly recaps the SDE and ODE formulation of diffusion models by Song et al. (2021b).
Euvp(u)
da
9ie (u)
= -Eunp(u)
Aï
<(1)
The forward-time SDE of the diffusion process can be de- rived by taking an infinitesimal step size At when applying the transition Gaussian kernel in Eq. (I), which yields:
(u((u)
= A Eupp(u)
7(())
= 0,
(29)
dæ = fix; dt + 9r dw:
(22)
where fa = 1 da: standard Wiener process. The corresponding reverse-time SDE is:
dt + 9: /
d dAt A and is the
and
OLE DEurp(u)
log q(u))
011 Eupp(()
dæ; = (Jrz, s.(e)) di + gt. dw.,
(23)
0 log g(v))
with the score function (x,) Va, log l(x)
0u
(u)
Ret Oraaam and the reverse-time standard
of
Eurp(u)
((u)
Dx [mg w)
Wiener process w. Using the Fokker-Planne equation, we can prove that the time evolution of the PDF p(æx)) described by the SDEs is equivalent to that described by an ODE:
1/2
[u((u)
(x(uu)
E
Bump(u)
q(u)
q(u)
1 -1 di, dæ, -froz
0.
(30)
(24)
From the above, follows that
which maps the noise o; to a deterministic data point æn.
[g(u))
A E Eump(u)
(31)
C.2. Proof of Theorem 3.1
q(u)
9a (u)) g(u)
(1(21)
Proof. Let p(u) be the PDF of an arbitrary distribution on RD.. and and
E Euvpp(u)
u
(32)
k= be a set of arbitrary D symmetric (
q(u)
16

---

Gaussian Mixture Flow Matching Models
Substituting Eq. (31) into Eq. (32), we have
which is convolution between p (.)) and
p( Sampling from p( (x))) can therefore be simulated by first sampling zo p( and then sam- pling 2i-At p(x:- Fo))
g qz((u) q(u) w
Al = pu)p(
(33)
Summing over all k, we conclude that
In general, given any empirical denoising distribution ( (z)), we can perform stochastic denoising sampling by substituting p(Zu(zz)) R qo (zolæ;) into the above sam- pling process. Specifically for GMFlow, (0 xxt) is a GM PDF D.3 shows that the convolution of GM and Gaussian is also GM with analytically derived param- eters. Therefore, GMFlow models the reverse transition distribution as a GM (= given by Eq. (9).
Σ -= (x(
Ag Eump(u)
u
(2u)
(34)
C.3. Derivation of the Reverse Transition Distribution
Given the ground truth data distribution p(mo) and the for- ward diffusion Gaussian p(z,(=0) = . arEoo of I), the ground truth denoising distribution rived using Bayes' theorem:
C.4. Additional Analysis of the Reverse Transition GMM
can be de-
By computing the first-order Taylor approximation of Eq. (9) w.r.t. A, we can re-write the GM reverse transition distri- bution as:
p(Z0) = p(zo))((z))))
(35)
p(x,))
Go (xt- ( 211
Conversely. let us assume that we know the ground truth denoising distribution p(zg a.) By rearranging Eq. (35),
K
AnN AnN(x -(fea: - (p At, ot AtI)
we can derive the data distribution as:
km
p(x,)=
p(ax (p(ar)
(36)
+ ((tt)
pl a (2)
With the data distribution, can apply the forward diffu- sion process and derive the noisy data distribution at At:
. I1-
fizr - 9 Σ Aktlak
At. St. ArI
+ +(At),,
(41)
{ p(z1- (x0) ((00) dæy
p() = Jgp
p(x p(21-11120) (z0))
pl (xx )p(pppp
dæy. (37)
where Pss - (a,, + +gf4)) The term Σ Agpis a. represents the model's prediction of the score function (xx), and is a linear transformation of the predicted mean velocity Σ AspuR- Recursively sampling Zr-At using the first-order approximation in Eq. (41) is equivalent to solving the reverse-time SDE in Eq. (23) (with predicted score) using the maler-Maruyama method, simple first- order SDE solver.
RD0
p(z,|az))
Finally, the reverse transition distribution p(x;: can be derived using Bayes' theorem:
p(ac-A1 xx))
p(x,-Ar)) p(x, Ar)
p(x;)
p( xc At (p(xt-Ae)220)
Eq. (41) is consistent with the standard diffusion model assumption that the reverse transition distribution is approx- imately Gaussian for small At. Even with GM parameteri- zation, SDE sampling is primarily influenced by the mean prediction when At is sufficiently small. This underscores the significance of mean alignment not only for ODE solv- ing but also for SDE solving.
