GLASS Flows: Efficient Inference for Reward Alignment of Flow and Diffusion Models
Authors: Peter Holderrieth
Uriel Singer
Tommi Jaakkola
Ricky T. Q. Chen
Yaron Lipman
Brian Karrer
Venue: ICLR 2026 Oral (2026)
Keywords: Flow Matching; Diffusion Models; Reward Alignment; Reward Adaptation; Inference-time scaling; Feynman-Kac Steering; Markov transitions; Sampling methods
Primary area: generative models
TL;DR: We improve inference-time reward alignment of flow matching and diffusion models by proposing a novel sampling paradigm that enables more efficient exploration.

Abstract: The performance of flow matching and diffusion models can be greatly improved at inference time using reward adaptation algorithms, yet efficiency remains a major limitation. While several algorithms were proposed, we demonstrate that a common bottleneck is the *sampling* method these algorithms rely on: many algorithms require to sample Markov transitions via SDE sampling, which is significantly less efficient and often less performant than ODE sampling. To remove this bottleneck, we introduce GLASS Flows, a new sampling paradigm that simulates a ''flow matching model within a flow matching model'' to sample Markov transitions. As we show in this work, this ''inner'' flow matching model can be retrieved from any pre-trained model without any re-training, effectively combining the efficiency of ODEs with the stochastic evolution of SDEs. On large-scale text-to-image models, we show that GLASS Flows eliminate the trade-off between stochastic evolution and efficiency. GLASS Flows improve state-of-the-art performance in text-to-image generation, making it a simple, drop-in solution for inference-time scaling of flow and diffusion models.

Published as a conference paper at ICLR 2026
GLASS FLOWS: TRANSITION SAMPLING FOR ALIGN- MENT OF FLOW AND DIFFUSION MODELS
Peter Holderrieth', Uriel Singer", Tommi Jaakkola', Ricky T. Q. Chen?,
Yaron Lipman?, Brian K MIT CSAIL,, 2FAIR, Meta
ABSTRACT
The performance of flow matching and diffusion models can be greatly improved at inference time using reward alignment algorithms,, yet efficiency remains a ma- jor limitation. While several algorithms were proposed, common bottleneck is the sampling method these algorithms rely on: many algo- rithms require sample Markov transitions via SDE sampling. which is signif- icantly less efficient and often less performant than ODE sampling. To remove this bottleneck, we introduce GLASS Flows, new sampling paradigm that sim- ulates a "flow matching model within a flow matching model" to sample Markov transitions. As we show in this work, this "inner" flow matching model can be re- trieved from pre-trained model without any re-training, combining the efficiency of ODEs with the stochastic evolution of SDEs. On large-scale text-to-image models, we show that GLASS Flows eliminate the trade-off between stochastic evolution and efficiency. Combined with Feynman- Kac Steering, GLASS Flows improve sfate-of-the- art performance in text-to-image generation, making it sim- ple, drop-in solution for inference-ting teteence-timeeeee scaling of flow and diffusion models.
demonstrate that
INTRODUCTION
Flow matching and diffusion models have revolutionized the generation of images, videos, and many other data types (Lipman et al., 2022 Albergo et al. 2023 Liu et al. 2022 [ong et al. [0206] Ho al. 2020). They convert Gaussian noise into realistic images or videos by simulating an ordinary or stochastic differential equation (ODE/SDE). Trained on large web-scale datasets, these models can generate highly realistic images or videos unprecedented quality. Due to diminishing returns of pre-training these models, many recent works propose methods to improve models at inference- time, i.c. by optimizing additional objectives commonly referred to as rewards (Uchara et al. 2025) 2025
These reward alignment algorithms allow to enhance text-to-image alignment (Zhang al.
solve inverse problems (Chung et al. 2022 He et al. 2023), and improve molecular design (Li et al., 20256). However, these algorithms achieve higher performance at the expense of more compute, efficiency remains a major challenge for deploying reward alignment algorithms.
Inference of flow and diffusion models has so far followed one of two sampling paradigms: (1) ODE sampling, as used in flow matching or the "probability flow ODE" in diffusion models, and (2) SDE sampling. Empirically, it is well-known that ODE sampling is significantly more efficient and is therefore the main choice for deployment of large-scale models (Karras et al. [022]] Esser et al. 2024). However, useful characteristic of SDE sampling is that it is random, 1.c. future point X is not determined by the present X, but characterized by transition probabilities
P (2,) = P = [x = =x1, (I,, Jp E Rd st
(1)
where petp is called the transition kernel. Many reward alignment algorithms rely on sampling from Prit- For example, search methods use samples X₂ ~ (- as branches of a search tree (for ODE sampling, there would be only one branch). This creates a dilemma: So far, it is not known how to obtain samples X₁ ~ Peje( (= using ODEs. Therefore, one has to switch from ODE to alignment algorithm that is meant increase it.
SDE sampling, losing efficiency in order to
In this work, we present a method to sample from transitions pe using ODEs: Gaussian Latent Sufficient Statistic (GLASS) Flows. GLASS Flows combine (1) the high efficiency of ODEs with (2) the controllable stochastic evolution characteristic of SDEs. GLASS Flows construct an "inner flow matching model" to sample from pele (see fig. Tp. Crucially, this inner flow matching can be

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Flow
GLASS (Ours)
Prompt
Start I s=0
Velocity Us (z
Preinened
U
photo of backpack
Ct
Ut
Denciser
Sample. (5 la)
Dr
photo of black kite and i green bear
Suffiient alsll sii (az T.)
photo tennis racket
Figure 1: GLASS Flows overview. Left: Sampling transition Peje(Iee (rz) with GLASS Flows. Initial Gaussian samples Σz=0 are evolved from inner time = to = via the velocity field u, (2, t) that is obtained by transforming a pre-trained flow matching model. Right: Reward alignment with GLASS Flows improves text-image alignment.
easily obtained from pre-trained flow matching models without any fine-tuning this transforma- tion relies the concept of a sufficient statistic, fundamental tool in theoretical statisties (Fisher 1922]. Hence, GLASS Flows are a simple plug-in for any algorithm relying on SDE sampling. We apply GLASS Flows to reward alignment and improve the state-of-the-an performance in text-to- image generation. To summarize, we make the following contributions:
We introduce GLASS Flows, method for efficiently sampling flexible Markov transitions via ODE's leveraging pre-trained flow and diffusion models.
2. We demonstrate that GLASS Flows can sample Markov transitions with significantly higher efficiency and lower discretization error than SDEs.
3. Text-to-image generation via GLASS Flows is shown to perform on par with ODE sam- pling, indicating GLASS Flows have eliminated the efficiency and stochasticity tradeoff.
4. We show significant performance improvements for text-to-image generation zero cost by plugging GLASS Flows into Sequential Monte Carlo and reward guidance procedures.
2 BACKGROUND AND MOTIVATION
In this section, we introduce the necessary background on flow and diffusion models. We follow the flow matching (FM) framework (Lipman et al., 2022 Albergo et al. [023]] Liu et al. [022). yet everything applies similarly to diffusion models (see appendix 3). We denote data points with E IRd and the data distribution with Pdata- Here, to data (Pdata). To noise data 2 E Rd we use Gaussian conditional probability path pr(ze(2):
corresponds to noise (A Ia)) and t
2, 2+2 + O2C, (0, Ia)
(=(2,12) =Nxx;; (22,ooLa)
(2)
where ag, o 0 are schedulers with og σ1 0 and a1 do and a (resp. o1) strictly monotonically increasing (resp. decreasing) and continuously differentiable. With & ~ Pdata ran- dom, this induces marginal probability path pe(2))) Ez-puan [pe (1a]a=)) which interpolates Gaussian noise po = N NO, Ia) and data p1 = Pdata- FM models learn the marginal vector field:
Ue (xx) = f
P1]((=(x)) Pe
(3)
(r()))
where un (2xl2) is the conditional vector field (see eq. the marginal vector field from initial Gaussian noise leads to a trajectory whose marginals are pt:
(3 for formula). Simulating an ODE with
d X. = uX(.)
X po:
Xe
dt
In particular, X1 Pdata returns sample from the desired distribution. This sampling method is commonly called ODE sampling with a flow matching model. In the diffusion literature, sampling in this way called the probability flow ODE (Song al. (02I). In addition, one can also sample
using the time-reversal SDE (Song et al. 2021 given by
22
V VlogPe(Xx) dt wdW, vî 2 it of 20,0: (5)
Xo NN1, Ia): dx, (X.)
ai
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where V log (2)) is the score function and o, 0, 0, are the time-derivatives of the schedulers (sce eq. (33) for a derivation). As this is the limit process of DDPM (Ho et al.) 2020), refer to this DDPM sampling in this work, regardless of the schedulers used. As the score function V log Ps is just a reparameterization of 0e (see appendix A.1), this SDE can be simulated using the same neural network. While every it 0 results in valid sampling procedure (Karras iet al., [2022] Albergo et al. 2023; Lipman et al., (2024), we restrict ourselves to the choice of Pr corresponding to the time-reversal SDE (DDPM sampling) as this is most commonly used.
MOTIVATION: EFFICIENT TRANSITIONS FOR REWARD ALIGNMENT
Inference-time reward alignment considers that the data distribution Pdota is not the "desired dis- tribution" that the model should sample from. To align models better with our goals post-training. one uses Pdata only prior distribution and steers samples from the model to maximize user- specified objective function r IR + R called the reward function. This goal formalized as sampling from the reward-tilted distribution
(=) =
Zr Pantaa (2) exp(r(=)) (Z, 2 0)
(6)
Note that the likelihood p" (z) is high if Pdata (z) is high and r(z) is high. We briefly review three of the most common reward alignment algorithms and how they rely on stochastic transitions Pric-
Sequential Monte Carlo (SMC) methods (Wu et al. 2023a Singhal et al. 2025; Skreta et al.
2025 use a transition kernel Pejt as a proposal distribution. They evolve particles 24 via
z Pejo(-(a() (0 St < 1' S 1, I,...,K) K)
The particles are then evaluated via potentials G(za, xe) that guide the particles towards the desired tilted distribution, e.g. G(I,, zF) = exp(r(x=)) r((x))) Subsequently, the particles are resampled:
(Gt æb), G(I xt)
(k = 1, K)
sample indioes
reussign particles
Here, SMC sequentially replaces "unpromising" particles by "promising" ones.
Search methods (Li al., 2025b Zhang et al., (2255) consider DDPM sampling as a rollout of a
search tree with branches coming from samples from Pelee Beyond sampling branches of the search
tree, search methods use approximations of the value function (Li et al. 2025b) defined via
(z,) = log Ez-p1j (xxpp//((()eepp(()))))))))))))))
where Pije(2 [mr) = (daaaa))) (x)
(8)
to evaluate nodes, i.e. to select nodes in the tree. Estimating the value function V, relies on the flow matching posterior P1iz- This is special case of a DDPM transition (Song al. 2020b):
(1]((( = DDPM (X1 = == =|X,- = x..)
As sampling from Pıje(z(zz)) is only possible with the SDE far and therefore inefficient, most search methods use approximations of this function (Li et al. 2025b Zhang et al. 2025). Similarly, approximations of the value function (zx) can also be used to define potentials in SMC procedures.
Guidance methods. Guidance methods (Skreta et al. 2025 Chung et al., 2022 He et al. 2023]
Feng al. 2025) modify the vector field u4 of the flow matching diftusion model using an intermediate reward function r : IRd + RR. such that ri r=())= = =(=)::
(x(x)) u, (x) + (x)
(c: 20
(10)
Again, ideally rx(x)) (x)),, which is computationally heavy to estimate for the same reasons. Instead, one can define re(rr) via simple approximations and potentially correct using SMC and SDE sampling (see e.g. (Skreta et al. 2025 Proposition 3.4)).
GLASS Flows motivation. Instead of proposing another reward alignment algorithm, we take a complementary approach: We optimize the transitions these methods rely on. Specifically, we aim to (I) improve how to sample the transitions; and (2) extend the space of transitions that we can sample from. As most deployed models use ODE sampling for efficiency, the reliance of inference- time reward alignment on stochastic transitions from SDEs is a common handicap making them slower and less performant. This motivates our goals:
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Goal 1: Simulate transitions Pelt in efficient way (via ODEs); and such that they are stochastic (i.e. emulates DDPM sampling).
Goal 2:
Extend the space of transitions Pric to allow for more effective reward alignment (e.g. SMC or search).
4 GLASS FLOWS
In this section, we present GLASS Flows, a novel way of sampling transitions from pre-trained flow and diffusion models. We begin by explaining the core idea.
