PairFlow: Closed-Form Source-Target Coupling for Few-Step Generation in Discrete Flow Models
Authors: Mingue Park
Jisung Hwang
Seungwoo Yoo
Kyeongmin Yeo
Minhyuk Sung
Venue: ICLR 2026 Poster (2026)
Keywords: ReFlow
Flow matching
Rectified flow
Primary area: generative models
TL;DR: We introduce PairFlow, a lightweight preprocessing step for training Discrete Flow Models (DFMs) to achieve few-step sampling without requiring a pretrained teacher.

Abstract: We introduce $\texttt{PairFlow}$, a lightweight preprocessing step for training Discrete Flow Models (DFMs) to achieve few-step sampling without requiring a pretrained teacher. DFMs have recently emerged as a new class of generative models for discrete data, offering strong performance. However, they suffer from slow sampling due to their iterative nature. Existing acceleration methods largely depend on finetuning, which introduces substantial additional training overhead. $\texttt{PairFlow}$ addresses this issue with a lightweight preprocessing step. Inspired by ReFlow and its extension to DFMs, we train DFMs from coupled samples of source and target distributions, without requiring any pretrained teacher.
At the core of our approach is a closed-form inversion for DFMs, which allows efficient construction of paired source–target samples. Despite its extremely low cost, taking only up to 1.7\% of the compute needed for full model training, $\texttt{PairFlow}$ matches or even surpasses the performance of two-stage training involving finetuning. Furthermore, models trained with our framework provide stronger base models for subsequent distillation, yielding further acceleration after finetuning. Experiments on molecular data as well as binary and RGB images demonstrate the broad applicability and effectiveness of our approach.

Published as a conference paper at ICLR 2026
PAIRFLOW: CLOSED-FORM SOURCE-TARGET COU- PLING FOR FEW-STEP GENERATION IN DISCRETE FLOW MODELS
Mingue Park
Jisung Hwang"
Scungwoo Yoo"
Kyeongmin Yeo
Minhyuk Sung
KAIST
Mrickkicik,40113 dreamy 1534, aaaaa, mhsung) @kaist.ac.k
ABSTRACT
We introduce PAIRFLOW, lightweight preprocessing step for training Discrete Flow Models (DFMs) to achieve few-step sampling without requiring pretrained teacher. DFMs have recently emerged as new class of generative models for discrete data, offering strong performance. However, they suffer from slow sam- pling due to their iterative nature. Existing acceleration methods largely depend on finetuning, which introduces substantial additional training overhead. PAIRFLOw addresses this issue with lightweight preprocessing step. Inspired by ReFlow and its extension to DFMs, we train DFMs from coupled samples of source and target distributions, without requiring any pretrained teacher. At the core of our approach is closed-form inversion for DFMs, which allows efficient construction of paired source- source-target target samples. Despite its extremely low cost, taking only up 1.7% of the compute needed for full model training. PAIRFLOW matches or even surpasses the performance two-stage training involving finetuning. Furthermore, models trained with our framework provide stronger base models for subsequent distilla- tion, yielding further acceleration after finetuning, Experiments on molecular data well as binary and RGB images demonstrate the broad applicability and effec- tiveness our approach. Project htge::htps::: /ttp///aaiiieeeeee flow. github ic
INTRODUCTION
Discrete Flow Models (DFMs) (Campeell et al. 2024 Gat al., 2024) have recently emerged as promising class of generative models, extending the idea of Flow Models (FMs) for continuous data the discrete domain. By adapting flow-based principles to categorical structures, DFMs provide principled and efficient way to capture complex discrete distributions through iterative sampling. They have shown success across a variety applications, particularly scientific domains such as molecule generation (Ramakrishnan al. 2014 Irwin et al.) 2012). where DFMs offer natural framework for modeling chemical structures and generating novel candidates.
Analogous to FMs in the continuous domain, key challenge of DFMs is the long computation time for generation due to their iterative sampling nature. Recent work (Deschenaux & Gulcehre 2025 Hayakawa et al. 2025; Sahoo et al., 2025 Yoo et al. 2025) have sought to accelerate the generative process through distillation-baeed finetuning, which builds on ideas originally developed for continuous flow matching. Notably, ReFlow (Liu Gong 2023 is well-known technique for FMs that pairs samples from the source (prior) distribution and the target distribution by running the generative process of pretrained model and using the resulting pairs for finetuning. Recently, this idea has also been extended to DFMs (Yoo et al., (025) to reduce conditional total correlation through finetuning, thereby enabling few-step generation.
Despite these promising results in acceleration, distillation-ban based methods incur substantial finetuning overhead, amounting about 10-20% of the time required to train the base model from scratch. In other words, the gain in generation speed comes at the expense of considerable additional training cost. To our knowledge, no prior work has addressed this training-time cost when pursuing inference-time
"qqual contribution.

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acceleration. This raises natural question: can achieve speedups comparable to distillation- based approaches while requiring only a lightweight preprocessing phase that requires orders of magnitude less compute, the order of tens of GPU minutes?
We propose PAIRFLOW, a training framework for DFMs that enables few-step sampling by con- structing paired source target samples using closed-form velocities. While inspired by ReDi-style coupling-driven training, our approach eliminates the need for pretrained teacher by using closed-form formulations and achieves acceleration without finetuning. The algorithm for com- puting souree-target pairs is fully parallelizable and requires at most 1.7% of compute needed for full model training. Despite relying only on a lightweight preprocessing step. PAIRFLOW attains performance comparable to, or even superior to state-of-the distillation -assdd techniques, which can require up to 143 times more computation. Furthermore, models trained with our technique provide stronger bases for subsequent distillation, delivering additional performance gains while incurring only minimal preprocessing cost.
At the core of our framework is the simulation of probability paths connecting source (prior) and target (data) distributions in discrete spaces, made possible by closed-form expressions of velocities. While closed-form forward velocities have been studied for flow models in continuous domains (Karras et al., 2022, Bertrand et al. 2025). they have, to the best of our knowledge, neither been explored for DFMs nor applied to identif ying suitable source souree-target pairs in the context of distillation- -balaa accelcration, as in ReDi (Yoo et al.) 2025). In this work, we investigate this idea for the first time. For DFMs, with a particular focus uniform-state models (Sahoo al. [025 Schiff et al., 2025) equipped with a self-correcting mechanism, we show that the closed-form forward velocity is determined by the Hamming distance, which measures the number of differing tokens between two sequences.
Using this velocity, samples from the source (latent) distribution can be mapped to given target samples. However, because multiple source samples may map to the same target, covering all targets through coupling would require an impractically large number of source samples. To overcome this, we derive the corresponding backward velocity in closed form and leverage to simulate backward probability paths that efficiently map data points to source points, making pair discovery computationally efficient.
In our experiments, we show that the proposed framework enables few-step sampling across diverse discrete domains, including molecular data (Ramakrishnan et al., 2014: Irwin et al. 2012) and 2D
images, exemplified by MNIST- ANIST-Binary (LeCun et al. 2002) and CIFAR 10 (Krizhevsky et al. 2009).
On the QM9 RRamakrishnan et al. 2014) and ZINC 250k (Iwwin et al. 2012) datasets, not only improves the base model but also performs comparably to, even better than, distilled models that require up to 143x more compute during finetuning, compared to our lightweight preprocessing algorithm. Similar improvements are observed on MNIST-Binar
where models paired with PAIRFLOW achieve performance comparable to those using DCD (Sahoo et al. (025) and ReDi (Yoo al. 2025), while being up to 35x faster. Furthermore, after subsequent distillation, base models trained with pairs generated by our method consistently achieve higher performance, underscoring the importance of well-constructed source-targed pairings.
2
RELATED WORK
2.1
DISCRETE FLOW MODELS
The concept of Flow Matching (ipman et al., (023) has recently been extended to the discrete
flow-based models (Gat et al. 2024 Campbell et al. 20244 Sahoo ct al. 2024| Schiff et al. 2025).
demonstrating its flexibility across high-dimensional and structured data (Bai al. 2025
Chang
et al. 2022 Weber et al. 2024 Arriola et al., 2025 Nie et al.) Yu et al.) 2023 Lee et al.
2025
Campbell et al.] 2024 Wang al. 2025). Among these, uniform-state models (ahoo al.
2025
Schiff et al. 2025) have been studied for their self-correcting properties, which enable recovery from errors introduced during parallel decoding. However, their performance degrades markedly in few-step settings, posing a key limitation for efficient generation under tight compute budgets.
2.2 ACCELERATING DISCRETE FLOW MODELS
Several approaches have been proposed to accelerate sampling with DFMs. Park et al. (2024) directly optimize sampling timesteps for improved parallelism while mitigating decoding errors. Hayakawal et al. (2025) highlight the importance of capturing dimensional correlations for faster sampling and
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introduce mixture models to this end, the cost of additional loss terms that complicate optimization. Another line of work adapts techniques from continuous domains, as in Sahoo et al. (2025). that
propose discrete analogue of Consistency Distillation (CD) (Song al. [022) by leveraging the
duality between uniform-state and continuous Gaussian models. Most relevant to our approach is
ReDi Yoo et al. 2025), which draws inspiration from the concept of straight flows in ReFlow (Liu & Gong 2023 and iteratively optimizes pairs of data and noise samples.
3 PRELIMINARIES
In this section, we provide a brief overview of flow matching for generative modeling of discrete data (Sec. 3.T.. followed by rectification technique Yoo et al. 2025) that enables faster generation (Sec. 3.2 by reducing total correlation errors.
3.1 DISCRETE FLOW MATCHING
The goal of Discrete Flow Matching (DFM) (Campbell et al. 2024 Gat al. 20244 is to learn
probability path (+). that connects known, casy-to-sample source distribution p(æ) to an unknown target distribution (ar), both defined over discrete state space. Once p-(-) is known, samples from can be generated by drawing To ~ and transporting it along the path.