p( xx)) dxp,
(38)
JRD
p( ( (0))
where p(zz/(:--e) is the forward transition Gaussian de- fined in Eq. (1). The term p
ma) be fused into one Gaussian PDF using the conflation operation described in D.I, which yields:
p(az jmp)
p(#;- A1 kr, σo) = N( + c2 cal), (39)
C.5. Derivation of (<za)=>)
with the coefficients C1 C22 c3 defined in $ 3.3. Therefore, Eq. (38) can be simplified into:
Let us assume that we know the ground truth denoising distribution at 24, i.e., p(zo/=z)) Eq. (36) shows us how to derive the data distribution p(z.) from p(z)) From p(xa), the denoising distribution at x, can be derived using
p(#;-A((0,) = / p( To)p((olar) dæy, (40)
17

---

Gaussian Mixture Flow Matching Models
Bayes' theorem:
D.3. Convolution of a Gaussian and a GM
p(zo)z;)) p(x0)) (p xn)) p(x0r)
Let p Ni + cØ2, 2) be conditional Gaussian PDF, where E R is linear coefficient, and
p(xx))p(( w ( )p(p) (20)
p(22) Σ
-=l (x2( Mg. E&) be GM PDF, where
pl a zen }p xx7)
A = Et-1 exp
exp
with logit ag. Their convolution yields
p [., (m1)
the marginal PDF Ø1:
pl (2)))
(42)
Z p(xx///))
J p(ar (22))(()))))))) dæz. RD
p(ar)
(51)
where Z 2( is a normalization factor. Substituting p(
p(zoo<z;) xx)) 2 go (2 (z)) into the above equation yields the
Since the GM PDF is a sum of Gaussians, the convolu- tion of a Gaussian and GM expands to a sum of convo- lution of Gaussians, which simplifies to a sum of Gaus- sians. Therefore p(x1) can also be expressed as a GM Σ A,N(2; 14, Σx), with the new parameters:
denoising distribution conversion rule in Eq. (10), The conversion involves conflations of Gaussians and GM, which can be approached analytically as shown in $ D.
D. Gaussian Mixture Math References
Σg = Σ + e² Ek.
(52)
D.1. Conflation of Two Gaussians
u = μt + cpi.
(53)
Let Pı (z) N(x; p1, E1), p2(x) Nix; p2, Σz) be two multivariate Gaussian PDFs, their powered conflation is defined as:
p'(x) = p7 (x)pp
(43)
Z
where 71. 72 E R. assuming 71 n, + 12 is positive definite, and fg p3" (m))p² 3 (z) dæ is a normalization factor. It's easy to prove that, p=x can also be expressed Gaussian NV (a: /r', E',, with the new parameters:
'' = (7) E 12Σ
(44)
= (' M4 + 72 1 p2)
(45)
D.2. Conflation of Gaussian and GM
Let (i Na u4, Σ) be Gaussian PDF, and (2
Σ AnN(x; ua, Σx) be a GM PDF, where A
expas with logit ag. Their conflation is defined as:
ΣR ax
P1 (1(x))2(x)
p'(x)
(46)
where Z = Sgo Pı (2) dæ is a normalization fac- tor. Since the GM PDF is a sum of Gaussians, the con- flation of a Gaussian and a GM expands to a sum con- flations of Gaussians, which simplifies to a sum of Gaus- sians. Therefore, pz(x) can also be expressed
GM
ΣK= A2;(2;; u E/,), with the new parameters:
Σ/ 1E-1+ET
(47)
1/ E M 4 ΣØ pk),
(48)
exp d'
A
(49)
Σ exp djp
where the new logit dg. is given by:
1 de az 7/2 (4 mw
(.
(50)
18

---

Gaussian Mixture Flow Matching Models
E. Notation
Table summary of frequently used notations.
Notation
Description
D
Data dimension.
t
E [0,
Diffusion time. Flow matching models define := 1.
Oz
Noise schedule coefficient. Flow matching models define dy 1 t.
Noise schedule coefficient. Flow matching models define dz == t.
Standard Gaussian noise.
2, Zo
E D
Data.
a. drE
Noisy data.
p(x)),((au)
PDF of data (ground truth).
p(a,)
PDF of noisy data (ground truth).
= =N of1)
PDF of forward diffusion distribution.
place ae)
=N(x6 on vo. (e,
of forward transition distribution.
01-2: of
Be,se
of- of
Variance of forward transition distribution.
p(
plana)piz,, lota) plate) p(m: (p)_) p(a.)
PDF of reverse denoising distribution (ground truth).
PDF of reverse transition distribution (ground truth).
Neural network parameters.
qo (mo))>))
PDF of reverse denoising distribution, predicted by network.
go (0x u
PDF of reverse transition distribution, predicted by network.
Random flow velocity.
E plo-ppall
Mean flow velocity at æe (ground truth).
p(u());)
PDF of velocity distribution æ (ground truth), derived from p(a
go( (u(w)))))
PDF of velocity distribution at æt, predicted by a network.
(x(x)
Mean velocity at æ,, predicted by a network.
K
Number of Gaussian components.
k
Index Gaussian components.
A
exp _x- exp da.
Mixture weight of the k-th Gaussian component.
Mean of the k-th Gaussian component.
Ex
Covariance of
Gaussian component. GMFlow defines Σ, == S
Pre-activation logit.
5
Shared standard deviation.
Mean of the k-th Gaussian component after u-to-2p conversion.
8x
0,8
Shared standard deviation after u-to-my conversion. Condition.
w
E (00)
CFG scale.
ú
E 1)
Probabilistic guidance scale.
a
Intermediate sample of denoised data.
NFE
Number of function (network) evaluations, a.k.a. sampling steps.
n
Number of sub-steps in GM-ODE solvers.
Sub-step diffusion time in GM-ODE solvers.
q( xx))
PDF of reverse denoising distribution, derived from go (wo)=;) (Eq. (10)..