Let us be given point X = I2 in a flow matching or diffusion trajectory. Given a time 1' from transition kernel Perr- We can consider this as
our goal is to sample X Parje(ael conditional generative modeling problem in itself. In other words, the variables Zt, are the variables condition (i.e. "prompts") and we want to sample ze. To do this, can, in turn, construct an inner flow matching model u, (T. [r, t) with a new time variable s (0 I 8, t < 1, Es, Te E IR() that is supposed model the transition kernel of Prlc-- Specifically, we want construct U, (f, such that after sampling from this model via
Xo
Pinit
X. = ux, + X1 Pe = 21)
(11)
ds
get samples from the transition kernel Prjz at for appropriate initial distribution Pinit- Note that with this approach, we achieve stochasticity by sampling the inner initial condition Xo- while the subsequent evolution follows a deterministic ODE. In contrast, SDE transitions have de-
terministic initial conditions but the increments are stochastic. We present a simple algorithm to obtain u. (,,/ t) that we explain in this section (see algorithm
Algorithm Transition sampling with GLASS Flows (with Euler ODE integration)
1: def D(z,, t):
>> FM denoiser
Input:
Start time end time t', current position Tr, pre-trained FM model u,, schedulers Olt, de, Ola, o. correlation p. number steps M
2: ut (xx)
> neural call
3:
return a (oqu )
4:
5:
def D(z Te, M4 Σ):
> GLASS denoiser
Output: Sample Xe Peje(Felzz)
6:
If 0: return De(x:) --z- S(x) + -. T,]
1: 7 p0rlop
2: Sample E ~ NOO, La)
7:
8:
t' + g ((1 111
3: Xo s 9x, ODE
4: $ s 0
9:
return D(ar (t))
5: h 1/M
10:
6: for m m= -,...., M do
def
us uz(T [(, l):
GLASS velocity
12:
+ d²
0.7 0²
+ of
ā. + 50t
Ot
7:
8:
X. + Xa hv
"
(X, p,, Call function
$ S h
13:
+ D Fo, ,4 Σ)
10: end for
14:
<- 0,0, 2012 <- 0,0, 0g Willly <- -5
11: Return X1
15:
return Wi F, + 8
4.1 GLASS TRANSITIONS
We first define the family of transitions Peij sample from. To define transition kernel Peje in flow matching model, we want Xx, X to have marginals given by the probability path:
X, ((a,2,07 Ta),
(z Paata)
Therefore, given a data point E Rd, the respective mean and variances of Xx, are fixed. However, have degree freedom set the correlation p between X, and Xp. Specifically, we define mean scale μ and covariance Σ as
H1 122
04 ae
Σ1 E2
Σ12
ot
pargy ot
Σ
Σn
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where the correlation -1 S P I 1 the degree of freedom that choose. Then let define the tuple X (X,, Xx,) and define the joint distribution as
d X ~ Pr. [ = II N (xx j=1
(z=(2".... (4))" ~ Priata)
(12)
Each coordinate is noised identically and independently we only allow for correlations across time, not across coordinates. Every joint distribution pe, pe,((X, xx) also defines a conditional distribution
Pele Pejo (X+ (X,, (x()
GLASS transition
(13)
which defines the GLASS transition. This is large family of transitions where p controls the similarity between X₁ and Xyr includes the important example of DDPM transitions:
Proposition 1. For
we get that: (X,, = Pe Xe) i.e. DDPM
transitions are special case of GLASS transitions.
See appendix A.2 for proof. Note that p defined like this valid correlation coefficient (i.e.
lpl & 1) because de' < 1 and Sir. de < by monotonicity of the schedulers.
4.2 CONSTRUCTING THE VELOCITY FIELD
In this section, we show how construct u. (T, (2,, t) to sample from the GLASS transition Pric from pre-trained flow matching and diffusion models without any re-training or fine-tuning. A fundamental concept we use is denoiser model D, defined as the expectation of the posterior:
D.(r)= = 2P1p (2/p(=/ 6 a,Ø;
(a - dz
(14)
The second equation shows that
easily obtain the denoiser by reparameterizing the velocity field uy (see appendix [A.1] for derivation). In the following, we use the same reparameterization idea but the other way around: To construct ú. (2,, t), (1) derive denoiser model for Markov transitions and (2) reparameterize to obtain the velocity field u. (T.
4.2.1 GLASS DENOISER
We begin by extending the idea of denoiser Markov transitions from F; to Ty- In eq. 11). have defined a joint distribution over X = (X,,Xp) Xy.) specified by some mean scale and covariance Σ. Therefore, define the GLASS denoiser as the expected posterior given both 2, and ze:
(x,x(x)= = 2p(Z = x x = (Iz, Te It, We E Rd
(15)
Here, it is instructive to think of 2; as a noisy measurement of a parameter 2. The "standard" de- noiser D, represents the mean of z given one Gaussian measurement wt, while the GLASS denoiser Dμ, represents the mean of given two Gaussian measurements (T,, Our core idea is that we can effectively "summarize" two measurements (Z,, 22y into a single variable via the transformation
S(x) 11 - 1'''-pp' x (x) e 2xd
sufficient statistic
In theoretical statistics, S(x) is called sufficient statistic (Fisher 1922 describing the idea that S(x) carries much information about the latent Z = 2 as does x. This is
Casella & Berger 2024
intuitive: S(x) is weighted average of øt, Zy the weight higher the more informative an element is about = (lower variance and higher scale factor p)). Finally, define invertible function g(t) of o as the effective noise scale defined by flow matching schedulers at, Ct. We get:
Proposition 2. Let x = (21, with X; E Rd and t* to (M, Σ) '((mE'11 (1) Then:
Dx,x(x)
Dr- (ar S(x))
"standard" pre-trained denoiser with reparameterized input and time where D, is defined in eq. (14) and a, is the scheduler in eq.
GLASS denoiser
So, the GLASS denoiser be obtained by single function evaluation of pre-trained model (see algorithm|] l). See appendix A.3 for a proof. We note that g a simple analytical formula depending the choice of (4, at (see appendix for specific formulas).
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4.2.2 GLASS VELOCITY FIELD
We now derive the GLASS velocity field (ā Zr. t) as a reparameterization of the GLASS de- noiser. Since Pr, (Tt, Tel [2) is Gaussian (see eq. 12 also the conditional distribution is Gaussian
(e k2,, 2) =N(ze;ā2 , dd)
(16)
where 7 = â ar you,
の² of
(17)
Therefore, we can construct conditional and marginal Gaussian probability path by
/ ps (T, [zz,
(a 2) =N (2 2 52,, 52,,0,1a). (e(Ea(21)
(18)
for schedulers a., o. such that ōg = 0, = ā, = o, o > 0. These conditions ensure that the marginal probability path interpolates noise and the GLASS transition:
= 0: (x) =V Ta, Ta), ==1:
pi (x)) = Prjo(X 5. (z+))
A natural choice of schedulers are ones such that (ā. [r. z) is the optimal transport path (CondOT schedulers (Lipman et 2022)), i.e. ā₂ sā, 0, (1-s)0g + sō. We present the following result (see appendix A.4 for proof):
Theorem 1. Let us be given two times t < t', starting point 2t. and correlation parameter
defining the GLAASS transition Pric eq. 13)) Then we can sample from Pr p (-12x) follows:
Define the GLASS velocity field as the weighted sum of F., and the GLASS denoiser
u. (,, 1) ( + w2(6) ((())) ((e) (Iz, F.) ) + wa wa)x))
(19)
with weight coefficients w1 w1(s), (s), w2(s), ws((s) E R and time-dependent mean scale and covari- ance pls), E(s) given by
0xs + j(z) ) E(s) -0= o 0² +
o² of?
u(s)
at
(20)
0,0.
w/ (s)
w2(s) 0,ā a, (s),
Wz(s) -w (s)
(21)
0x
where a., T. are chosen as in eq. via the ODE
18) Then the final point X2 of the trajectory X, obtained
Xu ~N(T2, Ia.
dx, u, (X,
(22)
ds
is sample from the GLASS transition, ie. X1 (r))(-(ze). More generally, X, p.(1) for all 0 < s I 1.
This theorem shows that any flow matching or diffusion model contains an "inner" flow matching model u. (i,[2,, |x,, t) that allows to sample GLASS transitions. By proposition] 2 no further training is required. As this result relies the idea of using the sufficient statistic of Gaussian measurements to infer a latent 2, we coin these flows Gaussian Latent Sufficient Statistic (GLASS) Flows. In algorithm [ rameterizations and x 2 matrix inversions are negligible compared to neural network evaluations. Therefore, the complexity algorithm pre-trained velocity field uz (I)),, i.c. the number of simulation steps M.
we describe pseudocode sample a transition with GLASS Flows. Note that the repa-
[ governed by the number of function evaluations of the
Sampling with GLASS Flows. To generate data point X1 Pdata. we set the number K of transitions and transition times ( S to < t1 <... < tx and initialize X,- X ~ ~N(0, Iz). For every A 0....
we sample the transition from X2, X1242 using algorithm T (set t= ta and t' = 1) with a choice of a correlation parameter that we can choose freely (note that can also vary across transitions). Assuming no discretization error and perfect training, X,, distributed according to the probability path, also the next step will have marginals specified by the probability path by theorem !
X1. Phu
Xtats Ptets

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In particular, it will hold that Xix X1 P1 Pdata- i.. the endpoint is valid sample from the desired distribution. This preservation of marginals holds for any p (not limited to the one corresponding DDPM transitions). Therefore, GLASS is novel sampling scheme resulting in Markov chains preserving the marginals of a pre-trained flow or diffusion models. The total number of function evaluations is K M. For K = only one transition, we recover standard flow matching as the conditioning for to = 0 is simply ignored. Further, for M = L, one simulation step for the inner transition, we recover DDIM sampling (Song al. 2020a) (see appendix B.2 for derivation).
Implementation. For numerical stability, we need to account for the cases when s 0 in al- gorithm ! We derive this edge case in appendix [.3] In appendix [B.]] we also discuss other techniques to make the implementation numerically stable. Further, all current large-scale flow matching models use classifier-free guidance (CFG) (Ho & Salimans] 2022) to condition on a prompt c. To use CPG with GLASS Flows, we treat the classifier-free guidance vector field uy (x(cc (1 (xlc)) (z) ie. all calculations are done with this vector field. Finally, it is well-known that there are many equivalent parameterizations of the vector field ue (e.g. via the score function or directly via the denoiser) and also diffusion models in discrete time. We discuss in appendix Ba.nddxxx.4JJoo [B.4]how to construct GLASS Flows with these alternative parameterizations. Further, we provide minimal implemen- tation of algorithm att githh.......... wttt://toooeeeeeeeeeeeee derrieth/eeeeeeee gl glass_ssfff.... wasssslflowl ows_tutor
wuy(x) (x) as the ground truth vector field for the same weight w 2 O,
4.3 INFERENCE-TIME REWARD ALIGNMENT WITH GLASS
Finally, we briefly explain how GLASS Flows can be applied to inference-time reward alignment, focusing on the algorithms discussed in section
Sequential Monte Carlo: we use GLASS Flows to evolve the particles with the proposal distribu-
tion Prij (see eq. (7) replacing SDE sampling (Singhal et al. (025)) with GLASS Flows.
Value function estimation: we estimate the value function V. as used in search methods (see proposition D. via samples from the posterior Pije replacing SDE sampling with GLASS Flows.
we can adjust the GLASS velocity field, analogous to eq. (10). apply GLASS Flows with reward guidance. Specifically, we add an appropriately scaled gradient of an analogous value function derived in appendix B.1
Reward guidance:
RELATED WORK
We discuss the most closely related work in this section and refer to appendix C for an extended discussion of other related methods. GLASS Flows operate in discrete-time, leveraging an under- lying continuous-timg model. Discrete-time diffusion (Sohl-Dickstein et al.) 2015) appeared prior to continuous-time diffusion, but such models are parameterized as 1st order approximations of the same ODE/SDE and are not qualitatively different. GLASS Flows instead consider sampling from latent Gaussian transitions for arbitrarily distant times. Recently, discrete-
transitions in FM models were also studied in Transition Matching (Shaul et al., 2025). In fact, the DTM supervi- GLASS transition with = (see appendix for detailed discussion). However, note that Transition Match- ing (Shaul et al. 2025) modifies pre-training and network architectures via patch approximations to sample transitions via flows, while our method focuses on inference-time modification post-training. Therefore, GLASS Flows and TM address different problems and lead different models that are theoretically related, yet practically different.
sion process in Transition Matching (see (Shaul et al.) 2025] equ. (1O)) corresponds
inference-time reward alignment methods are reviewed in (Uehara et al. 2025 They can be cate- gorized into single particle (i.e. guidance) and more general multi particle methods (i.e. Sequential Monte Carlo (SMC) and search). Guidance such as (Chung et al. 2022 Song et al. 2023b Ye al, 2024 Yu et al., 2023; Bansal et al., 2023 He et al. [023]]Song et al. 2023a Feng et al. (025) aims to approximate the difference between an existing velocity and an optimal velocity trained with a re-
ward. In addition to guidance, source-based methods specific to flows keep the velocity fixed while
altering the input noise distribution (Ben- Hamu et al. 2024| Eyring et al.] | 2024: Wang et al. 2025).
Multi- malti-particle methods evolye population of particles such SMC (Singhal et al. 2025
Skreta
et al. 2025
Wu et al. 2023a Mark et al. 2025
He et al. 2025) and search (Li et al. 20256
Zhang
al.
2025
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Recently, Chen et al. (2025b) propose Training-free Augmented Dynamics (TADA) introducing training-free free improvements of diffusion models using a similar mathematical principle as in this work. Specifically, they show that several recently proposed diffusion models with augmented state spaces (Dockhorn et al., [021] Chen et al. 2023) can be recovered from a pre-trained FM or diffusion model using Gaussian conjugacy/sufficien statistic argument (Chen et al. 2025b Proposition 3.1). Further, this principle can be extended to state spaces augmented with more than 2 variables. This allows them accelerate sampling significantly. While we design a different method designed to get fast stochastic transition samplers for reward alignment, both (Chen et al. (025b) and this work use the same mathematical principles to derive their respective algorithms. We discuss this in more detail in appendix C
Finally, reward fine-tuning methods based on GRPO (Xue et al. 2025 Li et al. 2025a Liu et al.). stochastic optimal control (L.iu et al. Comingo--Enrice et al.) (024), DPO (Wallace et al.. (022) or other reinforcement learning approaches aim to achieve the same goal as this work, i.e. to align a diffusion model with reward function r. GLASS has 2 different important synergies with these models: (I) Many of these algorithms require the use of DDPM/SDE sampling for exploration during training (Liu et al. 2025 Xue et al., 2025 Li et al. 2025a Domingo-Enrich et al.) 2024). However, this is very inefficient as discussed in this work. Therefore, one could potentially ac- celerate reward fine-tuning methods via GLASS Flows by replacing the slow SDE sampling with GLASS Flows. (2) One can also apply inference-time scaling with GLASS Flows to fine-tuned models. Many of these models learn a FM model of reward-tilted distribution. Hence, GLASS Flows is equally valid to be applied to reward fine-tuned models and can be used to improve these models as well. Therefore, both approaches complement each other.