Specifically, consider sequence (x x² ..... xN) of N tokens, where each token takes values vocabulary = {1, }) of size K. A sequence x then resides the product space V We denote by A {p R* Σ, 1, P 0} the probability simplex of dimension K - 1, on which distributions over are defined.
Given target probability path pc(x) V [0.1] (x))) DA x [0, + RNNK we introduce a network (x) : DN x [0,1] (A to
[0, 1] with associated velocity field
approximate uy (x)). Its parameters 0 are optimized via the DFM objective Gat et al. 2024):
(f) iE(I....,))
Σ
log pl (afl2),
(1)
where (=ilz) denotes the learned probability denoiser, which predicts the categorical distribution of the clean token oi given an intermediate sequence 2. Here, the conditional probability path Pe(2|=r,, 21) generates samples = by interpolating between data point 21 ~ q and source sample To p. Assuming independence across tokens in sequence the conditional density factorizes as
N
(2 x1) = II
(r(2" (c0.
1=1
As token-wise conditional paths (2(2" 220 x1), Gat et al. 2024 employ the mixture path of form:
(c x1) = -1- Kz)6 (1-x)) (2*) + Kzōz, (z'),
(3)
where the scheduler H2 k(t) is monotonically increasing function over t [0, 1] satisfying Ko = 0 and Ki = For notational convenience, given æ, y E V we define the Dirac delta (x) as
(x) = II (x*), where oy (x²) = {
N
x² #
We also use the shorthand (x()) by (x+). After optimizing 0, the learned model parameterizes an approximation of the marginal velocity field:
( 1- Ki
hic
-
5,(x)).
(5)
where K; to simulate trajectories along (-) and thereby generate samples. Each update step is defined as:
3xL This learned velocity field of (at 2) then transports samples over the interval [0. 1] öt
a ~ Cat (zithi 6.2(.) ()+ + h (f(ttt121)
(6)
where h > 0 is the step size.
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3.2 STRAIGHTENING PROBABILITY PATHS FOR ACCELERATED SAMPLING
The concept of straight probability paths was originally introduced in the continuous domain to enable accelerated sampling. Prior work (Liu & Gong. 2023) identified curved probability paths as key challenge in few-step sampling: when velocity fields are evaluated only at coarse time steps, numerical integration deviates from the true trajectories. Liu & Gong (202)) addressed this issue through "rectification," in which student flow model is trained on source target pairs generated by teacher model, effectively yielding significantly straighter probability paths.
In the discrete setting, this challenge of path curvature translates capturing the statistical correla- tions between tokens. Since DFMs approximate exponentially large joint transitions through fully factorized per-token updates, a mismatch inevitably arises between the true joint transition and its product-form approximation. This discrepancy becomes especially detrimental during few-step gen- eration, where highly correlated tokens must be updated simultaneously. To address this, prior works have primarily relied on distillation-bass
approaches (Hayakawa et al., 2025) Sahoo et al. 2025 Deschenaux & Gulcehre 2025). aiming better capture these correlations by explicitly transferring multi-step dependencies from stronger teacher model.
Yoo et (2025) formalized this factorization mismatch via conditional Total Correlation (TC), defined as:
TC (I,((a)))) = Exg DKL Pale(Ts
[2:)] II Paje (zr (mx)
2025 interpret this factoriza- tion error as the discrete analog of path curvature: minimizing TC is equivalent to "straightening" the trajectory by decoupling token transitions. Analogous to ReFlow (Liu Gong 2023), which rectifies continuous paths, they demonstrate that reducing Eqn. - iteratively refining the source-target coupling x ( ). To achieve this, they employ an iterative distillation process, alternating between generating improved pairs using the current model and optimizing CDFM. This procedure effectively finds a "statistically straight" coupling that enables efficient few-step generation.
which serves as metric for the factorization error. Crucially, Yoo
4 PAIRFLOW
For DFMs, ReDi (Yoo et al. 2025) improves sample quality in few-step generation by rectifying source-targe pairs. However, it relies on samples from pretrained model followed by costly retraining or finetuning. We take this one step further and pose the following question: What if these pairs could be generated directly from the data, without relying on pretrained model or sampling from it?
To address this question, we propose a principled approach for discovering well-aligned source-targed target pairs without relying on pretrained models, enabling models trained on such pairs to achieve strong performance with few-step sampling. Our method, termed PAIRFLOW, leverages closed-form velocity fields that can be computed directly from the data samples, requiring only prior knowledge of the source distribution. We assume this distribution to be uniform, choice extensively studied recent work (Sahoo et al.) 2025 Schiff et al. (025), as models trained under this prior naturally acquire self-correcting properties.
In Sec. 4. we introduce the closed-fome form forward velocity for discrete flow matching (Gat et al. 2024). In Sec. 4.2 we extend this to the closed-form backward velocity and propose an algorithm for discovering well-aligned source-target pairs during the preprocessing phase.
4.1 FINDING PAIRS VIA CLOSSED-FORM FORWARD VELOCITY FIELDS
As discussed in Sec. 3.I DFMs (Campbell et al. 2024 Gat et al. 2024) aim to learn a marginal velocity field (-) that induces a probability path pt (-), transporting the source distribution (- (-) to the target distribution q(-) (-)., which is unknown in practice. Instead, we only have access to a finite dataset of M samples {dm 1M -- This empirical distribution ((x) can be represented as a mixture of Dirac deltas centered at the observed samples:
M
((x) R q(x) = M
Σ m=l
Sam (x).
(8)
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t-1.0-
2
7
9
Algorithm 1: PAIRFLOW
Painflow (Ours)
t-0.8-
0
2
4
7
9
Input: Dataset {dm } Steps T Output: Pairs = { {Foom F1,m mm
t=0.0
0
7
2
9
3
4
4
3
Initialize π -)
for +- 10 M do
Average Hamning Distance:
F1,m s dm
t-1.0-
0
1
2
4
5
6
7
8
9
f 21,m
for t f 1-100 T do
UDLM
t=0.8-
@
1
0
6
5
6
7
4
5
Compute Poje(-12) and (x(; from Eqn.
Sample
t=8.0
6
6
4
3
~~ Cat(z; dz(+) - h (x())a))
Average Hamning Distance: 9.00
Figure 1: Illustrations of data inversion
Tumm f
in PAIRFLOW and the standard corruption pro- cess in UDLM. PAIRFLOW achieves lower average Hamming distance (6.47 vs. 9.0), pro- moting straighter paths during training.
12 f π U ((o 11m))
13 return π
For continuous domains, Karras et al. (2022); Bertrand et al. (2025) have shown that the velocity field transporting Po to q can be derived in closed form when both distributions admit tractable density expressions. To the best of our knowledge, this idea has not been explored in discrete domains; in the following, we derive the closed-form velocity field for discrete domains for the first time.
We base framework the assumption of uniform prior distribution over the discrete state space VA, defined as po(x) = 24 where 4 = Cat (-: k)) denotes the uniform distribution over the dictionary V. For the empirical target distribution ((x) introduced Eqn. we show in App. A.I A.I that the closed-form denoiser Pije(11]p (2f [2) and its associated velocity field 0g (ær 2) are given by:
DM dat, (x')) - hldm,))
Re
P1j((111>)
0r(x', 2) = 1 ( 62(x))) (9)
Mm ? (l(dm..))
K denotes the vocabulary size, and h(s, = de is the Hamming distance between sequences s and z, i.e., the number of positions at which they differ. The token-wise denoiser (=) (2) above is weighted mixture of Dirac deltas, where sequences closer to z under the Hamming distance contribute more. Intuitively. the forward velocity field (x(x", =) pulls each token toward those from dataset sequences most similar to z. The most direct way to construct source-target pairs using 0z (x²,, =2 is to sample xn po(zr) and evolve it along the velocity field until it reaches a dataset point ø1. In practice, however, the generated data points fail to fully cover q(z), requiring impractically large number of source samples to achieve sufficient coverage. Our empirical results, reported in App. C.T support this claim and motivate the exploration of more efficient alternative, which present the following section.
y
4.2 FINDING PAIRS VIA CLOSED-FORM BACKWARD VELOCITY FIELDS
We address this issue by backtracing trajectories along Pr(-). starting from q(x) and progressing toward the source distribution po(-). Unlike the forward construction in Sec. [.1 this guarantees that all data points in q(x:) are included in the resulting pairs by design. As illustrated at the top of Fig. [ PAIRFLOW inverts data samples toward the source distribution, assumed to be uniform. Unlike the standard corruption process used by UDLM (Schiff et al.) 2025) shown at the bottom of Fig. [ the source samples obtained by PAIRFLOW remain closer to the original data terms of Hamming distance. This helps the model learn recover data with fewer token transitions during training, effectively approximating the straight probability paths explored in ReFlow (Liu & Gong 2023) and ReDi (Yoo et al. 2025)
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Table Dataset and training statistics. N denotes the number of tokens per sample, K the dictionary size, [Xy] the dataset size, and T. the runtime of each method (in minutes, measured in wall-clock time with RTX A6000). The numbers in parentheses are the proportion of time relative to T Base.