((2ro) mx))
PDF of reverse denoising distribution, derived from go (ang At).
< (0
Transition ratio.
19

---

Gaussian Mixture Flow Matching Models
GM-SDE
DPM- SDE
SA Solver
DDPM small
DDPM large
GM-ODE 2
UniPC
DPM+ 2M
DEIS
Euler
Figure ff. Uncurated samples (at best Precision, NFE 32) from GMFlow and vanilla flow matching baselines using different solvers.
images generated by GMFlow exhibit more natural color saturation and, in some cases, improved structural coherence.
20


Reviews:
[{"review_id": "WRdXMh31Q3", "reviewer": "Reviewer_wKf5", "summary": "This paper proposes a Gaussian mixture (GM) flow matching (FM) model. The traditional FM model uses a Gaussian modeling velocity field, while the proposed GMFlow method in this paper uses a Gaussian mixture modeling velocity field. The author shows that GMFLow can produce better results with fewer steps. The author also proposes an SDE/ODE sampling algorithm suitable for GMFlow, as well as a probabilistic CFG algorithm, which can alleviate the over-saturation problem caused by traditional CFG.\n\n## update after rebuttal\nI thank the authors for their response and I will maintain my score as Accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review. We have uploaded a **revised manuscript** and essential **code** in this anonymous link (full code will be released upon publication):\n\nhttps://anonymous.4open.science/r/anonymous_gmflow-63FE  \nbackup: https://limewire.com/d/CgAn9#jkBxDmC3qh\n\n> Boundary of experiment in Table 2\n \nUnfortunately due to limited computing resources, we were not able to conduct experiments for K values beyond 16. That being said, as shown in Fig. 5 (or Fig. 7 in the revised manuscript). we have shown that the FID and Precision metrics generally saturate when $K\\ge 8$. The NLL values in Table 2 do not directly reflect image generation quality, and they are more relevant to some downstream applications (e.g., score distillation-like test-time optimization), which is not the main focus of this work. We are happy to add more discussions in the final draft.\n\n>  Discussion on the combination of GM with other generative models such as VAE\n\nThank you for the suggestion. In the revised manuscript, we have added a paragraph discussing GM GANs and autoregressive Transformers. In the context of modern generative AI, however, VAEs alone are often regarded not as generative models but rather as representation compressors. Some GM VAE papers are focused on clustering instead of generation [1, 2]. We are happy to add more related works in the final draft.\n\n- [1] Dilokthanakul et al. Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders.\n- [2] Jiang et al. Variational Deep Embedding: AnUnsupervised and Generative Approach to Clustering\n\n> FID in Table 3 \n\nIn Table 3, we aim to evaluate the saturation of different methods at their best precision, which typically requires large CFG values that are beyond reasonable ranges for FID comparison. The main comparisons on both FID and Precision are presented in Fig. 4 (or Fig. 6 in the revised manuscript).\n\n> In Line 70, $x_T \\approx \\epsilon$ does not take into account VE diffusion\n \nThank you for pointing this out. The manuscript states \"A typical diffusion model\", which does not cover all cases. In practice, mainstream diffusion models are trained with VP schedules, and they can be rescaled into VE diffusions during sampling (which is how the popular EDM Euler solver is implemented).\n \n> Will the results be better if $sI$ is replaced by a diagonal matrix?\n \nWe have tried predicting pixel-wise variances instead of the global $s$ in image generation. The cons outweigh the pros: it makes training less stable since it's more likely to have small variances in the denominator of the loss function, and the benefits for NLL can be equally achieved by increasing $K$.\n \n> Does pixel-wise factorization mean that each latent pixel is treated as a Gaussian component? How does this relate to K?\n \nYes. Each latent pixel (4 channels) is a 4-D GM of $K$ components. Please refer to the network architecture in Fig. 3 (or Fig. 4 in the revised manuscript).\n \n> Does pixel-wise factorization limit the method to a fixed resolution setting?\n \nWe don’t think so. We think the opposite might be true: without pixel-wise factorization, the entire latent grid would be treated as a high-dimensional GM, which would complicate diverse resolution generation due to varying dimensions of GM. We think pixel-wise factorization makes diverse resolution generation easier since the per-pixel dimensions are fixed.", "claims_and_evidence": "The paper claims that GMFlow has a lower number of sampling steps and that probabilistic guidance can alleviate oversaturation, which has been supported by experiments.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I have currently checked the correctness of  Theorem 3.1 in the main paper.", "experimental_designs_or_analyses": "The experimental design is mostly reasonable, especially the 2D experiment clearly shows the advantage of GMFlow in few steps.\nThe experiment in Table 2 is not perfect. The results show that increasing K has benefits, but the boundary is not seen.", "relation_to_broader_scientific_literature": "No.", "essential_references_not_discussed": "There is relatively little discussion of related work in this paper, and it is suggested to increase the discussion on the combination of GM with other generative models such as VAE.", "other_strengths_and_weaknesses": "Strengths\n+ The method in this paper is novel and promising to me\n+ The method proposed in this paper has many technical contributions. In addition to combining GM and Flow matching, it also proposes probabilities guidance and a new SDE/ODE sampler\n\nWeaknesses\n- Table 3 is the main experimental result, and FID is not used as a reference indicator", "other_comments_or_suggestions": "- In Theorem 3.1, w.r.t. with\n- In Line 70, $x_T \\approx \\epsilon$ does not take into account VE diffusion", "questions_for_authors": "- Will the results be better if $sI$ is replaced by a diagonal matrix?