6 EXPERIMENTS
Examples Posterior Sampling
Posterior Recovery
Value function estimation
100
1.0
Original
85-
GLASS 1-0.3
Comellatin
Nolsed
1-2
06AM--3
GLAG 1-6.1
DEAM wdd
GLAS 1=
Recovery (DOPA)
GLAS5
25-
1-32
Recovery [0ung)
0.
10 15
20
10
Number of simulation steps
Number of simulation steps M
Figure 2: Posterior sampling experiments. We noise images and then sample from the posterior P=j((-1z) via DDPM or GLASS Flows. Left:: Examples for t 0.2 and M 6 simulation steps. Middle: FID values for various simulation steps M and time t. Right: Estimation of the value function assessed by correlation with ground truth (200 Monte Carlo samples with M 200).
6.1 EFFICIENT POSTERIOR SAMPLING AND VALUE FUNCTION ESTIMATION
Posterior sampling. We begin by benchmarking the efficiency of sampling transitions with GLASS Flows (our method) vs SDEs (DDPM sampling). As an example transition of particular importance, use the posterior Pije of the probability path (see eq. (9) We use DIT/SIT models from (Peebles) & Xie 2023 Ma et al.] 2024), a competitive class-conditional flow matching model, trained on Ieage. Our experimental setup is follows: We sample data points from the ImageNet model (z ~ P.ata). noise them (x ~ p=())z), and then sample from the posterior via each respective method (z' (ije(-lz). For many simulation steps (M 200 in algorithm|
Dp. both GLASS Flows and DDPM sampling give high quality samples from the posterior (see app. fig. (O]. We then vary the number of simulation steps M to values ranging from M = 2 to M 50 and the time t. Note that the lower M and the lower the time f, the "harder" the task gets as we have more discretization error and have added more noise to the reference image. Figures] [] to 13 show that GLASS Flows return significantly higher quality samples for low M or f. To quantify this, we measure the image quality via Frechet Inception Distance (FID) using 50k images for both reference and each method (for each combination of t and M). GLASS Flows achieve significantly better FID than DDPM sampling for the same number of sampling steps (see fig. 2 Therefore, GLASS Flows represent a significant boost in efficiency when sampling from the posterior Pife-
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Value function estimation. Next, investigate whether better sampling from Pije also translates to better estimation of the value function V, (see eq. (8). As a reward model, we use ligeliiel of ResNet ImageNet classifier (He et al. 2016). We repeat the same experiment, i.c. noise image and sample from the posterior, but this time measure the correlation (or MSE) between the ground truth value function and estimators. The ground truth is measured by using M 200 simulation steps with the ODE/SDE and 200 samples. As one can see in fig. 2 GLASS Flows achieve signifi- cantly higher correlation for lower number of simulation steps. This demonstrates that the improved posterior via GLASS Flows translates to significantly better estimation of the value function.
6.2 NOVEL SAMPLING METHODS
ST
FLUX
0,70
DDPM
4.0
0.65
Centuul Spore
0
0.55
2.5
GLASS
0.30
0.45
1.0
0.40
DOPM
obt
GLASS (DD19)
GLASS (2
00PM
out
GLASS (DD19))
GLASS (p
Figure 3: Sampling from SiT/FLUX with various sampling methods. Left: Comparison with FLUX of images generated with DDPM vs. GLASS Flows. DDPM samples are more blurry and of lower quality. Middle: Results for SiT. Right: Results for FLUX. Prompts: "Carrots" and "Refrigerator".
We next investigate how GLASS Flows perform as a novel scheme to sample from flow matching and diffusion models. We use the DiT/SiT models and the FLUX model (Labs. 2024), a state-of- the-art text-to- text-to-image model generating high resolution images (size 768 x 1360). We use 50 neural network evaluations (default for FLUX model) for all methods. For GLASS Flows, we use equally spaced = 6 transition points. As one can see in fig. 3 ODE sampling vs. DDPM sampling have significant performance gap for the default FLUX parameters. However, GLASS Flows close this gap both for SiT on ImageNet (FID) and the GenEval benchmark on FLUX. In fact, GLASS Flows perform on par with ODE sampling, while having stochastic transitions. Therefore, these results demonstrate that GLASS Flows effectively remove the trade-off between efficiency and stochasticity for sampling.
In appendix D.3, we perform an ablation experiment over various correlation parameters p. Overall, we find that all values of P are numerically stable and lead to ODE-level performance with only minor differences. The choice of a constant correlation schedule of p 0.4 led to the best results for the FLUX model and we use this subsequent experiments. We note that the optimal choice of is dependent the data and model and may well differ for other settings.
6.3 SEQUENTIAL MONTE CARLO EXPERIMENTS
Next, we apply GLASS Flows to inference-time reward alignment via Sequential Monte Carlo (SMC), in particular Peynman-Kac Steering (FKS) (Singhal et al.] 2025 Skreta et al. 2025). We apply FKS on text-to-image generation using the FLUX model. We use different pre-trained model than (Singhal et al.) 2025) because FLUX is the current state-of-the-art model and our method re- quires a continuous-time model (the 1" map does not fall into a discrete set of grid points). Except that, we use the same hyperparameters as in (Singhal et al. 2025). Between resampling steps, we sample the transitions with either DDPM sampling like previous works (Singhal et al.) (025) or GLASS Flows. We also compare against Best-of-N baseline (BoN), i.e. where N images are sam- pled and the one with highest reward selected. To ensure that we do not overfit to single reward model, we run experiments for different reward models (CLIP (Hessel et al., (002I) Pick (Kirstain et al. 2023), HPSv2 (Wu et al. 2023b), ImageReward (Xu et al. 2023a)). Further, to avoid "reward hacking", we evaluate results also on GenEval (Ghosh et al. 2023). to measure whether we can effectively optimize reward without sacrificing GenEval performance. In table I summarize our results and we plot examples in fig. [4, The first observation we make is that FKS with vanilla SDE sampling does not outperform simple Best-of-N baseline the Best-of-N is sampled with
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Table 1: Sequential Monte Carlo via Feynman-Kae steering (FKS), Every reward model defines new experiment whose samples we evaluate on the same reward model and the GenEval benchmark. We set N 8 (number of particles). NFEs=400 for all rows except flow baseline (50 NFEs). BoN: Best-of-N. FKS: Feynman- Kac Steering.
Algorithm
CLIP
Pick
HPSv2
IR
CLIP
GenEv.
Pick
GenEv.
HPSv2
GenEv.
IR
GenEv.
Flow baseline
34.9
63.2
23.4
63.2
0.302
63.2
0.88
63.2
BoN-SDE
36.9
60.8
23.1
63.5
0.303
60.9
1.16
65.3
BoN-ODE
38.5
70.6
23.8
69.3
0.315
69.8
1.31
71.8
FKS-SDE (DDPM)
39.0
64.1
23.4
63.0
0.295
63.6
1.19
63.8
BoN-GLASS (DDPM)
38.6
70.8
23.8
71.5
0.316
68.8
1.33
71.8
BON-GLASS (p 0.4)
38.8
71.8
23.8
69.9
0.316
69.1
1.32
71.9
FKS-GLASS (DDPM)
39.7
70.5
24.1
68.8
0.317
68.3
1.37
68.2
FKS-GLASS (p 0.4)
39.8
72.6
24.1
72.2
0.318
70.3
1.40
74.3
ODEs, i.e. the performance gain by using ODEs compared to SDEs weighs more than using SMC vs. Best-of-N. However, replacing the SDE transitions with GLASS Flows (our method), we remove this trade-off. In fact, GLASS Flows combined with FKS leads to significant improvements for all 4 rewards models without sacrificing performance on GenEval. We repeat the experiment on the PartiPrompts benchmark Yu et al., 2022). Here, we optimize each reward model and eval- uate on all other models. shown in app. fig. 9 similarly FKS with GLASS Flows leads to significant improvements and also constitutes the best-performing method on PartiPrompts.
Combining GLASS-FKS with reward guidance. Fi- nally, we explore combining FKS-GLASS with reward guidance. We pick the best-performing reward from ta- ble ImageReward, and use it to compute the gradients in eq. (TO) (see appendix D.4 for details). Note that we decrease the resolution of the image to 672 x 672 as the high-resolution images generated by FLUX led mem- ory bottlenecks in the gradient computation. We present results in table 2 As one can see, guidance can improve results from FKS further increasing both the reward being optimized and the GenEval results. Finally, we note that we also explored reward guidance as stand- alone method, showing GLASS Flows led to an improved trade-off between reward optimization and image quality. As reward guidance is less commonly used to improve text-to-image -mage alignment as most improvements come from SMC (also in previous methods, e.g. (Singhal al. 2025)), present these results in appendix D.4
Table 2: Improving GLASS-FKS using gradient guidance. ImageReward (IR) and GenEval results. Note: benchmarks are slightly different table I as image resolution was decreased.
Algorithm
IR
GenEv.
Flow bascline
0.88
63.8
1.45
72.7
FKS-GLASS+V
1.52
73.1
CONCLUSION
We introduce GLASS Flows, a novel way of sampling Markov transitions in flow and diffusion models using an "inner" flow matching model. This inner flow matching model can be retrieved from existing pre-trained models using sufficient statistics. Here, we applied GLASS Flows to inference- time reward alignment: Traditional sampling procedures for flow matching and diffusion models are poorly suited for inference-time reward alignment, either having deterministic trajectories like ODEs or requiring many steps to accurately simulate like SDEs. By combining the efficiency of ODEs with the stochasticity of SDEs, GLASS Flows substantially improve prior algorithms for inference-time reward alignment that relied on SDE sampling. Hence, GLASS Flows serve as a simple, drop-in solution for inference-time scaling of flow and diffusion models. In the future, one could explore applying GLASS Flows to other methods relying on SDE sampling, e.g. some reward fine-tuning (Liu et al.] 2025 Xue et al., 2025 Li et al.) 2025a; Domingo-Enrien
al. 2024) image editing methods (Meng et al. 2021 Nie et al.] 2023). ). Further, one could explore learning or dynamically adjusting the correlation parameter p defining the GLASS transitions.
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REFERENCES
Abbas Abdolmaleki, Bilal Piot, Bobak Shahriari, Jost Tobias Springenberg, Tim Hertweck, Michael Bloesch, Rishabh Joshi, Thomas Lampe, Junhyuk Oh, Nicolas Heess, al. Learning from neg- ative feedback, or positive feedback or both. In The Thirteenth International Conference on Learning Representations.
Michael Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: uniffying framework for flows and diffusions. arXiv preprint arXiv:2303.0879) 2023.
Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Ap- plications, 12(3):313-326, 1982.
Arpit Bansal, Hong-Min Chu, Avi Schwarzschild, Soumyadip Sengupta, Micah Goldblum, Jonas Geiping, and Tom Goldstein. Universal guidance for diffusson models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 843-852, 2023.
Heli Ben-Hamu, Omri Puny, Itai Gat, Brian Karrer, Uriel Singer, and Yaron Lipman. D-flow: Differentiating through flows for controlled generation. arXiv preprint arXiv:2402.14011 2024.
George Casella and Roger Berger. Statistical inference. Chapman and Hall/CRC, 2024.
Hansheng Chen, Kai Zhang, Hao Tan, Zexiang Xu, Fujun Luan, Leonidas Guibas, Gordon Wet- zstein, and Sai Bi. Gaussian mixture flow matching models, 2025a. URL bttps://arxiv.
org/ abs/ 2504 05304
Tianrong Chen, Guan-Horng Liu, Molei Tao, and Evangelos Theodorou. Deep momentum multi- marginal schrödinger bridge. Advances Neural Information Processing Systems, 36:57058- 57086, 2023.
Tianrong Chen, Huangjie Zheng, David Berthelot, Jiatao Gu, Josh Susskind, and Shuangfei Zhai. Tada: Improved diffusion sampling with training-fres augmented dynamics. arXiv preprint arXiv:2500..1757 2025b.
Hyungjin Chung, Jeongsol Kim, Michael T Mceann, Mare Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. arXiv preprint arXiv:2209. 1468777709.1687, 2022.
Valentin De Bortoli, Alexandre Galashov, J Swaroop Guntupalli, Guangyao Zhou, Kevin Murphy, Arthur Gretton, and Arnaud Doucet. Distributional diffusion models with scoring rules. arXiv preprint arXiv:2502.02488. 2025.
Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-based generative modeling with critically- damped langevin diffusion. arXiv preprint 12.0022/ 2021.
Carles Homingo-Ennichh Michal Drozdzal, Brian Karrer, and Ricky TQ Chen. Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control. arXiv preprint 2rXiv:2409.0886) 2024.
Noam Elata, Bahjat Kawar, Tomer Michaeli, and Michael Elad. Nested diffusion processes for anytime image generation. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 5018-5027, 2024.
Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, Dustin Podell, Tim Dockhorn, Zion En- glish, Kyle Lacey, Alex Goodwin, Yannik Marek, and Robin Rombach. Scaling rectified flow transformers for high-resolution image synthesis, 2024. URL
1403.03200
Luca Eyring, Shyamgopal Karthik, Karsten Roth, Alexey Dosovitskiy, and Zeynep Akata. Reno: Enhancing one-step text-to-image models through reward-based noise optimization. Advances in Neural Information Processing Systems, 37:125487-125519, 2024.
Ruiqi Feng, Tailin Wu, Chenglei Yu, Wenhao Deng, and Peiyan Hu. On the guidance of flow matching. arXiv preprint 1500 2025.

---

Published as a conference paper at ICLR 2026
Ronald A Fisher. On the mathematical foundations of theoretical statistics. Philosophical transac- tions of the Royal Society of London. Series A, containing papers of a mathematical or physical character, 222(594-60443399-330 1922.