Dataset
|
NV
K
[X]
Tasee
TixD
TRaDi
T PairPlow
MNIST-Binary
768
60,000
80 (100.0%)
40 (50.0%)
49 (61.0%)
1.4 (1.7%)
CIFAR-10
3,072
256
100,000
6720 (100.0%)
360 (5.3%)
468 (6.9%)
20 (0.3%)
QM9
32
40
127,190
450 (100.0%)
115 (24,8%)
100 (22.2%)
0.8 (0.2%)
ZINC-250K
72 74
224.568
1,110 (100.0%)
211 (19.0%)
194 (17.4%)
13 (1.2%)
The remaining challenge is to derive the closed-form backward velocity that governs this process. This can be obtained by following a construction analogous to Sec. 4.1 Specifically, we first derive the closed-form noise predictor Poje(x"(2))
Po) = 52 (x') Nr (K6, (2+) - 1) Σ M
daz (2')) -R(dmmm))))
(10)
1 (K 1)ke
ΣM 7 h(dm:))
with detailed derivation provided in App. A.2 Substituting this into the definition of the backward velocity field from Gat et al. (2024)
6,(10, 2)
[62(x") Poje(a" [=][
(11)
Kt
we obtain the desired closed-form expression
Ni (K8, (2') EM
M m== des (=tdm-hidmms) h(dmis)
(e(x', 2)
(12)
M 1 (K - 1)88 Em=r ? h(dmmzz)
The second term in Eqn. computess the conditional likelihood of the i-th token taking value x' e V given the current sequence marginalized over all dataset {dm, J-M The contribution of each data sample dm, is determined by its proximity to under the Hamming distance hidm, 2), assigning higher weight to tokens with greater local consensus. Consequently, updating with 8, (x', 22) pushes the sample away from the data distribution and toward the source distribution Po(x). Using (x(x), ((x))))) 2). #1,m)
we construct pairs
by initializing from data points {d.,. Jm (equivalently, ( M ) and iteratively applying the backward update rule in Eqn. for a fixed number of iterations T. The overall procedure is summarized in Alg. [
EXPERIMENTAL RESULTS
We validate the effectiveness of the proposed method and the source-target pairs it discovers across several discrete generative modeling benchmarks involving molecular data and images. We first summarize the experimental setup in Sec. 5.1 In Sec. 5.2 and Sec. [5.]] we compare our method against baselines in molecular and image generation, respectively. In S.4. we further demonstrate that models trained with pairs discovered by our method not only achieve improved performance directly, but also benefit subsequent distillation phases, as the resulting base model provides stronger initialization for existing distillation techniques.
5.1 EXPERIMENT SETUP
Baselines. Across multiple benchmarks, we compare our approach against state-of-the-an discrete flow models, including MDLM (Sahoo et al., (024) and UDLM (Schiff et al. (2025). Since our method is based on a uniform source distribution, we adopt UDLM (Schiff et al. (025), the leading uniform- state model, as our base and denote UDLM trained with pairs generated by Alg. aas PAIRFLOW throughout the remainder of this section. In addition, we compare against these models augmented with distillation- based techniques that require additional finetuning, Discrete Consistency Distillation (DCD) (Sahoo al.) 2025) and ReDi YYo et al., 2025). denoted throughout this section by the suffixes "+ DCD" and "+ ReDi". The detailed training setups of these models, such as hyperparameters. are provided in App. B Additionally,
report the performance of the same base model trained on pairs formed by each data point and a source sample randomly drawn from the uniform distribution with detailed experimental results in App. D

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QM9 Validity (1)
QM9 Uniqueness (1)
QM9 Novelty (1)
1000
1000
200
500
MDLS
500
MDL.5
100
MOLSX
Pair
Poil Plass
Pair Flom 110EL 100
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Sampling Steps
Sampling Steps
Sampling Steps
Figure 2: Step-wise performance analysis on the QM9 dataset (Ramakrishnan et al. 2014). Each plot reports the number of valid (left), unique (middle), and novel (right) SMILES strings (Weininger 1988) out of 1,024 generated samples. Best viewed when zoomed in.
Benchmarks. We evaluate our method across a diverse set of discrete generative modeling bench- marks, covering both molecule and image generation tasks. For molecule generation, we experiment with the QM9 (Ramakrishnan et al., 2014) and ZINC-250K (Irwin et al. 2012) datasets. For image generation, we use the MNIST dataset (LeCun et al.] (2002) with binarized pixel values (denoted MNNST--Binary and the CIFAR- 10 dataset (Krizhevsky et al. (009), where pixel intensities are scaled to 8-bit integers, and horizontal flip augmentation is applied. Dataset statistics, including sample size, vocabulary size, and overall dataset size, are summarized in Tab.
Evaluation Setup. For molecular generation, we follow [chhff et al. (2025) and evaluate the validity, uniqueness, and novelty of generated molecules. Specifically, we sample 1,024 SMILES strings [Weininger] 1988), convert them into molecular graphs, and compute these metrics. All results are averaged over 10 trials, with further details provided in App. D For image generation, we report Fréchet Inception Distance (FID) (Heusel et al.) 2017) and Inception Score (IS) (Salimans et al.) 2016). FDD is computed with I. 1,000 images for MNIST-Binary, and both FID and IS are computed with 5,000 generated images for CIFAR-10. The training dataset is used as the reference for FID computation. Across all experiments, we vary the number of sampling steps to evaluate performance in both low- and high-NFE settings. In particular, we generate samples using 1 64 steps for molecular benchmarks (QM9 and ZINC-250K) and MNIST-Binary benchmark, and 8 1024 steps for CIFAR-10 (Krizhevsky et al. 2009), as models yielded excessively high FIDs under extremely low-step settings.
5.2 MOLECULE GENERATION
We begin by benchmarking unconditional molccule generation, where models are tasked with generating SMILES strings (Weininger 1988) that represent molecules. As illustrated in Fig. 2 and Fig. 33 which summarize validity (left). uniqueness (middle), and novelty (right), PAIRFLOW consistently improves upon its base model UDLM (Schiff et al. (2025), yielding substantial gains in few-step settings. It facilitates 1-step generation on QM9 (Ramakrishnan et al., 2014), a challenging setting that requires capturing all token-wise dependencies simultaneously. In this case, validity increases from 17.5 to 223.4, corresponding to a 12.8x improvement. Similar trends are observed in the 2-step and 4-step settings, with validity improving by 231% and 47.6%, respectively. As shown in Fig. 2 (left), this improvement is particularly significant: the 2-step and 4-step validities of PAIRFLOWW are comparable to the 4-step and 8-step validities achieved by UDLM (Schiff et al.,
2025). Comparable improvements are also seen on the ZINC-250K (Irwin et al. 2012) dataset.
Remarkably, PAIRFLLOw introduces minimal overhead-less than 2% of the training cost as shown in Tab. and requires no pretrained models, yet achieves performance comparable to, and in some cases surpassing, models distilled from the same base using DCD (Sahoo et al. 2025) and ReDi YYoo et al. (025). both of which rely on pretrained models and finetuning. On both QM9 (Ramakrishnan et al. 2014) and ZINC-250K (Irwin et al. 2012). PAIRFLOW consistently out- performs UDLM + ReDi across all few-step settings, achieving substantially higher 2-step validities on QM9 (232.4 vs. 416.0) and ZINC-250K (75.9 vs. 146.3). At the same time, PAIRFLOW matches the performance of UDLM + DCD, with comparable 2-step validities on QM9 (416.0 vs. 530.8). This is particularly notable given that the additional preprocessing cost of PAIRFLOW amounts to only 0.69% on QM9 (Ramakrishnan et 2014) and 6.16% on ZINC-250k (Irwin et al. 2012). relative to the full cost of DCD (Sahoo et al. 2025). Detailed numerical results with standard deviations are reported in App.
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ZINC-250K Validity
ZINC-250k Uniqueness
ZINC-250K Novelty (1)
500
500
500
MDLS
NDLS
Pair
Pair PairFlow
PairFlow UBLM
Une-
0
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Sampling Steps
Sampling Steps
Sampling Steps
Figure 3: Step-wise performance analysis on the ZINC-250K dataset (Irwin et 2012). Each plot reports the number of valid (left), unique (middle), and novel (right) SMILES strings (Weininger of 1,024 generated samples, Best viewed when zoomed in.
1988)
MNIST-Binary (FID
CIFAR-10 (FID J)
CIFAR-10 (IS 1)
10a
Pestbe
Peofiom
10a
100
16
16
64
128
256
512
1024
32
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1024
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Sampling Steps
Sampling Steps
Figure 4: Step-wise performance analysis on discretized image datasets. From left to right: FID on MNIST-Binary (LeCun al. 2002), FID on CIFAR-10 (Krizhevsky et al. 2009), and Inception Scores (IS) on CIFAR-10. 10. Best viewed when zoomed in.
MDLM
MDLM
UDLM
64 steps UDLM
PAIRFLOW
PAIRFLOW
UDLM DCD
MOLM
PAIRFLOW DCD
6
256 steps UDLM
6
UDLM
ReDi
1
PAIRE LOW
6
PAIRFLOW
ReDi
8
Figure 5: Qualitative results of 1-step generation MNIST-Binary (28 x 28; left) and 64-step (top right) and 256-step (bottom right) generation on CIFAR-10 (32 x 32).
5.3 IMAGE GENERATION
We further extend our experiments to image domains where each pixel has discretized intensities.
As in Sec. 5.2 we evaluate model performance across multiple sampling steps and summarize the
results in Fig. 4 Qualitative samples for MNIST-Binary and CIFAR- 10 are shown in Fig. 5 Both
qualitative and quantitative results show that PAIRFLOW improves the performance of UDLM Schiff
al. 2025 and, in few-step settings, achieves performance comparable to DCD (Sahoo et al. 2025
8

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QM9 Validity (1)
ZINC-250K Validit (1)
MNIST-Binary (FID 1)
1000
Parthor LC
300
500
PairFlew- ULD
PairFlos DED
10'
oe EILM neat PairFlowe
Bat:
0
Pairflow-
16
32
64
4
8
16
32
64
16
Sampling Steps
Sampling Steps
Sampling Steps
Figure 6: Step-wise performance analysis of distilled models on molecular and image datasets. From left to right: number of valid molecules on QM9, number of valid molecules on ZINC-250% and FID on MNNIT-Binaay.