\n- Does pixel-wise factorization mean that each latent pixel is treated as a Gaussian component? How does this relate to K?\n- Does pixel-wise factorization limit the method to a fixed resolution setting?", "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": "De8tTXr2Ou", "reviewer": "Reviewer_qy1B", "summary": "The authors present a new formulation of diffusion models, termed as Gaussian mixture flow matching (GMFlow). Unlike existing diffusion models, GMFlow models the PDF of velocity by predicting the parameters of a Gaussian mixture (GM) distribution. Based on this formulation, GMFlow can generate high-quality images with fewer sampling steps. To avoid over-saturation artifacts, the authors propose a probabilistic guidance via GM morphing. In addition, the authors designs specific SDE/ODE solvers for GMFlow. They validate the effectiveness of the proposed method on both 2D toy datasets and ImageNet.\n\n## update after rebuttal\nI appreciate their thorough response and the rebuttal has addressed my concerns. I raise my score accordingly.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review. We have uploaded a **revised manuscript** and essential **code** in this anonymous link (full code will be released upon publication):\n\nhttps://anonymous.4open.science/r/anonymous_gmflow-63FE  \nbackup: https://limewire.com/d/CgAn9#jkBxDmC3qh\n\n> Pre-defined number of mixtures can limit the types of distributions. Using an excessive number of mixtures leads to high costs and requires a large model\n \nWith pixel-wise factorization, a small number of Gaussians is enough to capture the distribution of a single latent pixel. The experiments also reveal that image generation metrics generally saturate at $K=8$.\n\nOn the other hand, GMFlow only expands the output channel of the final layer to accommodate the mixture components. With $K=8$, currently, the final layer has an output channel size of  $(4 + 1) \\times K = 40$, where 4 is the latent channel size. So the computational cost of increasing K is still minimal when compared to that of the remaining layers in DiT, especially considering the hidden dimensions of DiT are usually much larger.\n \n> typos\n \nThank you for pointing this out. We have fixed the typos in the revised manuscript.\n \n> Evaluating the proposed method across different model sizes and image sizes\n \nWe have added an experiment on unconditional CIFAR-10 image generation (please refer to our response to **Reviewer A5dP**), which employs different architecture (U-Net), model size (53M), and image resolution (32x32) from the DiT used in the paper.\nUnfortunately, training 512-res GMFlow and baseline methods are too computationally expensive and time-consuming, and is beyond the time frame of the rebuttal.\n \n> IS metric\n \nFollowing previous work [3, 4], we employ Precision as the quality measurement instead of IS. This is because IS metric does not capture the image quality under a large guidance scale---in particular, IS is not sensitive to over-saturation. In fact, both our GMFlow model and flow matching baseline reach IS > 500 under the highest guidance scale evaluated, despite that the baseline has clearly worse visual quality.\n \n- [3] Sadat et al. Eliminating Oversaturation and Artifacts of High Guidance Scales in Diffusion Models\n- [4] Kynkäänniemi et al. Applying Guidance in a Limited Interval Improves Sample and Distribution Quality in Diffusion Models\n \n> Discussion of GIVT\n\nThank you for suggesting this interesting paper. In the revised manuscript, we have added a paragraph discussing GIVT and other generative models using GMs.", "claims_and_evidence": "Strengths\n+ The claims regarding the limitations of diffusion and flow models are correct and wildly recognized in the field of generative models. \n+ The proposed method is reasonable  and easy to understand. It enhances the representation capacity of flow models and enables high-quality generation with fewer steps.\n\nWeaknesses\n\nNone", "methods_and_evaluation_criteria": "Strengths\n+ The proposed method is both intuitive and reasonable. Modeling the PDF of velocity enhances the representation capacity of flow models, leading to more effective modeling of complex distribution. Probabilistic guidance and SDE/DOE solvers are specifically designed to address the challenges of expensive sampling steps and over-saturation artifacts.\n+ The evaluation benchmark is reasonably appropriate for assessing the effectiveness of the proposed method.\n\nWeaknesses\n- The proposed method relies on a pre-defined number of mixtures, which can limit the types of distributions. In addition, using an excessive number of mixtures leads to high costs and requires a large model for effective representation.", "theoretical_claims": "I have check the correctness of any proofs for theoretical claims. And I do not find any major errors.  \ntypos: In Equation (6), $x$ and $\\mu_k$ are related to $x_t$, and since they are derived from the density of $\\mathcal{N}(\\mu; \\mu_k, s^2)$, it would be better written as $\\| \\mu(x_0, x_k) - \\mu_k(x_k)\\|^2$.", "experimental_designs_or_analyses": "Strengths\n+ The superior results on both 2D toy datasets and ImageNet demonstrate the effectiveness of GMFlow. This paper also provides exhaustive ablation studies to assess each key component in Table 4.