Stephanie Fu, Netanel Tamir, Shobhita Sundaram, Lucy Chai, Richard Zhang. Tali Dekel, and Phillip Isola. Dreamsim: Learning new dimensions of human visual similarity using synthetic data. arXiv preprint 2rx1w22306.09344 2023.
Dhruba Ghosh, Hannaneh Hajishirzi, and Ludwig Schmidt. Geneval: An object-focusee framework for evaluating text-to-image alignment. Advances in Neural Information Processing Systems, 36: 52132-52152, 2023.
Alexandros Graikos, Nikolay Malkin, Nebojsa Jojic, and Dimitris Samaras. Diffusion models as plug-and-play priors. Advances Neural Information Processing Systems, 35:14715-14728, 2022.
Jiajun He, José Miguel HemándezzLobatto Yuanqi Du, and Francisco Vargas. Rne: a plug-and. and- play framework for diffusion density estimation and inference-time control. arXiv preprint wrXiv:2506.05668,, 2025.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp- 770-778, 2016.
Yutong He, Naoki Murata, Chich-Hsin Lai, Yuhta Takida, Toshimitsu Uesaka, Dongjun Kim, Wei- Hsiang Liao, Yuki Mitsufuji, Zico Kolter, Ruslan Salakhutdinov, et al. Manifold preserving guided diffusion. arXiv preprint arXiv:2311. 1..31.1.644..... 2023.
Jack Hessel, Ari Holtzman, Maxwell Forbes, Ronan Le Bras, and Yejin Choi. Clipscore: A reference-free evaluation metric for image captioning. arXiv preprint arXiv:2104.08718 2021.
Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance, 2022. URL https://arxxi
org/ abs/ 220722272222 12598
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6855 2020.
Tero Karras, Miik Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. Advances in Neural Information Processing Systems, 15:26565-26577, 2022.
Yuval Kirstain, Adam Polyak, Uriel Singer. Shahbuland Matiana, Joe Penna, and Omer Levy. Pick- a-pic: An open dataset of user preferences for text-to-image generation. Advances in Neural Information Processing Systems, 36:36652-3666 2023.
Siddarth Krishnamoorthy, Satvik Mehul Mashkaria, and Aditya Grover. Diffusion models for black- box optimization. In International Conference on Machine Learning, pp. 17842-17857. PMLR, 2023.
Black Forest Labs. Flux. https:://itthi/ httt:s////tttthioooooo/oaasoss-s
2024.
Junzhe Li, Yutao Cui, Tao Huang, Yinping Ma, Chun Fan, Miles Yang, and Zhao Zhong. Mixgrpo: Unlocking flow-based grpo efficiency with mixed ode-sde. arXiv preprint arXiv.2507.21802, 2025a.
Xiner Li, Masatoshi Uehara, Xingyu Su, Gabriele Scalia, Tommaso Biancalani, Aviv Regev, Sergey Levine, and Shuiwang Ji. Dynamic search for inference-time alignment in diffusion models. arXiv preprint arXiv:2503. 020399 2025b.
Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint 20X00:22 2022.
Yaron Lipman, Marton Havasi, Peter Holderrieth, Neta Shaul, Matt Le, Brian Karrer, Ricky TQ Chen, David Lopez-Paz, Heli Ben-Hamu, and Itai Gat. Flow matching guide and code. arXiv preprint a 2xiv:24412.022 2024.
12

---

Published as a conference paper at ICLR 2026
Jie Liu, Gongye Liu, Jiajun Liang, Yangguang Li, Jiaheng Liu, Xintao Wang, Pengfei Wan, Di Zhang, and Wanli Ouyang. Flow-grpo: Training flow matching models via online rl, 2025. URL https://arxiv. org/abs/2505.05477
Jie Liu, Gongye Liu, Jiajun Liang, Yangguang Li, Jiaheng Liu, Xintao Wang, Pengfei Wan, Di Zhang, and Wanli Ouyang. Flow-grpo: Training flow matching models via online rl. arXiv preprint wrXiv:2505.054700 2025.
Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209. 2Et.::2209.30003,, 2022.
Nanye Ma, Mark Goldstein, Michael S Albergo, Nicholas M Boffi, Eric "anden-Eijndee and Sain- ing Xie. Sit: Exploring flow and diffusion-baset generative models with scalable interpolant transformers. In European Conference on Computer Vision, pp. 23-40. Springer, 2024.
Konstantin Mark, Leonard Galustian, Maximilian P-P Kovar, and Esther Heid. Feynman-kan kac--lowww Inference steering of conditional flow matching to an energy seeggyttilted posterior. arXiv preprint arXiv:2509.01543, 2025.
Chenlin Meng, Yutong He, Yang Song, Jiaming Song, Jiajun Wu, Jun-Yan Zhu, and Stefano Ermon. Sdedit: Guided image synthesis and editing with stochastic differential equations. arXiv preprint arXiv:2108.0 010773 2021.
Shen Nie. Hanzhong Allan Guo, Cheng Lu, Yuhao Zhou, Chenyu Zheng, and Chongxuan Li. The blessing of randomness: Sde beats ode in general diffusion-based image editing. arXiv preprint arXiv:2311.01410, 2023.
William Peebles and Saining Xie. Scalable diffusion models with transformers. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 4195-4205, 2023.
Ashwini Pokle, Matthew J Muckley, Ricky TQ Chen, and Brian Karrer. Training-fre free linear image inversion via flows. CoRR, 2023.
Neta Shaul, Juan Perez, Ricky TQ Chen, Ali Thabet, Albert Pumarola, and Yaron Lipman. Bespoke solvers for generative flow models. arXiv preprint arXiv:2310. 19075, 2023.
Neta Shaul, Uriel Singer, Itai Gat, and Yaron Lipman. Transition matching: Scalable and flexible generative modeling. arXiv preprini urXiv:2506.23560 2025.
Raghav Singhal, Zachary Horvitz, Ryan Teehan, Mengye Ren, Zhou Yu, Kathleen McKeown, and Rajesh Ranganath. À general framework for inference-time scaling and steering of diffusion models. arXiv preprint arXiv:2501.06848, 2025.
Marta Skreta, Tara Akhound-Sadege. Viktor Ohanesian, Roberto Bondesan, Alán Aspuru-Guzik, Arnaud Doucet, Rob Brekelmans, Alexander Tong, and Kirill Neklyudov. Feynman correc- tors in diffusion: Annealing, guidance, and product of experts. arXiv preprint arXiv:2503.02819, 2025.
Jascha Sohl-Dickster Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learn- ing. pp- 2256-2265. PMLR, 2015.
Jiaming Song. Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020a.
Jiaming Song, Arash Vahdat, Morteza Mardani, and Jan Kautz. Pseudoinverse-guided diffusion models for inverse problems. In International Conference on Learning Representations, 2023a.
Jiaming Song, Qinsheng Zhang, Hongxu Yin, Morteza Mardani, Ming-Yu Yu Liu, Jan Kautz, Yongxin Chen, and Arash Vahdat. Loss-guided diffusion models for plug-and-play controllable generation. In International Conference on Machine Learning, pp.
PMLR, 2023b.
Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:201 1. 134466, 2020b.
13

---

Published as a conference paper at ICLR 2026
Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- tional Conference on Learning Representations (ICLR), 2021.
Alessio Spagnoletti, Jean Prost, Andrés Almansa, Nicolas Papadakis, and Marcelo Pereyra. Latino-pro: Latent consistency inverse solver with prompt optimization. arXiv preprint arXiv:2503.12615: 2025.
Masatoshi Uehara, Yulai Zhao, Chenyu Wang, Xiner Li, Aviv Regev, Sergey Levine, and Tom- maso Biancalani. Inference-time aligmment diffusion models with reward-guided generation: Tutorial and review, 2025. htt.hhtps://arrr v.
09685
Bram Wallace, Meihua Dang. Rafael Rafailov, Linqi Zhou, Aaron Lou, Senthil Purushwalkam, Stefano Ermon, Caiming Xiong, Shafiq Joty, and Nikhil Naik. Diffusion model alignment using direct preference optimization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9228-8238, 2024.
Zifan Wang, Alice Harting, Matthieu Barreau, Michael M. Zavlanos, and Karl H. Johansson. Source- guided flow matching, 2025. URL wwt.:httt://axatte
wrg/aaaao /ltp////////../...//// 14807
Luhuan Wu, Brian Trippe. Christian Naesseth, David Blei, and John P Cunningham. Practical and asymptotically exact conditional sampling in diffusion models. Advances in Neural Information Processing Systems, 36:31372-31 2023a.
Xiaoshi Wu, Yiming Hao, Keqiang Sun, Yixiong Chen, Feng Zhu, Rui Zhao, and Hongsheng Li. Human preference score v2: A solid benchmark for evaluating human preferences of text-to- image synthesis. arXiv preprint vrXiv:2336609344 2023b.
Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, and Katherine Bouman. Princi- pled probabilistic imaging using diffusion models as plug-and-play priors. Advances in Neural Information Processing Systems, 37:1 (8389-188421 2024.
Jiazheng Xu, Xiao Liu, Yuchen Wu, Yuxuan Tong, Qinkai Li, Ming Ding, Jie Tang, and Yuxiao Dong. Imagereward: Learning and evaluating human preferences for text-to-image generation. Advances in Neural Information Processing Systems, 36: 36:15903-15935, 2023a.
Yilun Xu, Mingyang Deng, Xiang Cheng, Yonglong Tian, Ziming Liu, and Tommi Jaakkola. Restart sampling for improving generative processes. Advances in Neural Information Processing Sys- tems, 3366766066768888, 2023b.
Zeyue Xue, Jie Wu, Yu Gao, Fangyuan Kong, Lingting Zhu, Mengzhao Chen, Zhiheng Liu, Wei Liu, Qiushan Guo, Weilin Huang, et al. Dancegrpo: Unleashing grpo on visual generation. arXiv preprint arXiv:2505.072 0585000..88,,, 2025.
Haotian Ye, Haowei Lin, Jiaqi Han, Minkai Xu, Sheng Liu, Yitao Liang, Jianzhu Ma, James Zou, and Stefano Ermon. Tfg: Unified training-free guidance for diffusion models. Advances in Neural Information Processing Systems, 01-22370-22417 2024.
Po-Hung Yeh, Kuang-Huei Lee, and Jun-Cheng Chen. Training-free diffusion model alignment with sampling demons. arXiv preprint arXiv:2410.05760. 2024.
Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yinfei Yang. Bureu Karagol Ayan, al. Scaling autoregressive models for content- rich text-to-image generation. arXiv preprint arXiv:2206. 10789, 2(3):5, 2022.
Jiwen Yu, Yinhuai Wang, Chen Zhao, Bernard Ghanem, and Jian Zhang. Freedom: Training- energy-guided conditional diffusion model. In Proceedings of the IEEE/CVF International Con- 2023.
ference Computer Vision, pp.
Xiangeheng Zhang, Haowei Lin, Haotian Ye, James Zou, Jianzhu Ma, Yitao Liang, and Yilun Du. Inference-time scaling of diffusion models through classical search, 2025. URL https:// a arxi v. wwt.:or////as..//.o/ 23614
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A PROOFS
A.1 DETAILS ON FLOW MATCHING AND DIFFUSION BACKGROUND
Vector field and denoiser. We briefly present here the parameterizations between the vector field, denoiser, and score function. The conditional vector field uy (x/2) for Gaussian probability paths is
given by (see (Lipman al. 2022 2024
0t -x + (ås
at 0t
2
(23)
at.
Therefore, we know that
ux(
(24)
61 r + (àr - at 02
/
2p/n(2/=)d==
(25)
01
01
01 r + (àr
at à1
Di(z)
(26)
Rearranging this equation results in:
1
Dz(x) = =P/1(2//rdd=== â on0r2 (o, - Ōxxz)
(27)
Score function and probability flow ODE. For completeness, we re-derive here the known con- nection between the flow matching ODE and the probability flow ODE in the score-based diffusion literature (Song al. 2020b). We know that the score function is given by:
v log (<r1)
of
/
ar (x))
V logp:(1)
Vlog (olepaaeeepnl(tttabb=
Dz(x) _
v log (x))
Therefore, plugging this into the denoiser-vector field identity (see eq.
26) we get that
at ôt 0
(x(x)
ue
(ái
D((x)
(28)
-
x
+
ài
-5,00
V (z)
(29)
a4
12
+
v (z)
(30)
02
2
where vg = 2à /at - 2 as in eq. (5) Defining the forward drift function f(x, t)
6:1
and the forward diffusion coefficient as i/: 1/-1., then eq. (30) is the vector field of the probability flow ODE (Song et al. [20206] Equation (13)) of the forward diffusion process given by (note that the diffusion literature uses different time convention where t = 0 is Pdata and + corresponds noise):
Xo ~ Pdatt,
dx,= l)d + o,dW'
(31)
where W; is Brownian motion. Therefore, ODE sampling in eq. (4 is equivalent to the probability flow ODE for Gaussian probability paths (Song et al. 2020b Equation (13)) (up to differing time convention).
Time-reversal and DDPM. Further, it is well known the SDE in eq. (31) has a time-reversal given by the SDE with diffusion coefficient vi and vector field given by (Anderson 1982 Song et al.] 2020b)
(à at
+ vI log (1)))))))) dt ndW,
dx,
(32)
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Using eq. (30) convert this back to the following form:
dx,- (X,,)) 2 (oggoo(())))) dt v,dW;
(33)
This proves eq.
A.2 PROOF OF PROPOSITION
oed Proposition 1. For "y
get that: DDDM ( (x,) Pen((ee(X)), i.e. DDPM
transitions are a special case of GLASS transitions.
Proof. To show that it is sufficient to show that the joint distributions coincide
DDPM (X (x,) - Pepe (X,)
Prit
DDPM DDPM((,, (e) = Pt.t' (X,:
Pt.t'
In turn, enough to show that the distribution conditioned on z is the same
d
Pt,t Xe,
DDPM
Pt, (X,, P
II N ((X),
(34)
where we used eq.