On MNIST-Binary (Fig.P 4 left), PAIRFL.OW achieves an FID of 40.59 in the 1-step setting, equiva- lently a 68.9% improvement over UDLM (Schiff et al. 2025). Consistent gains are observed across other few-step settings as well: at steps, FID is reduced by 63.3% (15.61 vs. 42.54), and steps by 24.4% (8.51 vs. 11.25). On 10 (Krizhevsky et al. 2009), where FID (Heusel et al. 2017) and IS (Salimans et al. 2016) are reported in Fig. 1 (middle and right), PAIRFLOW likewise outperforms the base UDLM, validating the effectiveness of the discovered source-target pairs.
As in molecular generation (Sec. 5.2). PAIRFILOW performs comparable to distillation-based accel- eration methods. On MNIST-Binary, it achieves lower FID than UDLM+DCY in the 1-step setting (40.59 vs. 53.84) and comparable performance at 2 steps (15.61 vs. 16.09). Likewise, PAIRFLON performs competitively with UDLM+ReDi at 2 steps (15.61 vs. 10.36), while requiring substantially less compute than both. As summarized in Tab. I DCD (Shhoo et al. 2025) requires 40 minutes (TuCD) and ReDi (Yoo et al. (025) takes 49 minutes, whereas the preprocessing phase of PAIRFLOW completes in just 1.4 minutes (TPAIRii ow ). yielding 28.6x and 35x speedups. respectively. On the
CIFAR-10 benchmark, both DCD (Sahoo al. 2025) and ReDi (Yoo et al. 2025) degrade model per- formance, as indicated by the higher FID in Fig. (middle) and lower IS in Fig. 4 (right). The results in Tab. 13 suggest that, overall, acceleration methods do not work well on CIFAR-10. We hypothesize that this issue arises from the low performance of the teacher model, which negatively affects the student model when applying acceleration methods. Detailed results are reported in App. D
5.4 DISTILLING MODELS TRAINED WITH ALIGNED PAIRS
While PAIRFLOW alone achieves performance comparable to, or even exceeding, distillation- based techniques (Yoo et al. 2025 Sahoo et al.) 2025), as shown in Sec. [.2] and Sec. [5.3] we further emphasize that it also serves as a strong initialization for subsequent distillation, yielding even greater performance gains when combined with existing methods. Crucially, this incurs negligible additional cost relative to the overall time required for distillation.
We validate this by distilling PAIRFLOW, trained on QM9 (Ramakrishnan et al 2014), ZINC- 250k (Irwin et al.) 2012). and MNIST-Binary, using DCD (Sahoo et al. 2025) and ReDi YYo)) et al. 2025), and comparing their performance against distilled models whose teachers were the base UDLM (Schiff et al. 2025). As shown in Fig. 6 student models distilled from PAIRFLOW, denoted LOWFFLOWDDDD and LAIRFLOW+Reid achieved by distillation- based techniques. For example, on the QM9 dataset (Ramakrishnan et al.) 2014). FAIRFLLOWDDCD substantially improves validity over UDLM+DCD (453.8 vs. 323 for I step. 685.8 vs. 530.8 for 2 steps). A similar trend is observed for PAIRFLOw+ReDi on ZINC- 250k (Irwin et al., 2012). yielding higher scores in both 1-step (46.3 vs. 0.7) and 2-step (221.5 vs. 75.9) generation. Importantly, as summarized in Tab. [ these gains are achieved at only minimal additional preprocessing cost: 3.15% of the average runtime of distillation on MNIST-Binary. 0.77% on QM9, and 6. 6.42% on ZINC-250k.
push the frontier of performance previously
CONCLUSION
We have presented PAIRFLOW, a novel approach to accelerating the generative process of Discrete Flow Models (DFMs) through lightweight preprocessing step performed prior to training. Our preprocessing, which couples source and target samples, requires only up to 1.7% of the base model training cost, making at least 20x more efficient than finetuning, while still achieving comparable or even superior performance. The key enabler is the closed-fome inversion, which eliminates the need for pretrained teacher model.

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Acknowledgements. This work was supported by ITTP grants (RS-2024-00399117, RS-2025- 25441313, RSS2025-254433180 funded by the Korean government (MSIT); the Industrial Technology Innovation Program (RS-2025-02317320 funded by the Korean government (MOTIE); the National Supercomputing Center (KSC-2005-CRF--4455) and the DRB-KAIST SketchTheFuturn Research Center.
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A PROOF FOR CLOSED-FORM VELOCITY
In this section, we present the detailed derivations of the closed-form forward velocity Eqn. and backward velocity Eqn. 12 introduced in Sec. 4 Sec. A..1 provides the proof of the clomed-form forward velocity, while Sec. A.2 presents the proof of the closed-form backward velocity. Both derivations based on the assumption of uniform source distribution.
A.1 PROOF OF CLOSED-FORM FORWARD VELOCITY DISCRETE FLOW MODELS
Let æt, V be sequences of tokens x² 2' V for i E ,...., N, where each token takes values from the discrete vocabulary V {l...., Kj. We begin with the expression the forward velocity given in Eqn. 9
Ke 8e(x", 2) 1 Kr
-2))) 6x(2))
(13)
We first derive the closed-form expression for the probability denoiser Pije P1je(="|22) (z):
Using Bayes' rule,
(1je(='12)=
Σ
82f (x')
Pe (F 2112)
(14)
20,31
Sae (2rt) Pe x1,2)
Σ #0.30
(15)
(2(2)
Exown dzg (x*)
Pe (20, 31, Σzn,2y Pe( 31, 2)
(16)
We factor the joint as
(10,, & Po(zo) Pr (x1) (==
(17)
N
=
II
Ne (x') + (1 -Ke) و5 (=²)
(18)
f=1
and use the empirical target
1
(((1))
Σ
8am
(x1)
(Eqn.
(19)
M
m=1
Since po(o)) is constant, it cancels between the numerator and denominator of Eqn.
16 yielding
(1)((1112) = Exn,ey dof (a*) pr (11) (. (z
Enporr PI (x1) Pe Io,
ΣM En, Sx (x+) I [e d2 (2d) + ne) o. (x)] EM Eno I (K+ &eg, (z)) + (1 - -Ki) 5ug [21)
(20)
According this expression, the probability denoiser Pije (2) can be interpreted weighted sum over all data points Fi, given by the term
EI Jo
I (x0 kx)
(21)
Let fi(s, z) denote the Hamming distance between two sequences and defined as
h(s, 2) = Σ مe (z*),
(22)
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and let (s, 2) represent the similarity between the sequences, defined as
h+ 14(s 2) = Σ مes (c)(2²) = N h(s,2).
(23)
The weight is computed only when d'm coincides with the target token x' (i.e. dag (x") = 1). In this case, the term can be expressed as:
EII )
Nae din (2)) + -11))k)) ko)d (zt)
(24)
h (dmi)) (b+ (bw(()nz))) (1 re)) Nht (dmitt) ((K 1) ) (4
(
(25)
To understand this transition, we first note that dm and 2 are fixed in this scope, while zg is independent across each dimension and follows a uniform distribution. This implies that we only need to consider Tp. For an arbitrary dimension j. the cases can be divided into four possibilities, and the corresponding values of K (z²) + (1 ko) (2*) are as follows:
Case 1. d/, = 2², 2/ = 2,,
Kid mm
(2²) + (1 Ke)6 (z2)
=1.
Case 2. d/ = 21, ma + 2¹,
Ni di (z)) + (1 - 23) (x))
Kit.
Case 3. d/ + 23, 2 2²,
KeOy (2d) + (1 (1-x(16
ko)6, (3))
Ki--
Case 4. d # 23, 20 * 23,
Krd d/ (2d) + (1 - ne) (x²)
= 0.
Note that Case 4 makes the term inside the product [KoS + (1 (+)) equal zero. Thus, we only need to consider 2o for which no dimension falls into Case 4. Among the |K| possible choices of zo, only [K] (dm..) satisfy my 2 for all dimensions where dí * 21. We then classify the remaining cases according to the Hamming distance between To and dmn Note that the maximum value of h4 (zo, dm) is h4 (dms 2). Let & denote an integer in the range to h (dm 2). Then, the number of zig satisfying h4 (ro, dm) = k is (h~ (dm.))))) (K 1) and in this case the product term becomes
II 1 (2*) + (1 (1-))00)))
(dm)))+ ree) N-)
(26)
We can then arrive at the equation above by summing over all possible k. Resuming the proof, the term can be further simplified as follows:
EII
Ke (x') (1 kk)) (2²)
(27)
(dmiz)
(h+ (dm::))))
Σ
(dmii)) (1 rie) N- (dmitt) ((K 1)ree) 1)re)) (dmm))))
(28)
b
hy (dm,a)
(1 Nee) nxe)NN --- (dm
(dm,,)
hh (dms 2)) ((K 1)mc)*
(29)
k
k=
(K - 1)me + 1
hy (z)
(1 N
(30)
lit
by (d
(1 )
1
Kit K
(31)
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We define y := 1+ | K, and by substituting this simplified expression into the noise predictor above, we finally obtain Eqn. TO
Σm= da (x') Exro II, Fr (z + (1 Ke kz) (z1)
P1)((2112)-
(32)
Em=1 Exo I1g Ninoa, (2)) + (1 16p )6 (2)) he (1)
EM=1 Sap (x") (1 -1a) (1+
M
ne 1-ksy
(33)
Σm= (1 -ke)
N (1
I-me K) he (dmm)))
lep (cmmi)))
2m==1 Sasn (101) (1 I K)
M
(34)
he (dm;)))
MM 1
I K))
2m= de h(dmm)z)
M
(35)
EM= γ -h(dmme))
A.2 PROOF OF CLOSED-FORM BACKWARD VELOCITY IN DISCRETE FLOW MODELS
Similarly to the proof of the closed-form forward velocity in Sec. A.1 we start from the backward
velocity in Eqn.