\n+ The qualitative results across different sampling steps (Figure 2,6, and 8) make the efficiency of GMFlow more apparent and intuitive. \n\nWeaknesses\n- The evaluations on visual generation are insufficient. It is better to evaluate the proposed method across different model sizes and image sizes. For model architecture, DiT-L/2 is an option. And, the evaluating on ImageNet $512 \\times 512$ would help demonstrate the generality of the proposed method.\n- Altough Inception Score (IS) is mentioned in Sec. 4.2 (Evaluation protocal), no results are provided. It would be beneficial to include the IS metric to comprehensively assess the proposed method.", "relation_to_broader_scientific_literature": "To my knowledge, the proposed method in this paper is new.", "essential_references_not_discussed": "To my knowledge, there is a work that should be discussed. The authors should include a discussion of GIVT [1] in the related work.\n\n[1] Tschannen, Michael, Cian Eastwood, and Fabian Mentzer. \"Givt: Generative infinite-vocabulary transformers.\" European Conference on Computer Vision. Cham: Springer Nature Switzerland, 2024.", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "None", "questions_for_authors": "I acknowledge the novelty of the proposed method, but I also have concerns regarding the pre-defined number of mixtures and the insufficient evaluations. Thus, I am inclined to rate this paper as weak accept.", "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": "oXVRLFiP1G", "reviewer": "Reviewer_gZDF", "summary": "This paper proposes Gaussian mixture flow matching (GMFlow) model, which captures the flow\nvelocity distribution rather than only predicting its mean based on single-Gaussian assumption. In\naddition, the paper utilizes the Gaussian mixture sampling framework to provide probabilistic\nguidance via Gaussian mixture Morphing, which alleviate the image over-saturating problem by\navoiding global mean extrapolation. Corresponding SDE and ODE solvers are proposed based on\nanalytical velocity distributions. Finally, the paper experiments on the 2D checkerboard\ndistribution and on large-scale class-conditioned ImageNet dataset to show the effectiveness of\nthe proposed method.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review. We have uploaded a **revised manuscript** and essential **code** in this anonymous link (full code will be released upon publication):\n\nhttps://anonymous.4open.science/r/anonymous_gmflow-63FE  \nbackup: https://limewire.com/d/CgAn9#jkBxDmC3qh\n\n> The GM idea is adopted in [1]\n\nPlease note that [1] is essentially a moment matching method, which only converts its moment predictions into a bimodal GM during inference. This is fundamentally different from our GMFlow formulation, which directly learns GM parameters. In comparison, [1] employs three L2 losses for three moments, whereas we use a single loss for all Gaussian components. \n\nMoreover, our formulation can generalize to more GM components, and we further propose few-step ODE sampling with analytical substeps and probabilistic guidance.\n\n> Theoretical analysis of using GM\n\nGaussian mixtures are well known to yield better approximation with more components [2, 4]. In diffusion SDE sampling, it is also widely recognized that improving the accuracy of the reverse transition distribution can greatly improve few-step sampling. Taken together, these observations already support the claim that using more Gaussian components should lead to reduced SDE sampling errors.\n\nThe practical performance ultimately depends on how the network learns the GM parameters, a process that is difficult to analyze theoretically. Therefore, we believe that our empirical validation through experiments is more important here.\n- [2] Huix et al. Theoretical guarantees for variational inference with fixed-variance mixture of gaussians.\n- [4] Bishop. Mixture Density Networks\n\n> Minimum of the cross-entropy training loss \n\nCross-entropy equals the sum of the data entropy and the KL divergence. Although the data entropy is intractable, it is solely determined by the data and independent of the model, thus it can be ignored in the loss function. Note that Eq. (6) omits all irrelevant constant terms. \n\nOur goal is to minimize the KL divergence between the predicted GM and the ground truth distribution, which has a minimum of 0 when the GM perfectly fits the ground truth.\n\n> Flow matching loss in [3] achieves minimum 0 and reduces the error variance\n\nIn general, the flow matching loss does not have a tractable form. While [3] analyzes special cases with analytic velocity fields, our work focuses on practical applications. \n\nTheoretically, the \"error variance\" of stochastic flow matching loss equals the sum of denoising distribution variance and the squared error of the velocity. The former is determined by the data, whereas the latter can be reduced to near 0 with enough capacity. \n\nDifferent from standard flow matching, our goal is not just to learn accurate local velocity, but also to capture the underlying denoising distribution so that a global velocity field can be analytically derived for multi-substep GM-ODE sampling.\n\n> Spectral sampling for image generation is a detour\n\nWe want to clarify that spectral sampling is just an **optional extension** to standard diffusion sampling with **minimal overhead**.