12
For p
assumed, we obtain that
ay de
of on
at
E
ae Of
(35)
oze'
of ar
In turn, as the DDPM is the time-reversal of autoregressive forward process, it holds that for t < th: waddp (Ie, (,11))
Pte
ppur (xg/ær) )pr
waDp (xvlz)
where used that Per backwards time and it holds that
DDPM (xrx (ee,, 2) Pir
as the DDPM process is also Markov in DDPM parM ((w)
Pr (zel =N ar ov, Ib
DDPM
(re (xr) =N
at Te::
xv, (o?
ag o ?
Pre
Our
ay
DDPM is the transition kernel the forward noising where the second equation follows from the fact fact
process (see eq. (31), which is Gaussian Markov process discrete time (which is unique if we restrict have marginals given by Pt). Altemattvely,
also directly prove this by using that DDPM is the transition kernel of the forward process in eq. (31 and using the transition kernels of Pale Carmstein-Uhleneed processes, see c.g. Karras et al. 2022 equation (1)).
remains to work the mean and covariance of the joint distribution using classical rules for Gaussian distributions. Specifically, we can sample from DDPM DDDM(-122 (-12) by first sampling X₂ ~ Pe (zxlz) and then sampling
a
X,
X +
ooe, e~ N((,
Op
Therefore, holds that the conditional means are given by
0tt
E = 2] = arz, E[X, Z = 2] =
E[ 2 =
E
axr = 042
a+'
0ty'
and
ai Nor[XeXXyl22 = 2]+
a
Xol =
oi
o,
Cov(e,
Xy Z = =z]
ay
ay
2
Qi of ar
2
a
V = 2] =0
Var = =]
of
+ (0
o = oo
2
av
a
Therefore, that eq.
34) holds. This finishes the proof.
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A.3 PROOF OF PROPOSITION
We start by making a statement about summarizing 2 Gaussian measurements into measure- ment.
Lemma (Equivalent observations for multivariate Gaussian), Let R, let (#1, p22 E R² be mean vector and Σ E IR2x2 (positive definite) covariance matrix. Further, let X R be multivariate Gaussian random variable given by
X =
X2) (M(,, Σ)
Further, define the one-dimensional random variable Y via the mapping
,² x
Y S(X) where S(x)
(21, E IR
(36)
Then observing X at value X = x is equivalent to observing Y at Y = S(x), i.e. for any pior distribution Pdlata of = z holds that
p p
p = 5x)))
(37)
posterior with oiservation
X
posterior with observation
Equivalently, S(x) is sufficient statistic for X given z. Further, Y is again normally dis- tributed with
Y ~~N (z, 7
(38)
i.e. observing X1, X2 is equivalent to observing a single Gaussian measurement of z.
Proof. The fact that Y is again normally distributed follows from the fact that linear mappings of Gaussians are again Gaussian with mean and variances given by known formulas:
1
E|y] -T Elp El -
T
E
11 E
-0
4
-
1
V [Y]
(E' (u Eー'- 12
11a Σー ĐーンE
Further, it holds that
logp(X X = 2)
= - -2. E- 1-'--(X-- 1(X - -2-44
=C(X) ; =2p"² T - XT 1-1-
1. 22 T
2
=C(X) + 1ー11
1 1 (xS(X) + S((X)= - 22
2
=C(X) (z (2- -(X))
2(1TE-111)
C(() + logp(V = -= =)
where C(X) is an arbitrary constant independent of z. Therefore,
p exp(C(())(( S(X) 2)
Hence, we know that
x & PZZ)p x|2) x p(Z)p(Y S(X) & p(Z|Y = S(X))
where we dropped constants in Z. As both sides are distributions (i.e. integrate to 1), they must be equal. This finishes the proof.
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Proposition Let x = (21, x2) with x, E Rd and t* t" (1, = ((wEI p)) 1))
Then:
(x,
De (ar 5(x))
GLASS denoiser
"standard" pro-trained denoiser with reparameterized input and time
where De is defined in eq.
and ar is the scheduler in eq.
Proof. By lemma I we know that
D2,I(x)= = f 2p(Z = 2 2|X x)d2 = ff zp(2 2|Y S(x)) / zp(2 2/a, Y a,S(x))dd
for any $ 1. We know that arY given has distribution
a₂Y ~ ~N a12, E
a
Now the right-hand side, want to coincide with time point in the Gaussian probability path, Le. such that
of
N a12, 11'
(aazz
This is equivalent to
o²
9(()
E-1
Now, by assumption ar is strictly monotonically increasing and is strictly monotonically decreas-
ing. Therefore, the function is invertible and we can simply set 1" accordingly as stated in theorem. Then, get:
Du,z(x)= = 2p(2 2p(2= 2|Xr ar- S(x))d= D₁- (a. S(x))
A.4 PROOF OF THEOREM
Theorem 1. Let us be given two times < t', starting point 4r, and correlation parameter p defining the GLASS transition Prc in (13) Then can sample from Pe (-1a(((1;)) as follows:
Define the GLASS velocity field as the weighted sum T.. 2t and GLASS denoiser
u. u.( =w (s) + w2(s) (())))())))))))))))))))))))))))) (1) T.) wa(s)
(19)
with weight coefficients w/ (s), w2(3), (s)))) E R and time-dependent mean scale and covari- ance (uss), E(s) given by
m(s)=
i1s + 7(v ) E(8)
aq
o²
oft
(20)
0.7
72?
0.0.
w (s)
w2(s) =daā, a.Wi (s),
wa(s) = - (w/(( (s)
(21)
where o.,, J. 7 are chosen as in eq. via the ODE
18 Then the final point X, of the trajectory X, obtained
d _x us (X,lorr))
Xo ~o(22, oll)),
(22)
ds
a sample from the GLASS transition, ie.. X, ~ (p))(()o0)) More generally, X, ~ p.(.1x))
for all 0 < < 1.
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Proof. We can obtain samples X, from the probability path Po(X, N + yr,, o Ia) by
x, = Z + xX, + osc, c~ c~N(0, ~ Pdata
(39)
Therefore, the derivative with respect to is given by
O.X. -.ū.2 + 0,
(40)
Now, we can reparameterize into X, and X,
1
[X. aāz 7X1
02
Inserting this into eq. (40) we get
1 [X, - a,B - [X,] 0,X, -0, + 0,0, 0.
[s) (2) + w3y(s) (s)
where W1, W2, Wy are as in eq.
21
Taking the conditional expectation, get:
B 2₁, X, ø..
=wi (s) + {s X.] + wy(s)
=wi(s) + (22(2)) (8) D ((()) (2,, F.) + ws(s))
where Ug (Z, It, t) is defined as in the theorem. It remains to show that the left-hand side of the equation fulfills the continuity for the probability path ps ( l). Let f Ied - R be an arbitrary smooth function with compact support (test function). Then we have
/ f(x (E t)dē
-0. (1
Xx mi)
) = =x)
(IV Xs, Xx m.)]
(IV 1) .z+]
/ Vf(a))u, u, t),, (o)x,, t)dæ
= / f( (u.(F) t)ps (ā t))] dē
where we used partial integration in the last step. As f is an arbitrary test function, we obtain that both sides also coincide for each point:
O.p. (F () - (u. (Z/ze, ()ps (5) ())
This shows that the continuity equation is fulfilledd (see e.g. (Lipman et al. 2022 2024 ). This implies that the trajectory X, obtained via the ODE
Xo ~N(sn,, TT
ds X, u, (X,
(41)
has final point X, that is sample from the GLASS transition:
X, ~ p. (- xx)
(42)
for all 0 S 1. This finishes the proof.
n
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B ADDITIONAL GLASS DISCUSSION
B.I GLASS FLOWS AND REWARD GUIDANCE
We explain how (gradient) guidance aimed at sampling from the reward-tilted distribution as defined Section can simply applied to GLASS Flows. We first recall the construction for "standard" FM and diffusion models and then show how to translate it to guidance for GLASS Flows.
Guidance for "standard" FM models. Recall that to sample from the reward-tilted distribution ("() in our setting, the tilted vector field uf can be written in terms of marginal vector field ue( and the value function Vi(r))
(x) = ux(x) + e VV)()),
(43)
where Ct = à: - oz Equivalently, in the denoiser parameterization
(x) = D of VVi(t).
(44)
ay
In practice, Vz(r) and therefore (x) is often approximated via (Chung et al. 2022
Vi(x)) (rr(D(a)))
(45)
where r, as in eq. 10 and B, 2 is a hyperparameter (theoretically, 3r would be ideal, is common to tune this hyperparameter however). Therefore, the final approximated guidance vector given by
(T(x)) = (
Guidance for GLASS Flows. To derive guidance for GLASS Flows, we translate the same principles to GLASS Flows. For this, let (x)))()) be the denoiser for the reward-tilted distribution. Then we know that:
(()))),()) = Dj. (xx))
(46)
Further, using the same upproximation in eq. (45) and inserting into eq. (44). we get:
(1))),(1)) =D,- (ax- (x))
~D+ (a,- S(x)) + Br. of
V,
rD: (v)))1=a,,
at
= =D + 3,- of
v (D: (1) (v))
ag-
Finally, we can insert this identity into the formula for the tilted GLASS velocity field (see theo- rem
W[(Ta, 1)
(47)
-wi (s) (v)(s D((e) (xr, wa(s)a.
(48)
= =w + w (2 (s) (1) F.) + wy + Br of wz))s (Dr()())))))) -=a
(49)
=u, (E, se.f)
? (Z, t) + B,.
u2 uz (yr
(50)
ac.
where u, (7. Fe. t) is the GLASS velocity field for the (non-tilted) distribution Pdata- Theoretically. 3, = would be optimal for perfect estimation of the value function. However, because of the approximation eq. (45), we recommend tuning 3, 2 0 as done already for previous guidance
methods (Chung et al. 2022 He et al. 2023
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B.2 M = GLASS FLOWS
GLASS Flows generate data point using K transitions and M simulation steps per transition. For K 1, GLASS Flows are equal to standard flow matching integration of an ODE performed over M simulation steps. In this section, we instead consider when M 1. Let e ~ (0, Ia), where ãg q + oge. Then for one-step integration using the CondOT schedulers, get
As described in Section 4.2.2
ū = Zg + uo x2,
+ w1 (0) + wy (2), + wy (0)xz
.o + wn (0) + w2 (.) + (0)22)
Zo + m (0)5. + u2 (0) u2(()z;
00
-0000_
Zo +
To (z:)
= 5.t + ( .C.
(51)
Comparing with the conditional Gaussian probability path evaluated at 1, pi (Xi 21 (X = x/x,, 2), we note that GLASS Flows for M = 1 samples transitions via
Xy (x Dz(xt)).
(52)
This is identical the Gaussian transition kernel parameterization typically used in discrete-time diffusion models. So at M = transitions from GLASS Flows match a discrete-time diffusion model parameterized this fashion with the same Gaussian kernel and denoiser.
GLASS Flows are exactly equal to denoising-diffusion (DDIM) (Song et al., 2020a) for particular GLASS parameters and the same pre-trained denoiser. We begin by noting that DDIM uses a model parameterization that inserts the denoiser for 2. Next, we demonstrate that the z-conditional transition kernels are equal for particular GLASS parame- ters. DDIM uses conditional parameter per transition from to t', of,r, and marginal parameters op, where < of < and S (aP,)) < ap, and superscript denotes DDIM. From Equation 16 an arbitrary GLASS transition kernel can be written
In fact, M
diffusion implicit models
oe (m- 012), - (1-b)).) P),))) Pepp(Fe 2) Na 2 + Pv,
(53)
ot
where 0 s př., 1 and p's explicit dependence on é and t' is included for clarity. Now set
0 = ap
0 = - -at
(o
Pt.
(54)
1- a
where we note that 0 S p < is satisfied due to the constraint on o recover the DDIM transition kernel (see Eq. in (ong et al. (2020a).
Inserting and rearranging,
Vafz
DDIM Peit
(xel0), = afiz V of - -(o
V1 of (oP.))1).
(55)
B.3 NUMERICAL STABILITY
In the following, we show that is simple to ensure numerical stability in algorithm Generally, recommend performing all operations in algorithm 1 except the neural network evaluation in higher precision (float64). This has minimal overhead compared to large-scale neural network evaluations and minimize errors from reparameterization. We now discuss more specific steps.
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0 edge case. For 0, it holds that
Xo= = 7.X, + Tge
(56)
One could think of this 2-step Markov chain, i.c. it holds p(Xo]X,,,)) 2) p(Xo(_,,). It is
classical property of Markov chains that the posterior then also depends only on the state Xy:
p( Xo) - p(=(X;)
One can prove this directly by applying Bayes' rule twice and using the Markov property:
p(z|X,, Xo) (x X=)=)p(=)
(57)
- (Ko (x)())))
(58)
- xXXx
(59)
xp((x(2)p(z) xp((|Xi)
(60)
(61)
As the first and last term both integrate to I (as they are probability distributions over 2), they must be equal. As the posteriors are the same, also the denoisers are the same:
D z()) (X, Xo) Dx,),)
In algorithm 1
we use this fact to ensure numerical stability at 0.