Rss
de 2)
[0x - (Poje(z122)
(36)
Ke
We derive the closed-form noise predictor as follows:
Poje(x1|2) = Σ
dag (x (e 2112)
(37)
Eagg (xr) (1,0,,11,2
(38)
(r((2)
00.30
Expres &eg (xx )pe (10 x1, 2)
(39)
(z(2)
Engins 8eg (x (p x1,
(40)
Σxnr2 Pe (10 21
The last expression is further expanded to:
Eagoon Ong 21,
Poje Po)((1'122 (z²112)
(41)
Exo.,, Pi(20, 21,
Engao Oeg (x) pi (xi) pr
(42)
Exorr P1 (I1) Pt (c (0 211)
D== Eno dag (x')
M
[nve det (2)) (1 k2) Kz) 0gg(3()
(43)
Em=1 Σ 2p 11,- Re 8 de (2d) + (1 rie) (2))
MM
For the denominator, we use the same formula as in Eqn. 31
M
N
M
h (dms+)
Σ
Re65e, (+') + (1 -ke))_ ne)5 = (1 Re) 1+ 1- Rit
Rit K
(44)
m=
1=1
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Next, for the numerator, can rewrite it as:
(x*)
I Ni dez (z)) + (1- Kz) dag [20)
(45)
M
Σ
II
Rnd di (z))+ (1-Kk)
mz))
(46)
==
1
with
J
The i-th index should be considered separately as sb is set to be equal to xf. Separating the j = i term from the product yields
Σ
Kedas (2+) (1- ke)6ez (z')
II
Keddi (x²) +11)++11111 2zy))
(47)
Wi
Big with 1ig
if
[nedas (21)+ +
1ix )5)) (2t)
Ned di (2 (1 1z) (1) (2²)
(48)
2g mane
M
(nodan
Σ with jzf
II
Kedgg (+')+ (1-++k)kk 1000))(('))
(49)
m=1
26g =
Since the i-th coordinate of zo is fixed to x", the summation over so with zi = x² no longer depends on this index. Consequently, when consider the summation only over the remaining coordinates j * i, the resulting expression takes exactly the same form as the computation presented in Sec. Sec.[A.] (Eqn. Eqn. 31).. The only differences are that the effective dimensionality of the product reduced from NV to N - 1. and (ii) the matching count term must exclude the i-th coordinate, yielding h4 (dm, 2) to h+ (dm, 2) 6e., (z*). Reflecting these adjustments, we obtain
h- (dm.:)) -6g (5*)
Σ II 3o with i71 x/=="*
Knd Ke (x²)+ (1 x2)0 (1- -x) (1 11
Ki K
Ket
(50)
and
Σ
[K,da (2+) + (1 (x))
Σ II
No de (2²) + (1 kz)d (z²)
(51)
no with
M
1xt (dm 20) 50
Kgrga (2*) + (1 - kz) (x*)] )] (1 kx)) N- (1+
Rit K5
(52)
Ki
m=1
Kit
5 (2t) + 8ad (2+) (1 k2) (1 (1t
1+
Kt K
b (d) -)) (3*)
(53)
Ki
1
Kit
m=1
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Using this nominator, we can denote the closed-form noise predictor:
1 2) =)- (2)
Σ m=1
Ki I-Re
Saz (2*) + 8gn (2*) (1 - -mma (1
me K
1 1-ki
(=))(<c1==) =
(54)
Σ Σm= (1 - -Ke)
(1+ I-ke
K)
h4 (dmm)))
h+ (d)mm))))
6ap (2))
M
(I+ 1 K
Rit
de (2t) + (x0)
(55)
Kit
Σm (1+ k K)
bn (d
m=1
Krn daz (2') ] + (K - 1)me
g n (dmm)))
hit
8 (2") + (2))
(56)
Kit
Ms
Kit (g8g(22) -1)
M
7/1 (dmm)))
= 6g (x+)
Σ
dxm (2))
(57)
+ (K -1)rs
Em + (d)
Ad
m==
where y = 1 61 K. 1-k1
We can then express the closed-form backward velocity as follows:
0y (axx (x) = h; [6 (x*)) [=])(2"|=)]
(58)
Re (K5 (2¹) -1) M
h- (dm:))
Σ
dog (2*)
(59)
1+ 1+ (K - 1))itt
m'=1 7 (dm))s)
M
m==
ke (K6 (2*) -1
M
h(dmm:))
das
(60)
(K 1)re
2 mm/=1 7 (i(dm))))))
Since = 1 fort = 1, then + co. So this equation is formally is not defined at f = 1. Nevertheless, as lim;-11, the weighted sum over power of 7 is dominated by the maximum term, which converges to 1. Hence, the expression can be rigorously interpreted as lim 62 2), and in practice, this limiting value is used for sampling at
B EXPERIMENT DETAILS
In Tab. 2. we summarize the hyperparameters used in the experiments presented in Sec. covering both training and finetuning configurations for each dataset. All reported samples were generated using the greedy-tail denoiser described in (Sahoo et al. 2025). We employed implementation of the closed-form backward velocity that is optimized at the CUDA level.
For the CIFAR-10 dataset, follow the same setting as baseline (Schiff al. 2025 Tab. 3 reports the FID and IS of baseline and PAIRFLOW measured with 1,000 steps and 50K samples, which are consistent with the results reported in Tab. of (Schiff al. 2025) and PAIRFI.Ow outperforms it.
C ADDITIONAL EXPERIMENTS
C.1 COVERAGE OF TRAINING DATASET BY SAMPLING WITH FORWARD VELOCITY
As discussed in Sec. constructing pairs using the closed-form forward velocity with a training dataset of size [Xi incurs significantly higher cost achieve full coverage compared to using the backward velocity. Let ls denote the number of source samples drawn from the source (prior) distribution, assumed to be uniform in our work. The probability that given element in the training
set is selected is (1- {
Accordingly, we denote by the number of unique samples among
the k draws, whose expectation is: _ (1 - = |x. (1 1 In addition,
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Table 2: Summary of the training settings used in Sec. 5 Specifically, "Sampling Steps" under PAIR- FLOW and ReDi (Yoo et al. 2025) indicate the number of steps taken to generate pairs, "Teacher Update Period" under DCD (Sahoo et al., 2025) denotes the number of fine-tuning iterations between updates, when the teacher model is replaced by the current student model. "# Pairs" under ReDi Yoo
al. 2025) denotes the number of pairs for the fine-tuning.
MNIST-Binary
QM9
CIFARIO
ZINC-250K
Training Iterations
10K
50K
300K 32 x
200K
Data Dimension
28 x
72 256
Batch Size
256
1024
512
Network Architecture
U-Net
Transformer
U-Net
Transformer 92.4M
Parameter Count
25.8M
92.4M
35.8M
PAIRFLOW
Sampling Steps
20
20
64
DCD (Sahoo et al.
2025
Training Iterations
5K
10K
50K
30K
Teacher Update Period
1K
2K
10K
5K
ReDi Yoo
2025)
Training Iterations # Pairs Sampling Steps
5K
10K
50K
30K
10K
20K
10K
20K
256
256
1024
256
Table 3: FID (Heusel et al. 2017) and
Table 4: Total Correlation measure with pairs sampled from UDLM (Schiff et al. 2025) and PAIRFLOW trained on QM9 (Ramakrishnan al. 2014
IS Salimans et al., 2016) of UDLM (Schifi)
al. 2025) and PAIRFLOW the CIFAR-10 dataset (Krizhevsky et al. 2009).
FID
IS
Total Correlation
UDLM
33.65
6.96
UDLM PAIRFLOW
31.87
PAIRFLOW
28.07
7.37
30.72
Table 5: Summary of training set sizes |X,] for each dataset, the number of unique samples k obtained by simulating paths using the closed-fome forward velocity in Eqn. D and the corresponding coverage values: empirical (COV) and theoretically predicted (COVPrea).
QM9
ZINC-250K
MNIST-Binan
CIFAR-10
X11 k
127,190
224,568
60,000
100,000
77,104
140,779
37,711
63,117
COV
60.62%
62.68%
62.85%
63.11%
COV Frod
63.21%
we define the coverage as the ratio between the number of unique elements obtained through this sampling procedure and the training set size: COV = |x/-
To validate our claim in Sec. 4.1 we sample independently drawn from the uniform distribution, along the velocity field defined in Eqn. 9 Using these samples, we evaluate the coverage following the definition above. The dataset sizes, number of unique samples among the generated samples, and the empirical and theoretical coverages are summarized in Tab. 5 These findings indicate that, even when sampling the same number of points the training set size, only about 63% of the training distribution can be recovered in practice. Achieving full coverage would therefore require a substantially larger number of samples, introducing significantly higher computational cost. Motivated by this finding, we instead propose tracing backward from data samples, using a closed-form velocity field that we derive for this purpose (Sec. 4.2
|X,] data points 21 by transporting source samples
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C.2 TOTAL CORRELATION ANALYSIS OF CLOSED-FORM VELOCITY
As in Sec. 3.2 Yoo et al. (2025) demonstrated that iteratively refining the joint distribution of source- target pairs discrete flow models improves few-step performance by reducing the total correlation of the model. In this section, we measure the total correlation following their methodology. Specifically, we perform sampling with neural networks, including UDLM (Schiff al. 2025) and PAIRFLOW trained on QM9 (Ramakrishnan et al. 2014), starting from identical initial states To but with varying random seeds. We randomly select 20,000 initial states z0, and for each zo, we generate 10 samples with step size of 256. As shown in Tab. 4 PAIRFLOW achieves lower total correlation, consistent with the improved performance observed in few-step sampling, as discussed above.