\n- **Optional**: It's not used in our ODE solvers. In our SDE solvers, it's an optional trick to improve FID, whereas Precision is not affected, as shown in our ablation studies (Table 4).\n- **Extension**: When setting $K=1$ and using a uniform spectrum, spectral sampling is equivalent to standard diffusion sampling. Therefore, **we consider it a more general approach instead of a detour**.\n- **Minimal overhead**: The total overhead of GMFlow is less then 2% in inference time.\n\n> Choosing suitable K\n\nWith pixel-wise factorization, multimodality is indeed less apparent. From a practice point of view, we choose the best $K$ based on the evaluation metrics of interest. We also found that early-step training NLL can serve as a potential indicator.\n\n> Why the best Precision drops from K=8 to K=16\n\nWe believe this is due to numerical errors in spectral sampling, which grow larger with increasing $K$. We have tested the case without spectral sampling, where both $K=8$ and $K=16$ yield the same Precision of 0.946, with NFE=8.\n\n> Results of K=2\n\nWe have added an experiment using $K=2$ on the CIFAR-10 dataset (training on ImageNet is beyond the time frame of rebuttal), and our few-step results are much better than GMS. Please refer to our response to **Reviewer A5dP**.\n\n> Inference time\n\nSorry for the confusion. We have rewritten this paragraph for improved clarity: GMFlow adds only 0.005 sec of overhead per step (batch size 125, A100 GPU) compared to its flow-matching counterpart, which is minimal compared to the total inference time of 0.39 sec per step---most of which is spent on DiT.\n\n> Scale the experiment to large dimensions and preserving spatial correlations\n\nThis will be one of our future directions. We have recently conducted preliminary experiments using patch-wise factorization instead of pixel-wise factorization, which show promising results.", "claims_and_evidence": "- The novelty of the paper appears to be limited. The GM idea to approximate the reverse\ntransition kernel of diffusion process is adopted in [1], which is the case of K=2.\n- The theoretical analysis of using Gaussian mixture compared to single and bimodal\nGaussian remains to be established. The discretization error reduction could be further\nexplored with details including the approximation effect of the number of mixing\ncomponents K, similar to the Theorem 7 of [2].\n- The minimum of the cross-entropy training loss function (Equation (5) and (6)) is not 0\nand unknown due to the intractability of the entropy term. The error variance of this loss\ncan be sub-optimal compared to the modified flow matching loss in [3] that achieves\nminimum 0 and reduces the error variance.", "methods_and_evaluation_criteria": "See the Questions part.", "theoretical_claims": "I've read the proof for most theorems but not checked them carefully.", "experimental_designs_or_analyses": "Also see the questions.", "relation_to_broader_scientific_literature": "The paper applies the GM framework to GM morphing, which is an appealing direction in\nCFG to mitigate OOD problems.", "essential_references_not_discussed": "No, they are adequately discussed.", "other_strengths_and_weaknesses": "Strenghts:\n- The paper is well organized with clear motives and detailed implementation\nconsiderations.\n- The approach to capture the multimodality of velocity distribution by using GM structure\nand KL (with the cross-entropy part) training objective is straightforward and analytically\nfeasible.\n- The comparisons and ablation studies are sufficient in small and large scale experiments.\n\nWeaknesses:\n- To avoid mode collapse problems of GM models, spectral sampling for image generation\nis a detour compared to standard diffusion generation.\n- The codes of experiments are not released.", "other_comments_or_suggestions": "The theoretical and empirical understanding of choosing suitable number of mixing\ncomponents K remains unknown. The theoretical analysis of using Gaussian mixture\ncompared to single and bimodal Gaussian remains to be established.", "questions_for_authors": "- In Fig. 5 and Table 2, could the authors provide some explanations on why the best\nPrecision drops from K=8 to K=16? It seems to me that larger K should increase the\nvelocity distribution approximation capability and obtain better results. In addition, why\ndoesn’t the authors report the results of K=2, which correspond to the case of [1]?\n- The effect of larger K is evident in the 2D checkerboard experiment presented by Figure\n8(a), while Fig. 5 showed that increasing K has limited positive and even negative effect. Is\nthis because multimodality is not apparent in the large-scale class-conditioned ImageNet\nexperiment? If yes, is there any indicator/scheme (heuristic or analytic) for choosing\nsuitable K in experiments?\n- Which “base model” is GMFlow compared to in the “Inference time” paragraph (line\n376)? What does “simple” exactly mean in “based on simple arithmetic operations” (line\n374), since the calculations in other SDE and ODE solvers are also based on arithmetic\noperations that are not complex?\n- Is it possible to design another modified flow matching loss similar to [2] that achieves\nminimum 0 and lower error variance?\n- Is it possible to scale the experiment to large dimensions and preserving spatial\ncorrelations in a more natural way without using spectral sampling?\n\n\n[1] Hanzhong Guo, Cheng Lu, Fan Bao, Tianyu Pang, Shuicheng Yan, Chao Du and\nChongxuan Li. Gaussian Mixture Solvers for Diffusion Models. In NeurIPS, 2023. arXiv\npreprint arXiv:2311.00941.