Numerical stability of matrix inversion for E(s) and weight coefficients. given as:
The covariance matrix
o²
o7 82 + 7 0
of O.O
{(s)
deaip 0² p'o
(62))
o.7
where we inserted pov pov/o /or. Then
det (s)) (o, + p2o0t0o - oioo
Therefore, det [s)) > and Ess) invertible whenever of > 0 and o² 0. We now discuss is equivalent
when this might not be the case. First, of decreasing with 01 = 0 by assumption. Hence, ae > O always holds practice as we never take a transition starting the final time t 1. However, it is important that the operation would not be well-defined in this case. Second, o. is also positive for < by assumption and it fulfills at s +1, it would hold that ō 0. Hence, for L' = or = 1,, the matrix E(s) would not be invertible. However, in algorithm I we always simulate and take velocities for < Therefore, everything is well-defined and we observed that the inversion of (s) did not constitute a numerical problem even for +1. In fact, even for / +1, the samples we obtain are of high quality (see experiments in appendix D.3). Further, one can add a small value to the diagonal matrix to make it invertible: E(s) <- Σ(s) + + for < >> 0 to account for s close to 1. Similarly, the weight coefficients for the GLASS velocity field are given by:
< 1 as is strictly monotonically
that oं = ot (1 g²). Therefore, for either /' = 1
0,0. W1 (8)
w2()) = 0,0, - CrgWi (8), wa((s) = - (s)
(63)
for 7- por We sample transitions for t' S Therefore, < 1 and also de 0 and therefore 7 is well-defined. Further, for well-defined as T. = However, as before, algorithm only uses time steps S 1 1---2 1/M and therefore we did not encounter any numerical instabilities. As mentioned above, recommend performing all operations in algorithm - except the neural network evaluation higher precision (float64). This will have negligible overhead compared to neural network calls.
and p +1 or l' 1, it holds that wi (8) is not
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B.4 OTHER DIFFUSION PARAMETERIZATIONN,
Other vector field parameterizations. In algorithm 1 we assume that the pre-trained flow or diffusion model is given in the velocity parameterization (x(x) used in flow matching. it well- known that diffusion models can be equivalently parameterized via the score function V log or the denoiser (x or the noise predictor (also called e-predictor):
(x = [[xo = x1]
To use model trained with different parameterization for GLASS Flows, we simply reparameter- ize them into the denoiser parameterization (x).). In algorithm [1 we have presented this for the reparameterization of u, into D, for other models one might simply have to use the corresponding
reparameterization See e.g. (Lipman al. 2024 Table 1) for reparameterization formulas.
Discrete-time parameterizations. Our derivations assume that we have model (x) trained in continuous time 0 I t S In contrast, discrete-time diffusion models (Sohl-Dickstein al. 2015 Ho et al., 2020) are trained with a different time reparameterization, We discuss how to use GLASS Flows for these models. Let us assume that the discrete-time diffusion models is given in the shape of a denoiser model D& (x) with discrete time steps k = 1,... .N and discrete-time schedulers āg. kk We can map these discrete-time k -,... .N into a grid {ty]y=1,.. ,N of continuous IN via
time points t 82
=
where g(t) = of / as before for continuous-tinn timm schedulers Ctt St. Further, define the denoiser model D, the grid points
Dig (x) = D2
Ö x
Note that (x) is a valid denoiser model for t E in the grid and schedulers 022 same as before. However, there is one important difference: querying (z) for & 9 would correspond to an invalid input to D2 or, at least, and out-of-domain query of the neural network. Naturally, we want to avoid such out-of-domain queries. Specifically, during simulation of the GLASS Flow for transition from t to t', the denoiser model is queried at times t" (M(s), E(s) given by
1" (m((),, E(t)) =9 ((n(+)E (b))())) -
p(s) = Cis you,
at
Ess) = o + Po?
It holds that to (a()), Σ(0)) t (see appendix B.3) and therefore in particular restrict transitions
to only appear from grid points to grid points, I.c. t, t' E G. To choose "inner" grid points 0 = 8g < restrict ourselves to the set T {s [0, 1] (u(s), E(s) G), iee choosing 8; E T. In general, there is no closed-form for T as we allow for general sched- ulers Qt, d., Gg. A simple numerical approach to (approximately) obtain T is always valid. If os, G. G. have simple analytical formulas, we might also obtain closed form for T. Therefore, GLASS Flows can be applied discrete-time diffusion models in the same way and all of our re- sults equally hold with the only difference that the time points in algorithm [T are constrained to E T. Note that the above procedure is not an approximation but is exact: Simulating GLASS Flows with a reparameterized discrete-time diffusion model or a continuous-time diffusion model leads to identical results as long as time steps 8 E TT (assuming both models have no training error). Of course, there is still an error in the discretization of the simulation of the ODE which would be identical for both however. Therefore, GLASS Flows can also be applied to discrete-time diffusion models in the same way with constrained set of valid inner grid points in algorithm
81 <.... < SM 1,
B.5 ROLE OF
We briefly discuss the role of and how it determines the stochastic nature of the GLASS transition perje (e (z:). Now that the GLASS transition can be written as:
f (z,,,))) 2)
Peje(Iz (z,) f
P1](2(11)) dz
GLASS (dopends on #) posterior (indop. of p)
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Note note that for p the conditional transition Peje(re|az,. 2) becomes deterministic. How- ever, (1]r(= is independent of and is not deterministic but rather a proper posterior distribution. Therefore, even for / 1, the GLASS transitions are stochastic (in particular, P does not correspond to ODE sampling). Therefore, P determines Peij (az, z), in particular how correlated that we add to a single data point but only partially determines the probabiiitiic/soohastii transi- tion. We found that the time difference t' - (determined by the number of transitions K) and the variance of the posterior Pije are equally important factors determining the variance the transition distribution (ejo(eela
B.6 ANALYTICAL FORMULAS FOR g=
We derive specific formulas for g. 9.9- for various schedulers at, 0z. The inversion g is used in algorithm T Recall the definition:
g
For choices of oe ot. can derive 9 analytically, We present the derivation for the most com- mon schedulers.
Linear schedule.
Let us set
04 = I 0==1-
Then:
g(t) (1 t)2
2
- -1)
t2
(v) + Vu
Variance schedule.
Let set:
a: = v1- 1 1,
at = Vi
Then:
g(t) = 1 1
t
9 (y) =
1 +
Variance-explodingg schedule.
Let us set:
Vt, at
Then:
g(t)
(w) y
C EXTENDED RELATED WORK
Other applications. A wide range of reward guidance methods have been proposed (Chung et al. 2022 Abdolmaleki et al., Ye et al.) 2024, Yu et al.] 2023 Bansal et al. 2023 He al.) 2023 Graikos et al.) 2022), particularly focused on solving inverse problems such as Gaussian deblurring or inpainting. Many of the methods can seen as various approximations of the posterior Pitt- Here, we give new way of sampling from Piμt, potentially also opening new possibilities for these type of problems. However, note that the text-to-imate setting we consider in this work comes with various challenges and constraints that are different than many of the settings these works consider: The reward models are neural networks themselves, i.c. we cannot query them out-of-die out-of- -fdiiitributtin, and their gradients might not be informative. Further, the reward models are highly non-convex.
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Other approaches to reward alignment. We briefly discuss other methods. The LATINO sam- pler (Spagnoletti et al., 2025) devises a scheme that iteratively noises a data point and then maps it back to a clean data point with a one-step sampler. The DEMON method (Ych et al. (024) uses stochastic SDE based sampling of diffusion models and considers the noise that is added as part of the SDE in search space, i.e. they find the optimal noise to be added to an SDE. We could apply GLASS Flows also to this setting. Wu et al. (2024) introduce an auxiliary variable that is effectively noisy version of a clean image and then sample both jointy via Gibbs sampler. KKrihnamoorthy et al. (2023) train a classifier-free guidance model where the conditioning variable y is the reward or objective value. By setting that reward to be high, they then approximately sample from the distribution of high values.
Other posterior approximations and related sampling methods. Previous works have explored sampling from the posterior Pult. i.e. instead of just learning the mean via the denoiser actually sample from the posterior (De Bortoli et al.) 2025] Elata et al.) 2024] [Chen et al.] 2025a). For example, De Bortoli et al. (2025) explore learning the posterior posterior P1 is via a generative model trained via scoring rules alwinnng it to sample discrete-time transitions. In our experiments, we have also explored first sampling 2 P1te via GLASS Posterior Flows and then taking a conditional transition (ate 2). However, we found it to lead to significantly worse performance than going "directly" to Ty by sampling a GLASS transition (note that our setting is different as we reparameterize an existing model instead of training
one as in (De Bortoli et al. 2025 Shaul ct al, (025 Similarly, Gaussian mixture flow matching (Chen et al. (225a) approximates the postenoor Pilz via a Gaussian mixture. This induces more stochasticity into flow model and is shown to lead to performance improvements. Further, our method shares ideas from restart sampling Xu et al. (2023b), in that variable is noised and then denoised via flow again. However, our method is different in that the new noisy variable F. still depends on Ie up to = 1. Further, our method does not really go back in time (i.e. does not restart, the time t* where the neural network is queried can monotonically increase). Finally, our method is not approximate but theoretically exact.
Scheduler changes for Gaussian probability paths. Proposition does not require the desired scheduler to match the praitrdined denoiser's s scheduler. Scheduler changes have been studied pre- viously, where reparameterization recovers the denoiser in this more limited setting of a single mea- surement (Lipman et al.) 2024 Karras et al. 2022 Shaul et al., 2023 Pokle et al. 2023). GLASS Flows substantively extends this to multiple correlated Gaussian measurements.
Detailed discussion of Transition Matching (TM) (Shaul et al., 2025). TM is a general pre- training framework for learning inner flow matching models. Due to changes in the neural network architecture and the focus on pre-training, TM and GLASS Flows are different, complementary methods. However, the theoretically optimal transitions are indeed closely related as we explain here in more detail. Specifically, we discuss here the DTM supervision process (Shaul et al. (025). We write it here via a continuous time variable 0 < t < 1 using the convention of our work. In TM, the intermediate times are sampled via the CondOT probability path
X₁ = t- X1 + (1-11 t) X1 = 2 ~ Pdata: Xo ~ NO Ia)
The training target is Y = X1 Xo. i.e. TM learns a flow matching model to sampling from the conditional distribution of Y Xx x4. Note that for fixed X, it holds that
It tX1,
- It. Y= X,
X₂
1-t
1-t
Hence, sampling Y or sampling X1 is equivalent- - one can transform each variable into one another. Further, let us define the probability path for the TM model conditioned on 24:
X, - Y + U.E
6 6~ ~~N10, La)
à X1 Ft +.CC e~ IO
1
σ St.
Xi 5.E
ā.
X,
St
/.
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where Cgg 0. schedulers for the inner TM model, i.c. Shaul al. (025 choose d, and
o, = 1 8. The variable X, is distributed according to the GLASS probability path in eq. (18)
chose t' and a₂ 0,/(1 t) and 0₂ O3:
ps (T (,,
where used that 0 in this case. Therefore, if we simulate GLASS trajectory with these parameters (t' = 1, 0g = os 0₂ (1 t)), we obtain that the transformed GLASS trajectory
X, X,
a trajectory obtained from TM/DTM. This elucidates the connection between DTM TM and GLASS Flows.
Detailed discussion of TADA (Chen et al. 2025b). We provide an extended discussion of how the TADA method relates to this work. Specifically, we rewrite simplified argument from (Chen et al. 2025b) in the notation of this work to showcase the connection. Specifically, we focus here on simple case of
2, i.e. augmenting the data space = E IRd with a single variable p E R and where the terminal point x1 = (2, P1) in the state space is sampled independently. i.e. 2 ~ Pdata and p ~ (O, Ia) (for the argument full generality, we refer to (Chen al. 2025b)). The probability
path in this case is given by
(14 T2)3 + (Ie Ta)e, (= 61, (2, NVO,
The core realization connecting it our work is now that that X; = (Ii, pt) consists of two com- ponents 22z and pt that are both themselves noisy (correlated) Gaussian measurements of z. This follows simply from the fact p1 is Gaussian and of Gaussian are again Gaussian. Hence, we are in a similar setting as in section 4.2. In fact, Chen ct al. (2025b) applied similar argument to recover the denoiser from pre-trained flow matching or diffusion model as in section 4.2.1 (see (Chen et al. 2025b] Proposition 3.1)). This showcases the close connection of the mathematical principles enabling (Chen et al.) 2025b) and this work.
(c1, 62) is Gaussian and linear transformations
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D EXPERIMENTS
In this section, we provide details for experiments and present further experimental results.
D.1 SAMPLING FROM THE POSTERIOR AND VALUE FUNCTION ESTIMATION (SECTION 6.
10000
0:
0
0-03 5
03
t=0
60
40
20-
30
Figure 4: Detailed results for fig. 2 (Middle). Comparing the performance of sampling the posterior Pije via GLASS Flows (Ours) and SDE (DDPM) sampling. Ablate over different times t and sam- pling steps. GLASS Flows achieve significantly lower FID for lower number of sampling steps than DDPM sampling.
GLASS Flows (Ours)
SDE (DDPM)
640
-1.0
X/
8:20
<75
0/C
644
6:08
3:4
F11
8:35
624
0.04
67
8.3
621
6/2
3/5
1.0
654
67
644
664
301
113
G/C
6/4
Correlation
04
5.39
651
04
ERT
JUS
8:21
032
6:2
3.44
22
6.2
671
50
60
22
M sampling steps
M sampling steps
Figure 5: Detailed results for fig. (Right). Comparing the performance of estimating the value function Vz(z) via sampling the posterior P1tt via GLASS Flows (Ours) and SDE (DDPM) sampling via correlation. Experiment performed for different times t and sampling steps M. GLASS Flows achieve significantly higher correlation for lower number of steps than DDPM sampling. Ground truth is measured via 200 samples with 200 simulation steps of ODE/SDE.
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D.2 SAMPLING RESULTS
Table 3: GenEval results
Algorithm
Overall
object
objects
colors
color attr.
position
counting
ODE Sampling
0.6327
0,9843
0,8005
0.7154
0,4313
0.1900
0.6719
SDE (DDPM)
0.4435
0.6938
0.4596
0.5186
0.2720
0.1200
0.5969
GLASS (DDPM)
0.6357
0,9812
0.8030
0.7074
0,4746
0.1475
0.7031
GLASS (p- 0.4)
0.6304
0.9844
0.7904
0.7101
0.4449
0.1400
0.7125
Table 4: Sampling evaluation for SiT and FLux models using various sampling algorithms intro- duced in this work. We use 50 total neural network evaluations for all experiments and transitions (i.e. 10 simulation steps for each transition).