D DETAILED EXPERIMENTAL RESULTS
In this section, we provide the detailed experimental results corresponding to those summarized in Sec. [ For the molecular datasets (QM9 (Ramakrishnan et al.] 2014) and ZINC-250K (Irwin et al. 2)12)), ) we generate 1,024 samples across varying timesteps and evaluate validity, uniqueness, and novelty. Reported values averaged over 10 trials, with standard deviations also included. For the image domain, we report FID on MNNISTBBinar (LeCun et al., 2002). and both FID (Heusel et al.)
2017) and IS (Salimans et al. (016) CIFAR-10 (Krizhevsky et al. 2009). Detailed experimental settings are provided in Sec. 5.1
Results QM9 presented in Tables 6 and 8 reporting validity, uniqueness, and novelty, respectively. Corresponding results on ZINC-250k are shown in Tables 10 and [1] Finally, results for the image datasets summarized in Tab. T2 (FID MNIST-Binary) and Tab. 13 (FID and IS on CIFAR- 10). The FID measured on MNIST MNIST-Binary (LeCun et al. 2002). FID and IS measured on
CIFAR-10 10 (Krizhessky al. 2009). are summarized in Tab. 12 and Tab. 13 respectively.
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Table 6: Validity scores (1) on QM9 (Ramakrishnan et 2014) for various methods across different steps. Best values per column are highlighted in bold.
Method
2
4
16
32
64
MDLM
51.41557
80.0+11.1
154.8+100
341.5411.
130.0
16..9=16.5
16..
UDLM
17.5+32
125.5=118
497.648.1
826.6:10.
953.5+6.1
991.9444.2
1000.1
Random
47.145.7
194.6482
554.2+13.3
858.3+175
962.0+67
989 9776
698.64453
PAIRFLOW
223.4:12.77
416.01124
734.9
921.5:11.0
977.1::39
990.91:599
1000.2:14.5
UDLM DCD
323.0=19.5
530.8
816.6+144
941.1
981.0+448
993.0::3.6
999,
PAIRFLOW DCD
453.8=16.4
685.8=16.7
891.6+1199
963.1
983.7+85
989.3:3.55
993.245.7
UDLM ReDi
59.71888
232.419.2
588.4115.8
849.6114.2
940.518.
967.5.15.2
978.8
PAIRFLOW ReDi
361.01115.2
512.6144.2
775.7 11000
929.1111.6
976.2.145
985.6.16.7
993.117.1
Table 7: Uniqueness scores (1) on QM9 (Ramakrishnan et al. different steps. Best values per column are highlighted in bold.
2014) for various methods across
Method
2
4
8
16
32
64
MDLM
18.9+3.0
28.8+3.3
87.4+84
254.7=10.3
143.8+164
591.0=18.6
666.9+17.5
UDLM
17.5+32
125.4
495.0+8.2
819.5+114
943.0+5.7
990,0#44.7 97..1+5.0
Random
47.1
194.5+8.1
551.2+125
853.1
953.5478
981.1+6.7
989. 989.6+5.7
PAIRFLOW
223.0-112.3
414.7:12.0
731.4169
917.4111.8
971.61143
986.2.15.3
994.815.2
UDLM DCD
320.9:18.77
528.0
808.5112.7
932.3.18.1
970.9
978.31149
087.21443
PAIRFLOW + DCD
451.8:15.7
681.6:11.55
886.5+12.
957.8:17.6
978.7:9.1
983.4+3399
989.0+53.
UDLM ReDi
59.7188
231.61955
581.0115.1
834.71114
917.5199
944.8:16.2
956. 1
PAIRFI.Ow + ReDi
359.5
507.7
765.348.8
913.5+10.2
959.745.8
968.818.4
973.049.
Table 8: Novelty scores (1) on QM9 (Ramakrishnan et al. 2014) for various methods across different steps. Best values per column are highlighted in bold.
Method
2
4
8
16
32
64
MDL.M
15.4
23.642..
56.34552
127.4194
172.2:12.4
206.547.
193.6=10.4
UDLM
13.8#29
52.0+8.2
120.043.8
152.419.1
144.2+1224
147.249.7
145.11190
Random
26.915.0
66.517.1
128.8+13.0
156.8:10.6
145.5:191
146.5117.0
150.148.7
PAIRFLW
68.817.88
85.61110.
109.2:174
96.81999
106.5
108.9494
110.0199
UDLM + DCD
145.9+7.9
137.5499.
185.5+10.9
186.2+13.7
173.41155
168.0=10.4
165.4=11.6
PAIRFLOW DCD
110.316.3
136.1
146.2112.9
139.2.19.6
131.811177
140.1
133.2:179
UDLM ReDi
31.418.0
73.3177.
110.4:1222
126.8
116,3184
120.2.19.0
117.617.1
PAIRFLOW ReDi
84.2+1113
92.0+6.9
101.1496
98.6484
98.8+13.9
95,8+8.3
98.9+7.5
Table Validity scores (t) on ZINC-250K (Irwin al. 2012) for various methods across different steps. Best values per column are highlighted in bold.
Method
2
4
8
16
32
64
MDLM
15.0+440
79,4+4.7
194.6+15.1
351.3+20.6
463.7+17.7
15.79=10.7
610.1
UDLM
0.3105
65.2.182
435.71144
775.1
887.3
921.5
937.3139
Random
0,6119
68.3
351.21158
569.4.1666
611.0.11.3
602.4
571.0.13.2
PAIRFLOW
9.942.3
146.3
533.9
799.419.
873.2:14.1
901.0.14.2
907.8177
UDLM DCD
25.7+447
323,9
718.2+13.5
873.5+8.4
919.8=1000
933.145.9
942.24433
PAIRFLOW DCD
114.9+14.3
436.3-11.5
725.1+11.5
858.2+10.0
896.5+8.2
900.949.5
907.1+13.7
UDLM ReDi
0.7408
75.947.
424.8
734.44186
856.3411.
892.2-10.4
100.1
PAIRFLOW ReDi
46.31663
221.5-:11.0
562.8+12.7
793.6:884
869.3+14.2
897.4:9.1
907.0=11.5
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Table 10: Uniqueness scores (1) on ZINC-250K (Irwin et al., 2012) for various methods across different steps. Best values per column are highlighted in bold.
Method
2
4
16
32
64
MDLM
7.4
32.742.
105.148.
18.3
16.6
14.0
582.4-117.3
UDLM
0.3405
65.2488
435.7=14.4
1955
12.
921.548.5
937.2433.8
Random
0.6209
18.3
351.2215.8
169.4116.6
611.0=16.3
602.4=13.3
571.0=13.2
PAIRFLOW
9.9
146.3:10.4
533.9113.9
799.41922
873.2:11.11
901.0=14.2
907.817.7
UDLM DCD
25.7+
323.9
718.2+13.5
873.5+844
919.8=10.0
933.1+5.9
942.2444
PAIRFLOW DCD
114.9+14.3
436.3=11.5
725.1+11.5
858.2+10.0
896.5+8.2
900.949.5
907.1
UDLM ReDi
0.7108
75.9.7.
424.8
734.318.7
856.3
892.2.110.4
900.011009
PAIRFLOW ReDi
46.3
221.5.11..
562.8 112.7
793.6188
869.3
897.4.19.1
907.0 10.5
Table 11: Novelty scores (1) on ZINC-250k (Irwin al.
2012) for various methods across different
steps. Best values per column are highlighted in bold.
Method
2
4
8
16
32
64
MDLM
3,8+2.7
24.2+226
84.5+7.
228.3+17.11
372.0=15.2
494.5=15.2
569.3+17.1
UDLM
0.3+0.5
65,2+882
435.7=1444
175.
887.3=12.7
921.348.8
936.9+4.11
Random
0,640.9
68.3+11077
351.2+15.8
569.4416.6
611.0=16.3
602.4=13.3
571.0=13.2
PAIRFLOW
9.9
146.3:10.4
533.9413.9
799.419.
873.2:14.11
901.0=14.2
907.84777
UDLM DCD
25.7
323.9:12.
718.2413.5
873.5484
919.7+999
933.015.8
942.2114
PAIKFLOW DCD
114.9:14.3
436.3+11.5
725.1+11.5
858.2+10.0
896.418.3
900.949.5
907.1+13.7
UDLM ReDi
0.7408
75.9479
424.8416.6
734.348.7
856.3411.
892.1
900.0+10.9
PAIRFLOW + ReDi
46,3+63
221.5::11.0
562.8+12.7
793.5
869.3:14.2
897.449.1
907.0+10.5
Table 12: FID (L) on MNIST-Binary (LeCun et al. Best values per column are bolded.
2002) for various methods across different steps.
Method
4
16
32
64
MDLM
204.66 64
159.26
103.74
54,41
28.51
12.31
7.01
UDLM
130.57
42.54
11.25
5.70
4.69
4.77
5.01
Random PAIRFLOW
128.57
36.59
9.41
5.60
5.00
5.10
5.19
40.58
15.61
8.50
5.97
5.55
5.24
5.17
UDLM DCD
53.84
16.09
8.06
7.46
7.12
6.52
6.65
PAIRFLOW DCD
19.51
14.20
11.47
9.28
7.75
7.82
8.42
UDLM ReDi
18.44
10.35
8.11
6.56
6.49
PAIRFLOW + ReDi
12.90
9.38
8.54
6.85
6.79
6.96
6.94
E EXPERIMENT ON CONTINUOUS FLOW MATCHING
Alongside our main experiments in the discrete setting, also demonstrate the potential of our method to extend to continuous domains, as illustrated by the toy experiment presented below. Here, we denote by PAIRFLOW continuous flow model trained on source-target pairs constructed using the continuous variant of the algorithm described in Sec. 4.2
E.1 CLOSED-FORD VELOCITY IN CONTINUOUS FLOW MATCHING
Setup. Let Xo Po (source), X1 q (target) be independent random variables in N and consider the linear probability path
X, (1 1) LX1, t EE [0,
(61)
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Table 13: FID (U) and IS (f) on CIFAR-10 (Krizhessky al. 2009) for various methods across different timesteps. Best values per column are bolded.