\n\n[2] Tom Huix, Anna Korba, Alain Durmus and Eric Moulines. Theoretical guarantees for\nvariational inference with fixed-variance mixture of gaussians. In ICML, 2024. arXiv\npreprint arXiv:2406.04012.\n\n[3] Gleb Ryzhakov, Svetlana Pavlova, Egor Sevriugov and Ivan Oseledets. Explicit Flow\nMatching: On The Theory of Flow Matching Algorithms with Applications. arXiv preprint\narXiv:2402.03232.", "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": "RGk4rhwCPZ", "reviewer": "Reviewer_A5dP", "summary": "This paper introduces GM-Flow, a novel variant of flow matching that explicitly parameterizes the entire velocity distribution using a mixture of Gaussians, rather than learning only the mean velocity as in conventional approaches. Unlike CFG, which extrapolates class-conditional and unconditional velocities, GM-Flow employs GM morphing to obtain the class-guided velocity, effectively avoiding extreme cases that lead to oversaturation in generated images. Additionally, the paper proposes a GM solver for sampling. The method has been validated through large-scale experiments on ImageNet, demonstrating strong scalability.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review. We have uploaded a **revised manuscript** and essential **code** in this anonymous link (full code will be released upon publication):\n\nhttps://anonymous.4open.science/r/anonymous_gmflow-63FE  \nbackup: https://limewire.com/d/CgAn9#jkBxDmC3qh\n\n### **Weaknesses and Questions**\n\n> 1. Comparison with GMS.\n\nWe’ve made a direct comparison with GMS and reported the results on unconditional CIFAR-10 image generation in Appendix B.3 of the revised manuscript. We train GMFlow with $K=2$ from scratch using the same U-Net backbone as GMS. We choose CIFAR-10 because re-training an ImageNet model using the backbone of GMS is expensive and time-consuming.\n\nHere are the FID results. GMFlow significantly outperforms GMS and other moment-matching methods in few-step sampling.  The competitor results are from the original GMS paper.\n \n|**NFE**| 10 | 25 | 50 | 100 |\n|-|-|-|-|-|\n| DDPM large | 205.31 | 84.71 | 37.35 | 14.81 |\n| DDPM small | 34.76 | 16.18 | 11.11 | 8.38 |\n| SN-DDPM | 16.33 | 6.05 | 4.19  | 3.83 |\n| GMS | 13.80  | 5.48 | 4.00 | 3.46 |\n| **GM-SDE 2 (ours)**| 9.11 | 4.16 | 3.79 | 3.76 |\n \n> 2. Computational complexity of KR transport\n \nWe would like to point out that KR transport is highly efficient since it's basically 1D CDF mappings. In the paper, we stated that GMFlow incurs only 0.005 sec of overhead per step, which is minimal compared to the total inference time of 0.39 sec per step (most of which is spent on DiT). This includes the KR transport. The computation complexity of per-pixel KR transport is $O(K\\cdot C^2)$ ($C$ is the channel size). We have added more details on spectral sampling in Figure 9 of the revised manuscript.\n\n> Section 3.2 needs to be expanded\n\nThank you for the suggestion. In the revised manuscript, we have fully rewritten and expanded Section 3.2 to make it more clear. We kindly refer the reviewer to the revised manuscript for more details.\n\n> 3. Why construct a Gaussian mask using two surrogate Gaussians?\n \nThe reasons are twofold:\n- The division of a GM PDF by another GM PDF doesn't have a closed-form solution, and would require slower numerical approximations.\n- If a raw GM PDF is put into the denominator, it's likely to be unstable since the denominator can be very small in some regions.\n\nWe have already experimented with full GM formulations using adaptive importance sampling, and they are slow and unstable.\nIn contrast, using Gaussians is simple and performs well in our experiments.\n \n> 4. Intuition behind this design of guided Gaussian.\n\nIn the revised manuscript (Line 183–196), we have added two paragraphs explaining the intuition behind the design.\n\n> 5. Why do you scale the shift $\\Delta \\mu_n$ by $s_c$?\n \nThis is to satisfy bias–variance decomposition, such that when $\\tilde{w} = 1$, all the energy in the variance $s_c$ is converted into the bias $s_c \\Delta \\mu_n$ , making the hyper-parameter $\\tilde{w}$ more meaningful. Please refer to the revised manuscript (Line 183–189) for more details.\n\n>  6. How do you obtain $\\mu_c$ and $\\mu_u$?\n \nIn the revised manuscript (Line 174–175 and Appendix A.1), we added that we approximate the conditional and unconditional GM predictions as isotropic Gaussian surrogates by matching the mean and total variance of the GM. This is equivalent to minimizing the KL divergence.\n\n> 7. Orthogonal projection trick\n\nWe apply this to $\\mu_\\text{c} - \\mu_\\text{u}$, prior to normalization.\n\n > 8. Please provide the analytical solution of Equation (8).\n \nYour idea is correct. The appendix has discussed conflation of two Gaussians **with powers**. A Gaussian PDF divided by a Gaussian PDF is basically multiplication with power 1 and -1, and the result is still a Gaussian. The code is simply implemented as\n```\ngm_output = gm_mul_gaussian(gm_cond, gaussian_mul_gaussian(gaussian_guided, gaussian_cond, 1, -1))\n```\nwhere all the operations are analytical.\n\n### **Additional Questions**\n\n> 2. and 5. (Typos)\n\nThank you for pointing these out! These typos been fixed in the revised manuscript.\n\n> 3. Please explain how you apply DDPM sampling scheme to flow models in details.\n \nAdded in the revised manuscript (Appendix A.5).\n\n> 4. Is $\\mathcal{L}_\\text{spec}$ used for training even if GM-ODE will be used for sampling?\n \nYes, we use the same model for both ODE and SDE sampling. The spectrum MLP is completely isolated and does not back-propagate to the main model. Therefore, the loss weights do not matter.