Algorithm
SiT
Flux
FID
CLIP
Pick
HPSv2
IR
ODE Sampling
2.34
33.82
22.80
0.291
1.060
SDE (DDPM)
4.36
33.70
22.55
0.287
1.017
GLASS (DDPM)
2.58
33.81
22.73
0.273
1.079
GLASS (p- = 0.4)
2.54
33.90
22.72
0.293
1.049
Diverily Reau
03
dL clout4)
Figure 6: Evaluating diversity of samples from various sampling scheme. All 3 sampling ap- proachess (DDPM, ODE, GLASS) show very similar results for diversity. This is consistent with theory all 3 approaches should sample from the 100 total NFEs on GenEval prompts (we take more NFEs than in fig. reduce discretization error of the SDE). We take samples per prompt and measure the average DreamSim (Fu al.! (023) similarity between two samples (error bars equal average standard deviation of samples of prompt).
distribution. We evaluate the model with
D.3 ABLATION OVER CORRELATION PARAMETER
We perform further experiments in this section ablating the correlation parameter p. We choose two strategies for ablation: First, constant correlation parameter p chosen across all transitions from t + independent of f, l'. Second, we choose p as time-varying based on the DDPM schedule (see proposition
p
and we ablate over the parameter x 2 0. use the FLUX model and measure GenEval per- formance for prompt adherence/generaaion quality and DreamSim diversity (Fu et al. 2023) measure diversity. We present results in fig. 7 The most striking results is that GLASS Flows achieves ODE-level performance for almost all correlation schedules and differences of perfor- mance are relatively minor. A constant correlation schedule of p = 0.4 performs best in GenEval performance and has relatively high sample diversity. Therefore, choose p 0.4 in subsequent
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experiments. For the ablation K, DDPM (k 1) performs very high (just diversity is slightly higher for A = 2). Due to the wide use of DDPM and its theoretical importance as a time-reversal, also use DDPM in subsequent experiments.
Correlation Schedule
Exponentiated Correlation
Divenily
0.734
Preaetti Gurbball Score
Donbual
[Rhb]
30
35
2.5
2.5
4.0
Figure 7: Ablation experiment of correlation schedule p. Left:: Constant correlation p across all
transitions. Right: Time-dependent correlation schedule given by p di0yy note that corresponds to the DDPM schedule (see proposition I
D.4 REWARD GUIDANCE
In this section, we explore using reward guidance with GLASS Flows to improve text-to-image alignment and also discuss the challenges in assessing reward guidance in the context of text-to- image generation. We note that reward guidance has so far been mainly used for inverse problems (Chung et al. 2022 He et al. (023) and only few works explore it to improve text-to-image align- ment for scale models (Singhal et al. 2025 Eyring et al.) 2024), the application we focus on in this work. As an intermediate reward, we use the common model of rx(x)) r (V))(DD(z)) where VAE is decoder of latent diffusion model. Computing Vr.(x) via backpropagation led to out-of-memory errors (mixed precision on A100 80GB memory GPUs). To remedy this, we make two modifications: First, we use lower resolution images (size 676 676 instead of size 768 x 1360) to remove a memory bottleneck at the output the VAE. Second, we detach the denoiser Dz(x) from the computation graph and compute the gradient at that point. This is common technique in the context of linear inverse problems (He et al., 2023). This allows us to compute a gradient at every step with reasonable computational overhead (as the velocity field u, model is significantly bigger than the VAE or the reward model). Further, we then set the guidance strength c2 in eq. (TO) as a v 22 for a hyperparameter stability, i.c. we only apply guidance in the interval [0.2, In table we present the results for 0.4. However, the guidance strength À has varying effects on different methods. Therefore, we vary the guidance strength with ImageReward and plot effects on GenEval performance in fig, B As one can see, all methods can increase the RewgeReward value arbitrarily with artifacts appearing for high guidance strength. GLASS Flows achieves the highest performance GenEval for the same ImageReward values.
Further, we G 0 for 0.2 S 0.7 for numerical
D.5 FURTHER RESULTS FOR FEYNMAN-
In fig. 9 we plot results for Feynman-Ke Kac Steering with GLASS Flows the PartiPrompts bench- mark. This further confirms the results from section 6.3 that GLASS Flows improve the state-of- the-art performance.
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Table 5: Reward guidance results GenEval prompts. N 50 simulation steps. The best value in each column is bulded, and the second best is underlined. Reward guidance with GLASS Flows improves both GenEval score and the reward of interest, while flow guidance leads to decreased performance on GenEval.
Algorithm
CLIP
Pick
HPSv2
IR
CLIP
GenEv.
Pick
GenEx,
HPSv2
GenEv.
IR
GenEv.
Flow baseline
34.9
63.8
23.4
63.8
0.302
63.8
0.884
63.8
SDE baseline
34.7
57.0
22.9
57.0
0.280
57.0
0.621
57.0
Flow guidance
37.3
63.0
24.3
63.4
0.320
63.0
1.387
62.7
SDE guidance
36.9
60.1
23.5
61.2
0.294
60.9
1.267
61.3
GLASS guidance
36.6
63.4
23.9
63.6
0.314
63.9
1.315
64.7
artiiccts
0.675
ODE
GLASS
Gentuel Duerell D200e
0.625
0.600
0.550
0.6
3.4
1.6
1.8
2.0
Imagsileward
Figure 8: Varying reward guidance strength across different methods GenEval benchmark with reward ImageReward. By increasing the guidance strength, we can increase ImageReward. GLASS Flows has higher performance on GenEval performance for the same ImageReward value. High guidance strengths lead to image artifacts that are not properly captured by our metrics.
Average
Normalized Reward
3.0
2.5
1.5
0.4
0.2
0.5
0.0
PKS-SLASS (DOPM)
PK' (rho=044
Best OCN (ODE)
FIS SDE (DOPW)
PKS-GLASS ASS (DOPM)
PICS GLASS (mho-0.4
Best N (ODE)
PISS (DOPM)
Dest of-t-o-NN (SDC)
Best-af-t
Figure 9: Inference-time reward alignment results on PartiPrompts benchmark. For each reward model (Clip, Pick, HPSv2, ImageReward), we run reward alignment with difference methods and evaluate across all reward models (i.e. this gives us 16 4 4 values). Left: We take the 16 values, rank the methods, and take the average rank. Right: We take the average normalized reward value (normalized via min and max observed).
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50E
GLASS
camp
GLASS sample
GLASS
Figure 10: Various samples from the posterior Pıte via GLASS Flows using M 200 simulation
steps. Both GLASS Flows and the SDE sample from the posterior given the noisy image.
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GLASS
506
GLASS
50E
GLASS
GLASS
Figure 11: Posterior recovery for 0.05 for various number of simulation steps M. As one can see, GLASS Flows achieve significantly better performance for low M than the SDE/DDPM sampling.
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CLASS
GLASS
506
GLASS
Figure 12: Posterior recovery for 0.15 for various number of simulation steps M. As one can see, GLASS Flows achieve significantly better performance for low M than the SDE/DDPM sampling.
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roforencs
nolay image
GLASS 2.500
55 25002
28:05
50 26op2
CLASS
22000
528 51099
GLASS
20002
50
20 20095
ELA
SDE -9:35
Figure 13: Posterior recovery for - 0.7 for various number of simulation steps M. As is close to
1, the umcertainty/variance of Pıje is very low. Hence, also with low number of simulation steps, reasonable performance is achieved regardless of the method (best compared with fig. fig. 12
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Flow
FKS-SDE
PKS-GL FKS-GLASS
rgt a sandafch
photo
photo offour stop
STEP
STOP
STOP
STOP
STOP
STOP
STOP
SYOF
STEP
STOP
STOP
SIYY
STOP
racket and bicycla
photo of yollow lork
photo a blacc ic
photo of siring
srowboord
photo of an orange
Figure 14: Examples for reward alignment Sequential Monte Carlo experiment on GenEval (sce section 6.3)
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Reviews:
[{"review_id": "WbhBttNr1j", "reviewer": "Reviewer_dnmw", "summary": "Sorry for my late review!\n\nThis paper addresses the inefficiency of inference-time reward adaptation for flow matching and diffusion models, which currently require sampling stochastic Markov transitions \\(p_{t'|t}(x_{t'} \\mid x_t)\\) via SDEs even though ODE sampling is far more efficient but deterministic, thereby inducing a trade-off between stochastic exploration and runtime. The authors propose Gaussian Latent Sufficient Statistic (GLASS) Flows, which construct an “inner” flow matching model that realizes a rich family of Gaussian Markov transitions (including DDPM transitions as a special case) as ODE trajectories in an auxiliary time \\(s\\), enabling ODE-based sampling of stochastic transitions directly from a pre-trained model. Technically, the method treats \\((x_t, x_{t'})\\) as two noisy measurements of a latent variable \\(z\\), compresses them into a sufficient statistic \\(S(x_t, x_{t'})\\), and shows that the standard marginal denoiser \\(D_t\\) of the base model can be reused—with an appropriate input and time reparameterization—to define a GLASS denoiser and the corresponding conditional velocity field \\(u_s(x_s \\mid x_t; t)\\) without any retraining. The paper proves that integrating this inner ODE from \\(s=0\\) to \\(s=1\\) samples from the desired GLASS transition in the continuous-time limit, analyzes discretization error, and argues that GLASS Flows inherit the computational efficiency and stability of ODE samplers while providing controlled stochastic evolution comparable to SDEs. Empirically, the authors first show on base text-to-image generation with SiT and FLUX that GLASS-based transitions match ODE sampling in FID and alignment metrics and substantially outperform DDPM-style SDE sampling at matched compute, thus eliminating the efficiency–stochasticity trade-off at the base model level. They then plug GLASS Flows into Sequential Monte Carlo Feynman–Kac steering for inference-time reward adaptation on FLUX, across four reward models (CLIP, PickScore, HPSv2, ImageReward) and benchmarks such as GenEval and PartiPrompts, demonstrating that FKS with standard SDE transitions fails to beat a simple Best-of-\\(N\\) ODE baseline whereas FKS–GLASS consistently improves both the target reward and GenEval scores without extra cost. Finally, combining GLASS-based FKS with gradient-based reward guidance further boosts reward scores and alignment metrics, showing that GLASS Flows serve as a simple, theoretically grounded, and practically effective drop-in replacement for SDE sampling that enables more powerful and efficient inference-time reward alignment of large text-to-image flow and diffusion models.", "questions": "The weight function $w_1(s)=\\partial_s\\bar\\sigma_s/\\bar\\sigma_s$ can become numerically unstable if $\\bar\\sigma_s$ approaches zero. In your setup, $\\bar\\sigma_1^2=\\sigma_{t'}^2(1-\\rho^2)$, which remains positive unless $\\rho=\\pm 1$. Since the method allows $\\rho\\in[-1,1]$, the paper should warn users that $\\rho=\\pm 1$ leads to degenerate cases where $\\bar\\sigma_1=0$ and $w_1$ becomes large near $s=1$, which could be numerically problematic with Euler integration. In practice, should users avoid $\\rho$ values too close to $\\pm 1$?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The paper introduces a method for inference-time reward adaptation by reframing stochastic Markov transitions as an inner flow matching problem and leveraging a Gaussian latent sufficient statistic to re-use existing denoisers, yielding a unified ODE-based mechanism that subsumes DDPM-style transitions instead of proposing yet another bespoke reward adaptation algorithm.\n- The technical development is careful and well-grounded, deriving GLASS transitions from a joint Gaussian model, justifying the sufficient-statistic construction, proving that the induced velocity field satisfies the continuity equation, and supporting the method with extensive experiments on both base generation and multiple reward-adaptation setups, including compute-matched comparisons against strong ODE/SDE baselines.\n- The exposition is generally clear and well structured: the ODE–SDE trade-off is crisply articulated, the roles of marginal vs. conditional flows are distinguished, the GLASS denoiser is given an intuitive “two noisy measurements” interpretation, and the paper repeatedly connects theory to practical design choices (e.g., how to plug GLASS into FKS and reward guidance).", "weaknesses": "The core technical step in GLASS Flows—the use of a linear sufficient statistic over multiple Gaussian observations to recover the original denoiser at a reparameterized time—appears mathematically very close to the training-equivalence result in TADA (arXiv:2506.21757), in particular Proposition 3.1. In GLASS, Proposition 2 shows that the GLASS denoiser (D_{\\mu,\\Sigma}(x_t, x_{t'})) can be written as the original denoiser evaluated at a suitably reweighted statistic and an effective time determined by (\\mu^\\top \\Sigma^{-1}\\mu); TADA’s Proposition 3.1 establishes an essentially identical form for an augmented Gaussian state, again via a reweighted linear combination and an effective SNR. Since the underlying Gaussian-conjugacy/sufficient-statistic argument is the same, it would be helpful if the authors could explicitly cite TADA and briefly discuss how their Proposition 2 relates to Proposition 3.1 there, while emphasizing that GLASS applies this mechanism in a different context (Markov transitions and reward alignment).\n\n\n1) In Equation (1), the transition is defined as $p_{t'|t}(x_{t'}|x_t)=\\mathbb{P}[X_{t'}=x_t|X_t=x_t]$. The left-hand variable in the probability should be $x_{t'}$ rather than $x_t$.