2
8
32
64
128
256
512
1024
Method
FTD (L)
MDLM
407.31
359.97
340.43
340.98
321.79
228.34
131.63
77.44
52.89
42.88
UDLM
340.47
321.98
255.95
151.60
91.31
62.43
49.06
42.26
40.18
37.63
37.60
Random
328.83
314.64
220.16
127.66
82.76
58.42
45.18
39.50
36.78
35.16
35.07
PAIRFLOW
269.87
260.29
192.62
112.78
73.87
52.22
40.06
34.42
33.83
32.42
31.85
UDLM DCD
318.33
282.38
204.15
108.30
75.56
69.87
74.62
79.35
84.34
85.79
87.87
PAIRFLOW DCD
223.86
190.15
139.46
95.91
82.11
80.70
86.59
100.16
114.25
123.41
129.45
UDLM ReDi
251.02
212.20
171.24
148.18
138.18
131.94
125.81
123.77
123.39
121.52
121.06
PAIRFLOW ReDi
275.09
250.87
184.45
119.04
89.46
74.46
66.22
61.83
61.35
60.10
59.53
Method
IS (t)
MDLM
1,21
1.22
1.24
1.31
1.52
2.47
4.03
5.08
5.66
6.18
6.38
UDLM
1.32
1.48
3.54
4.85
5,90
6.25
6.66
6.87
6.81
Random
1.37
1.52
2.47
4.01
5.24
6.08
6.56
6.75
6.87
7.07
7.05
PAIRFLOW
1.72
1.80
2.70
4.27
5.61
6.23
6.86
7.00
7.12
7.14
7.33
UDLM DCD
1.49
1.60
2.23
3.90
5.09
5,45
5.42
5.16
5.09
5.07
4.84
PAIRFLOW DCD
2.21
2.38
3.23
4.38
4.99
5.13
4.97
4.48
4.26
3.82
3.81
UDLM + ReDi
1.85
2.35
2.96
3.37
3.52
3.67
3.86
3.96
3.93
3.99
3.94
PAIRFLOW ReDi
1.80
1.95
2.91
4.18
5.02
5.65
5.88
6.03
6.06
6.09
6.30
Table 14: FID of PAIRFLOw on MNIST (LeCun al. 2002) with continuous values, measured using FID) over 50K samples across various timesteps. Best values are bolded.
Method
16
32
64
CondOT 398.43
91.17
27.34
10.80
5.81
3.99
3.16
PAIRFLOW 74.89
14.40
6.89
3.78
2.70
2.42
2.37
CondOT+RE
32.70
9.12
5.46
4.24
3.93
3.56
3.24
PAIRFLO++RF
28.61
4.15
3.01
2.87
2.89
2.91
2.94
For flow matching with the linear path Eqn. 61
the optimal velocity field equals the conditional drift:
(x) = E [X+ Xx = x].
(62)
We derive a closed form of Eqn. 62 that is directly computable from po and
Derivation. By Bayes' rule with Dirac constraint for the linear relation Eqn. 61
p( Fı Xx x) po(ar)) g(x1) 8 (x (1 - (1-t)xo 1)xro-10 txx).
(63)
Integrating out uy using 8(Ay- - b) = |det A1 8 A-16) with A = -(1- 11-10 1)l gives
x l.r
p(a1 X,
((x1) 1)- no
(64)
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Published as a conference paper at ICLR 2026
Hence,
(x(x) // (x1 ro) p( 21 Xx = x)) drodr1
(65)
// p( Xe = x) dæo dx1
/| (I1 mo) po po( ( (xr (1 t) txx) drodæ
(66)
|| po po(31o) ((01)) 8 (xr ()50 121) dxn day
I g(01) po
- (I1
-
dæ
(67)
((x1)
( dx
Observing 21 - z-tz
I-t I- we obtain the compact form
=1-
/ ((x1)) 2-(xz
Po
I-t
(z)
x))
dr
(x(x)=
(68)
((01) po
dæ
I-t
When we have dataset with samples {dm 1m the target distribution is approximated by the empirical measure ((x1)
1 M Σm dd.,. (x1), then Eqn. 68 reduces as follow:
1 EM Po
2-1dm I-t
(dm x)
Mx(x)
(69)
1-t.
M
g-Idm
Em=11 Po
l-t
When Po is standard Gaussian, (1) (2m)
-D/2
D/2 exp (-
(1=11)1 the normalizing constants cancel
Eqn. 69 yielding the closed form velocity:
EM exp
- x Ida
1-1
(dm - x))
(x))
(70)
Σ ΣM exp
x- 1-t
2
This formulation has already been introduced in previous works (Karras et al. 2022 Bertrand et al. 2025); however, to the best of our knowledge, no prior work has extended this idea to designing couplings for accelerating flow models using the re-flow technique (Liu Gong 2023). In the continuous domain, the backward velocity can be obtained directly by flipping the sign of the forward velocity. In contrast, in the discrete domain, corresponding expression does not converge as lim,-1, and thus the backward velocity cannot be employed for sampling starting from data points. Therefore, in this section, we perform experiments using the forward velocity.
E.2 CONTINUOUS FLOW MATCHING ON MNIST
We train rectified flow models on MNIST (LeCun et al. 2002) using two pairing strategies: (i) independent pairing (baseline) and (ii) closed-form pairing described in Sec. E. We adopt CondOT (Lipman et al. (203)) as our base flow model, which is originally trained with independent pairing. We denote the variant of CondOT trained pairs generated by the closed-form forward velocity as PAIRFLOW. To enable a few-step sampling, we subsequently apply rectification distillation (ReFlow (Liu & Gong 2023) each pretrained model, denoted by the suffix "+RF".
We use an NCSN++sttyle U-Net backbone (Song et al., 202T) with base width of 64 and downsampling stages (doubling channels at each stage), optimized using Adam (Kingma & Ba 2014
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with learning rate of 2 x 10-4 The pretraining takes 500 epochs. The distillation stage requires 200 epochs with a learning rate of 2 x 10-5.
Tab. 14 summarizes performance various sampling steps. Without distillation, closed-form pairing PAIRFLOW) yields significantly better FID in the few-step settings and maintains the performance in the stny settings, relative to the baseline. With distillation (ReFlow (Liu & Gong (023), our method still shows better performance: PAIRFLOW+RF achieves a lower FID in every sampling budget than ReFlow applied to the baseline. These results show that closed-form pairing benefits both undistilled and distilled flow models, with especially large gains when the sampling steps are small.
E.3 CONTINUOUS RECTIFIED FLOWS ON DIMENSION- VARYING SYNTHETIC DATA
To assess scalability, we construct N-fold product of the standard two-moons distribution, yielding a dataset in R2N, We consider dimensions E {2, 4, 8, 16, 32, 64, 128, 256} (i.c., d = 2N) and train rectified flow models with and without closed-form pairing under common training setup. The architecture is a simple transformer-basee encoder with depth 8, where the hidden size increases with dimension as 32, 64, 128, 192, 256, 384, 512, 768, respectively.
For the synthetic experiments we report the Chamfer distance (log scale) between 50,000 training datapoints and 5,000 generated samples. Since the dataset is -folld product of 2D two-moons, Chamfer distance is computed using only the first two coordinates to keep the metric scale consistent across d and measure fidelity to the base 2D geometry.
Fig. 7 shows the quantitative results. At low dimensions, closed-form pairing yields substantial improvements over the independently paired baseline. However, as the data dimension increases, we observe that the magnitude of the improvement decreases. This trend suggests practical limitation of pairing for high-dimensional continuous data.
F ADDITIONAL QUALITATIVE RESULTS
In addition to Fig. 4in in the main paper, we further visualize the 1-step and 2-step generation results
for MNIST-Binary (LeCun al. (002) in Fig. 9 and Fig. [1] As discussed in Sec. 5.3 PAIRFLOw outperforms the base models (Shhff et al. 2025 Sahoo et al. 2024) and achieves comparable quality to the base models combined with acceleration methods (Sahoo al. 2025 Yoo et al. 2025).
Additional visualizations for CIFAR-10 (Krizhevsky et al. (2009) with 64- and 256-step generations shown in Fig. demonstrating that our method outperforms the other base models.
G REFLOW ITERATION RESULTS
This section details the results of applying the iterative reflow procedure (Liu & Gong 2023 Yoo et al. 2025) on the QM9 dataset (Ramakrishnan et al. 2014). We generated 1,024 samples across various timesteps and reflow iterations, assessing validity, uniqueness, and novelty. The configuration for subsequent reflow iterations follows the same protocol as our main experiment. As shown in Tab. 15 Tab. 16, and Tab. 17 all metrics are averaged over 10 independent runs with standard deviations provided. Additional reflow steps are denoted by "+ ReDi" (Yoo et al. 2025). We observed that iterative reflow consistently enhances few-step generation capabilities. Notably, PAIRFLOW demonstrates superior performance over the baseline; outperforms UDLM not only at equivalent iteration levels but also surpasses UDLM with multiple reflow iterations even when PAIRFLOW uses one no additional iterations.
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Table 15: validity scores (1) on QM9 (Ramakrishnan et al., 2014) for UDLM and PAIRFLOW across varying rectification steps and NFEs (1 to 64). The best and second-best values per column are highlighted in bold and underlined, respectively.