\n\n> 6. Is there any typo in the definition of ${\\mu_s}_k$?\n \nWe confirmed that it's correct. This is the same as converting the flow velocity to the score function, i.e., $s_t(x_t) = -\\frac{1}{\\sigma_t} ( x_t + \\alpha_t u)$.\n\n> Is $s$ shared across all spatial locations? What does \"the mean of per-pixel GM variances\" refer to?\n \nYes, $s$ is shared. The GM variance refers to the GM’s total variance divided by $D$, which is also dependent on $\\{\\mu_k\\}$. We have added more details in the revised manuscript (Fig. 9).", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "See Weaknesses", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "This method extends conventional flow matching and addresses a key limitation by explicitly parameterizing the velocity distribution, offering a more robust approach to handling class-guided velocity.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "## Strengths\n1. The GM morphing approach is quite novel.\n2. The performance of the method is quite impressive and the method seems to scale well to large-scale datasets and models.\n\n## Weaknesses and Questions\n\n1. This paper lacks a direct experimental comparison with GMS (Guo et al. 2023).\n2. When using GM-SDE and perform spectral sampling, it seems to entail inverse KR transport. If I understand correctly, this step needs to be solved numerically, am I correct? In this case, the numerical steps should also be considered as a part of the computational complexity, instead of simply use NFE as the complexity metric. Please provide more detailed explanation on how the spectral sampling is performed and how complex it is.\n\nMost of my weaknesses/questions come from the Section 3.2. This section needs to be expanded, or a section in Appendix should be added to further elaborate the content in Section 3.2 in details.\n\n3. Why constructing a Gaussian mask using two surrogate Gaussians instead of the full GMs? Is it due to computational complexity when utilizing the full GMs? \n\n4. In Equation (7), how to interpret $g(u)$ in the context of \"class-conditional guidance\"? I mean, CFG basically add a shift $w(\\mu_c - \\mu_u)$ to the unconditional \"basis vector\" $\\mu_u$, right? But in Equation (7), the \"basis vector\" is $\\mu_c$, and the shift is $\\tilde{w} (\\mu_c - \\mu_u) / ||\\mu_c - \\mu_u|| \\sqrt{D}$. What is the intuition behind this design of guided Gaussian?\n\n5. In Equation (7), why do you scale the shift $\\Delta \\mu_n$ by $s_c$? \n\n6. In Equation (7), how do you obtain $\\mu_c$ and $\\mu_u$ from the model? Do they obtained by minimizing KL divergence between the surrogate and the learned GM $q_\\theta(u|x_t)$? \n\n6. You said you use the orthogonal projection trick to obtain better sampling quality, is this trick applied to Equation (7)? Do you apply it on $\\Delta \\mu_n$?\n\n8. In Equation (8), why do they have $ \\mathcal{N}(\\mathbf{u}; \\mathbf{\\mu}_c, s_c^2 I) $ in the denominator? The Appendix C only illustrate how to compute the $\\frac{g(u) q _{\\theta}(u|x _t, c)}{Z}$. If there is a Gaussian PDF in the denominator, will Equation (8) still yield a GM, and still analytically computable? My thought is that $ g(u) / \\mathcal{N}(u; \\mu_c, s_c^2 I) $ is proportional to another gaussian, so in the end it is a gaussian multiplying with a GM, thereby producing another GM. Am I correct? **Please provide the analytical solution of Equation (8).**", "other_comments_or_suggestions": "Please see the section of [Other Strengths and Weaknesses].", "questions_for_authors": "1. **Please upload the code during the rebuttal.**\n\n2. In Eq (6), what is $\\mathbf{x}$, shouldn't it be $\\mathbf{u}$? and what is $D$, the dimension of $\\mathbf{x}$? In previous sections you use $d$.\n\n3. What do \"DDPM small\" and \"DDPM large\" mean in Figure 4 (a)? The ancestral sampling of DDPM with different standard deviations $\\sigma_t = \\beta_t$ or $\\tilde{\\beta}_t $ of the isotropic Gaussian? Please explain how you apply DDPM sampling scheme to flow models in details.\n\n4. Is $\\mathcal{L}_{\\text{spec}}$ used for training even if GM-ODE will be used for sampling? How are the two losses  $\\mathcal{ L } _ {\\text{trans }}$ and $\\mathcal{L} _ {\\text{spec}}$ weighted?\n\n5. In the paragraph \"$u$-to-$x_0$ reparametrization\", in Line 192-193, the second parameter of the GM component should be $s_x^2$. The same for the $q_\\theta(x_0|x_t)$ on the right hand side of the same page, in the paragraph \"GM-SDE solver\".\n\n6. In Appendix A.4, is there any typo in the definition of $\\mathbf{\\mu}_{s_k}$? Please double check it.\n\n7. In Figure 3, is $s$ shared across all spatial locations? If this is the case, then what does \"the mean of per-pixel GM variances\" refer to? Additionally, please explain the modelling process of $\\mathbf{s}_{\\text{F}}$ more detailed.", "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:

Meta-review: All reviewers have carefully assessed the paper and provided detailed comments on its contributions. The authors have also addressed all the concerns regarding technical details and references to prior work. The reviewers have acknowledged the rebuttals and integrated them in their reviews. The overall recommendation from all reviewers is to accept the paper. All acknowledge that the paper is a clear step forward with several innovations (eg, a faster generation process and a solution to the over-saturation problem caused by traditional CFG) to the training of generative models for flow matching.
Decision: Accept (poster)