\n\n2) In Algorithm 1 (left panel), the function `D(x,t)` returns:\n  - `(1 / (dot_alpha_t * sigma_t - alpha_t * dot_sigma_t)) * (sigma_t * u - dot_sigma_t)`\n- The factor of $x$ multiplying $\\dot{\\sigma}_t$ seems to be missing. The standard denoiser identity is:\n  - $D_t(x)=\\frac{1}{\\dot{\\alpha}_t\\sigma_t-\\alpha_t\\dot{\\sigma}_t}\\left(\\sigma_t u_t(x)-\\dot{\\sigma}_t x\\right)$.\n\n3) In the GLASS denoiser, $S(\\mathbf{x})=\\frac{\\mu^\\top\\Sigma^{-1}}{\\mu^\\top\\Sigma^{-1}\\mu}[x_t,\\bar x_s]^\\top$. With the natural (CondOT) choice $\\bar\\alpha_0=0$, we have $\\mu=(\\alpha_t,\\,\\bar\\alpha_s+\\bar\\gamma\\alpha_t)^\\top$. When $t=0$ and $s=0$, both $\\alpha_t=0$ and $\\bar\\alpha_s=0$, resulting in $\\mu=\\mathbf{0}$ and potential division by zero in both $S(\\x)$ and $(\\mu^\\top\\Sigma^{-1}\\mu)^{-1}$. This edge case arises in the first Euler call at $s=0$ in Algorithm 1 (right panel), and should be handled explicitly in the implementation. The correct limiting behavior at $s=0$ is that $\\bar X_0$ carries no additional information about $Z$ beyond $X_t$ (since $\\bar X_0=\\bar\\gamma X_t+\\bar\\sigma_0\\epsilon$ does not depend on $Z$ given $X_t$). Thus:\n  - $\\lim_{s\\to 0} D_{\\mu(s),\\Sigma(s)}(x_t,\\bar x_s) = D_t(x_t)$.\n\n4) Since $p_{t'|t}$ is constructed from a joint $p_{t,t'}$ with correct marginals $p_t$ and $p_{t'}$, for any fixed pair $(t,t')$:\n  - $\\int p_{t'|t}(x'|x) p_t(x)\\,dx = p_{t'}(x')$.\n- Therefore, under exact integration of the inner ODE, composing GLASS transitions across a grid $0=t_0<\\cdots<t_K=1$ should produce $X_{t_k}\\sim p_{t_k}$ at every step (and hence $X_1\\sim p_1$), for any $\\rho\\in[-1,1]$. This is an important property: GLASS sampling preserves correct marginals even when $\\rho$ differs from the DDPM value (though the multi-time joint may differ from that of a particular SDE/ODE). Since this property is central to GLASS as a sampling scheme, the paper would benefit from making it explicit with a brief proof in the main text or appendix.\n\n5) In Appendix A.1, the GLASS guidance is formulated as:\n  - $u_s^r(\\bar x_s|x_t,t)=u_s(\\bar x_s|x_t,t)+\\beta_{t^\\star}\\frac{\\sigma_{t^\\star}^2}{\\alpha_{t^\\star}}w_2(s)\\nabla_y r(D_{t^\\star}(y))\\big|_{y=\\alpha_{t^\\star}S(\\x)}$.\n- This follows from replacing $D_{\\mu,\\Sigma}$ with $D_{\\mu,\\Sigma}^r=D_{t^\\star}^r(\\alpha_{t^\\star}S(\\x))$ in the $w_2(s)D$ term, treating guidance as a \"denoiser-level\" correction. However, readers familiar with $u_t^r=u_t+c_t\\nabla V_t(x)$ in the standard ODE domain may find this formulation unclear. A brief derivation showing that GLASS guidance is the \"denoiser-based\" guidance analog of $D_t^r=D_t + \\frac{\\sigma_t^2}{\\alpha_t}\\nabla V_t$ transplanted into the $w_2(s)D$ structure would clarify the construction.\n\n\n6) For $\\rho=\\frac{\\alpha_t\\sigma_{t'}}{\\sigma_t\\alpha_{t'}}$, the paper claims that $p_{t'|t}^{\\text{DDPM}}=p_{t'|t}^{\\text{GLASS}}$. The proof in Appendix A.2 is sound, computing the conditional of the forward noising process (OU skeleton) and matching covariances. One minor note: the paper should explicitly state that $|\\rho|\\le 1$ holds in this case, since $\\alpha_t/\\alpha_{t'}<1$ and $\\sigma_{t'}/\\sigma_t<1$ for $t<t'$.\n\n\n\n\n\n7) Each GLASS inner step requires one neural network call to $u$ (then $D$ via the reparameterization) and a few $O(1)$ operations for the 2×2 inverse, $t^\\star$ lookup/inversion of $g$, and forming $S(\\x)$. This is comparable to the SDE per-step cost (one NN call), supporting the claim that GLASS can be swapped into SMC at essentially zero additional cost.\n\n8) Computing $D_t$ from $u_t$ requires $\\dot\\alpha_t,\\dot\\sigma_t$. Many deployed models expose continuous-time schedulers, but some libraries wrap discrete-time models and approximate derivatives. The paper should mention the requirement that $\\alpha_t,\\sigma_t$ be differentiable and clarify how discrete-time models are handled in practice.\n\n9) GLASS transitions achieve stochasticity by sampling the inner initial condition $\\bar X_0\\sim \\mathcal{N}(\\bar\\gamma x_t,\\bar\\sigma_0^2 I)$, while the subsequent evolution follows a deterministic ODE. Clarifying this distinction would help readers who might associate \"stochastic transitions\" exclusively with SDEs.\n\n[1] Chen, T., Zheng, H., Berthelot, D., Gu, J., Susskind, J., & Zhai, S. (2025). TADA: Improved Diffusion Sampling with Training-free Augmented Dynamics. arXiv preprint arXiv:2506.21757.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oKs4mv5TPh", "reviewer": "Reviewer_JMLt", "summary": "This paper introduces GLASS Flows, a new inference-time sampling paradigm that constructs an inner flow matching model within a pre-trained one to efficiently simulate stochastic evolution, combining the stochasticity of SDEs with the efficiency of ODEs without any retraining, thereby significantly improving the performance and efficiency of flow and diffusion models.", "questions": "1.  **On SDE Performance in Few-Step Sampling (Figure 2):**\n    Could the authors elaborate on the poor performance of SDE sampling in the few-step regime, as shown in Figure 2? Typically, reducing sampling steps in diffusion models results in blurriness or structural artifacts. However, the samples shown (e.g., in Figure 9, which seems related) exhibit high-frequency residual noise, suggesting the denoising process is incomplete rather than just imprecise. This seems counter-intuitive. [1] also analyzed the poor image quality produced by time-reversal SDE sampling and proposed an improvement that enables SDE sampling to generate high-quality images.\n\n\n2.  **Baselines for Few-Step Sampling:**\n    * For the experimental setup in Figure 2 (middle), what is the FID-vs-steps curve for standard ODE sampling? This baseline is crucial for contextualizing the performance of GLASS.\n    * How does GLASS compare against other state-of-the-art few-step solvers, such as DPM-Solver++ or UniPC? These methods are specifically designed for high-fidelity generation in the low-NFE regime, and a comparison of FID scores would be highly informative.\n\n3.  **Best-of-N Sampling (Table 1):**\n    The results for \"Best-of-N\" sampling using GLASS are not presented in Table 1. Given that Best-of-N is a common technique in reward alignment, could the authors provide the performance metrics for a Best-of-N(GLASS) approach?\n\n4.  **Applicability to Online RL Methods:**\n    Could the proposed GLASS flow be integrated into online reinforcement learning algorithms for flow matching models, such as DDPO[2] or Flow-GRPO[3]? A discussion on its potential applicability, or any inherent limitations for these types of methods, would add significant value to the paper.\n\n[1] Coefficients-Preserving Sampling for Reinforcement Learning with Flow Matching\n\n[2] Training Diffusion Models with Reinforcement Learning\n\n[3] Flow-GRPO: Training Flow Matching Models via Online RL", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper introduces an elegant sampling method (GLASS flow) that successfully enhances sample diversity without compromising quality.\n\nThis represents a fundamental improvement and holds significant potential for integration into a wide range of reward alignment frameworks.", "weaknesses": "The paper lacks a direct quantitative comparison of sample diversity between the proposed GLASS flow and standard SDE sampling. Providing metrics to substantiate the diversity claims would strengthen the paper significantly.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jtxtWIKI9E", "reviewer": "Reviewer_56nT", "summary": "The paper proposes **GLASS Flows**, a plug-and-play sampler for diffusion/flow-matching models that realizes stochastic transitions via an inner ODE (instead of an SDE). By reparameterizing a pretrained denoiser into a ``GLASS denoiser\" using sufficient statistics, the method constructs a transition kernel \\(p_{t'|t}\\) without retraining, unifying ODE-level quality with SDE-like randomness. This improves posterior sampling and downstream inference-time reward adaptation (e.g., SMC/FKS, reward guidance, search). At matched NFE, GLASS approaches ODE quality, outperforms SDE, and yields stronger prompt following/reward alignment on large T2I models.", "questions": "Please see the weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Theory to practice bridge: Closed-form construction of a stochastic transition via an inner ODE.\n* Drop-in practicality: No finetuning; works with CFG, SMC/FKS, reward guidance, and value-estimation pipelines.\n* Quality & alignment: Consistent gains on prompt-following and reward-driven metrics at the same NFE, while retaining randomness for search/SMC.\n* Clear complexity lens: Cost summarized by NFE = \\(K \\times M\\) (transitions × inner steps), enabling fair compute-matched comparisons.", "weaknesses": "1. **Choice of $\\rho$.**   \nThe paper fixes $\\rho = 0.4$  for most experiments, but this parameter directly controls the stochasticity of the transition. It would be helpful to show how varying $\\rho$ in $[0, 1]$ affects generation quality, prompt adherence, and diversity, and to explain why $\\rho = 0.4$  was chosen. In addition, please comment on whether different models (e.g., SiT vs. FLUX) show similar sensitivity to $\\rho$ or require different optimal values.\n\n2. **Under-evaluated relation to DTM/Transition-Matching.**  \nDTM is claimed as the  $\\rho = 1$ special case, but there may lacks head-to-head comparison.\n\n3. **Positioning vs post-training with reward models (RL/DPO/GRPO).**  \nRecent post-training approaches such as GRPO[1,2] or DPO also leverage reward models for alignment, but differ by updating model parameters rather than modifying inference. It would be helpful if the authors could clarify the conceptual connection and distinction between these paradigms, and possibly comment on whether GLASS could be complementary to such training-based methods or directly compared under similar reward setups.\n\n[1] DanceGRPO: Unleashing GRPO on Visual Generation\n\n[2] MixGRPO: Unlocking Flow-based GRPO Efficiency with Mixed ODE-SDE", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Zt9vWyFAKJ", "reviewer": "Reviewer_1Ktu", "summary": "This paper proposes **GLASS Flows**—a sampling paradigm that lets a pre‑trained flow/diffusion model simulate **stochastic Markov transitions** using only **ODE integration**. The key idea is to view a transition $x_t \\rightarrow x_{t'}$ as its own conditional generation problem and to construct an “inner” flow-matching model $u_s(\\bar x_s\\mid x_t,t)$ whose terminal state $\\bar x_{s=1}$ is distributed as the desired transition $p_{t'|t}(\\cdot\\mid x_t)$. Technically, the paper defines a **GLASS transition** by a bivariate Gaussian joint\n$(X_t,X_{t'})\\sim\\mathcal N(z\\mu,\\Sigma)$ with a tunable correlation $\\rho$ across time (no cross‑dimension correlation), inducing $p_{t'|t}$. In practice, GLASS is a drop‑in replacement wherever reward‑time algorithms need stochastic transitions: (i) posterior sampling and value‑function estimation $V_t$, (ii) Sequential Monte Carlo (SMC; Feynman–Kac steering), and (iii) guidance. Experiments on SiT/DiT ImageNet‑256 and FLUX (768×1360) show better few‑step posterior sampling (lower FID at small inner steps $M$), higher correlation in $V_t$ estimation, and SMC reward gains without sacrificing GenEval alignment; with $M=1$ the method reduces to DDIM.", "questions": "1. How does GLASS handle near-singular covariance matrices when $ |\\rho| \\approx 1 $? Are any clipping or jitter techniques applied to maintain numerical stability?\n\n2. Is $ g(t) = \\sigma_t^2/\\alpha_t^2 $ guaranteed to be invertible under learned or non-monotonic schedules? If not, how is $ t^\\* $ computed?\n\n3. Could $ \\rho $ be learned or dynamically adjusted during inference to balance exploration and stability, especially in SMC or value-guided sampling?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- The paper empirically demonstrates that GLASS transitions retain ODE‑level efficiency yet produce stochastic branches needed for SMC/search. This answers a real bottleneck noted in prior work: practitioners prefer ODE/probability‑flow sampling for speed (e.g., EDM design space) while SMC/search need sampling from $p_{t'|t}$ (typically SDE‑based).\n- Casting a transition $p_{t'|t}$ as an inner flow and showing that its denoiser is just the sufficient statistic re‑timing of the existing denoiser is elegant and broadly applicable. It explains in one stroke how to obtain stochasticity from an ODE‑driven model without re‑training. The mapping via $S(x)=\\frac{\\mu^\\top\\Sigma^{-1}x}{\\mu^\\top\\Sigma^{-1}\\mu}$ and $t^\\*$ is simple to implement and leverages any pre‑trained flow/diffusion network.\n- Plug‑and‑play application to Feynman–Kac SMC (GLASS‑FKS) and reward guidance improves reward metrics while preserving GenEval alignment, and does so on strong modern backbones (SiT/DiT and FLUX). This makes the method attractive to practitioners who already deploy ODE sampling but need stochastic branches.{index=6}", "weaknesses": "The statement that GLASS “eliminates the trade-off between efficiency and stochasticity” is strong, yet Tables 1 and 4 show roughly comparable rather than strictly superior performance. Missing wall-clock time and variance analyses weaken this claim.\n\nThe influence of correlation $ \\rho $ and inner-scheduler choices (CondOT vs. alternatives) is underexplored. Additional ablations could clarify their role in balancing stochasticity and sample quality.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)