Method
2
4
8
16
32
64
UDLM
17.5+3.2
125.5+11.8
497.6=8.3
826.6+10.3
953.5#:6.1
991.9:4422
1000. 1
ReDi
59.7188
232.419.2
588.4415.8
142
940.548.5
967.5455
978.815.2
ReDi
160.9-118
368.0+11.
673.3411.2
16.
945.4118.1
444.9474
975.8+8.0
+ ReDi
280.246.7
470.7+18.6
742.2+18.7
897.9=9.4
945.3455.
965.04:599
972.6+7.1
PAIRFLOw
223.4:12.7
416.0:1224
734,947.2
921.5+11.0
977.111399
990.91:5.9
1000.2:::.5
ReDi
161.0::11,2
512.6:144.2
775.7+110.0
929.11:116
976.2:14.5
985.6.16.77
197,1
ReDi
443.4=134
598.6+1844
823.0+16.0
935.2=777
969.0#5.7
98..1+600
989 0+
ReDi
529.2=10.6
688.0+11.3
863.6-9.5
945.9=4.5
975.7=6.1
982.6+7.1
990.2+7.2
Table 16: Uniqueness scores (1) QM9 RRamakiishaan al. 2014 for UDLM and PAIRFLOw
across varying rectification steps and NFEs (1 to 64). The best and second-best values per column are highlighted in bold and underlined, respectively.
Method
1
2
4
8
16
32
64
UDLM
17.5.332
125.4:11.7
495.018.2
819.5
943.0.15.7
979.1
990.0.1..7
ReDi
59.7+88
231.6#95
581.0+15.1
834.7=11.
917.5+99.
944.8+6.2
956.1+5.2
ReDi
159.4=10.9
363.3+12.2
657.249.3
846.3+118
910.8+7.3
930.549..
938 8111771117
ReDi
275.64772
456.3+17.7
712.1+15.7
856.8+10.3
899.0#4499
907.0=11099
918.9+3.7
PAIRFlow
223.0-12.3
414.7:11.0
731.41669
917.4
971.6.14.1
986.2.15.3
994.8
+ ReDi
113.
507.744430
765.3488
913.5+10.2
959.7445.8
968.8118.
973.0119.6
ReDi
437.9=13.8
586.7+17.9
801.0+155
906.5+10.1
939.1+6.2
955.947..2
960.2+66.5
ReDi
116.9
662.8+11.44
828.7=8.4
903.2=447
932.5#95
942.8+44.8
949.6:884
Table 17: Novelty scores (1) on QM9 (Ramakrishnan et al. 2014) for UDLM and PATRFI.Ow across varying rectification steps and NFEs (I to 64). The best and second-bess values per column are highlighted in bold and underlined, respectively.
Method
1
2
4
8
16
32
64
UDLM
13.8+229
52.0+
120.0
152.4=9.1
144.2=12.4
147.2:19.7
145.1
ReDi
31.41880
73.3
110.4::2.2
126.8 889
116.3:184
120.2::9.0
117.6.7.1
ReDi
61.11665
103.018.6
129.0
124.6:12.5
128.3:15.6
128.4410..
122.819.4
ReDi
91.2:192
116.4118.
127.2.18.3
124.2-9.3
121.9:16.0
117.0:14.9
121.818.00
PAIRFLOw
68.8+7.88
85.6+1000
109.247.8
96.8+9.9
106.5=12.5
108,949.4
110.0+9.9
ReDi
84.2.11..3
92.0.16.9
101.119.6
98.618.4
98.8
95,8
98.917.5
+ ReDi
100.8:17.7
109.5188.7
101.01668
101.1
94.5116.
95.6119.6
99.0
ReDi 114.619.44
108.01.5.9
106.51889
96.2.19.8
95.61190
94.8117.1
95.718.2
H SUBSET PAIRING RESULTS
In this section, we present comprehensive experimental results from applying our subset-partition pairing technique to the ZINC-250K molecular dataset (Irwin et al.] 2012). Following the same protocol as our main experiments, we generated 1,024 samples across varying timesteps and par- tition counts to evaluate validity, uniqueness, and novelty. The only deviation from the standard method the pairing strategy; here, we calculate the closed-form backward velocity exclusively within each subset to reduce the computational cost of Eqn. 10 All reported metrics, summarized in Tab. I8 Tab. 19 and Tab. 20 are averaged over 10 independent runs with corresponding standard deviations. Additionally, results demonstrate that our subset-pand pairing algorithm effectively reduces the computational time for pairing, while maintaining performance comparable the full-set baseline.
report the pairing- time cost for each subset-partine
configuration. The
24

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Published as a conference paper at ICLR 2026
Table 18: Validity scores (1) on the Zinc-250k dataset (Irwin al. 2012) evaluated across different subset partitions and NFEs (1 to 64). The best and second-best values per column are highlighted in bold and underlined, respectively. TAARFLOW denotes the runtime (in minutes) of each configuration, measured in wall-clock time using an RTX A6000 GPU.
Method
TRaRFlow
2
16
32
64
UDLM
0
0.3
65.2+88.
435. 134.7414.4 775. 19.5 12.7 8,1
337
Random
0
0.6:0.9
68.3+10.7
351.2+15.8 569,4 (1t.o6661...100 11.3
602.443333322223333333333 571.0#13.2
PAIRFlow (Full)
13m
9.942.3
10.4 799.419.2
873.2=
901.0:14.2
907.8 2.777
PAIRFloww (2-Sub)
6m
10.64228
882.647.1
132.4=13.911.554
PAIRFLW (4-Sub)
2.9m
12.1
142.5#5.2
780.94144
858.947.9
886.748.7
19.
PAIRFLOW (8-Sub)
1.5m
12.3:12.5
141.1=6.7
510.9417.2 16 8.. 886.949.6
69.,5
Table 19: Uniqueness scores (1) Zine-250k (Irwin et al. 2012) evaluated across different subset partitions and NFEs (1 to 64). The best and second-best values per column are highlighted in bold and underlined, respectively. TaAnFlOW denotes the runtime (in minutes) of each configuration.
Method
TPaRFLOW
1
2
16
32
64
UDLM
0
0.3:+0.5
65.2+t8.2
435.7
775. 19.5 887. 177.3:::2277
8.5
937 22
Random
0
0.6:10.
68.3+11.7
351.241558 569.4116.6
111
60..4113.3
571.0
PAIRFlow (Full)
13m
9.9+2.
5333.
799.4
873.2
901
14.2 907. 8
142
PAIRFlow (2-Sub)
6m
10.6#288
145.7411653306-2206b0000
802.5+6..
882.6+7.1
902.4=13.4911.544
PAIRFLOW (4-Sub)
2.9m
12.113.3
142.5
01.7111.07809910
858.917.
886.718.7
19.
PAIRFLOw (8-Sub)
1.5m
12.3
141.116.7
510.9117.2 766.8 857.5
886.919.6 896.516.3
Table 20: Novelty scores (f) on Zinc-250k (Irwin et al., 2012) evaluated across different subset partitions and NFEs (1 to 64). The best and second-best values per column are highlighted in bold and underlined, respectively. T TRARFLOW denotes the runtime (in minutes) of each configuration.
Method
TramFLOw
32
64
UDLM
0
0.3
65.2.18.2
775.1 887.3 12.7 921.3
936.9 1..1
Random
0
0.61099
68.3
351.2.1558 169.4116.6
602.4.13..
571.1111111110111 571.0
PAIRFLOw (Full)
13m
9.9423
146.3:114
799.4:922
873.2+14.1 901.0+11.22
907. 8777
PAIRFLOW (2-Sub)
6m
10.6+288
145.7:113.53066-222
802,5
882.6+7.1
902.4=134911.5--4
PAIRFlOww (4-Sub)
2.9m
12.1
142.5
509.7111.0
780.9
85..8 818..
886.74887
899.01 10.5
PAIRFlOw (8-Sub)
1.5m
12.3
141.1
510.9117.2 766.8112. 857.5
886.9196
896.516.3
APPLICATION FOR MORE COMPLEX SYSTEMS
In this section, we evaluate our method on nigher-dimensione dataset. Specifically, we use the FFHQ (Karras et al., 2019) dataset, downsampled to 64. Following the same protocol as in our main experiments, we generate 5,000 samples across varying timesteps and report the FID computed against the training set. The results of this experiment are provided in Tab, 21 All training hyperparameters are kept identical to those used in the CIFAR-10 experiments described in Section
Table 21: Comparison of FID scores (L) FFHQ (Karras et al. 2019) downsampled to 64 x 64 resolution across extended NFE steps (1 to 1024). Best values per column are highlighted in bold.
Method
2
4
64
128
256
512
1024
UDLM
403.04
399.26
363.97
273.31
153.71
97.87
74.85
63.93
59.28
55.99
55,30
PAIRFLOW
394.14
368.36
329.13
243.88
140.05
90.85
69.67
59.86
56.52
54.19
53.18
25

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We adopt the LMIB (Chelba al. 2014) dataset to evaluate method under substantially larger vocabulary size and training corpus. The text corpus is segmented into sequences varying lengths
(N 16, 32, 64, 128), while keeping the total number of training samples fixed ([X, x 3.5M). To assess generation quality, we compute generative perplexity using GPT-2 Large and entropy on 1,024 generated samples for each NFE setting. The results summarized in Tab. [2] and Tab. 23 For
training, we follow the network hyperparameter configuration of (Schiff' et al. 2025 ) modifying only the number of training iterations for each sequence dimensionality.
Table 22: Generative Perplexity (U) on LMIB (Chelba et al. 2014) measured with GPT2-large across varying lengths (N) and their corresponding training iterations (Iter.) over NFE steps 4 to 1024. Best values highlighted in bold.
Tter.
Method
4
128
256
512
1024
UDLM
299.18
225,92
207.17
195.82
200.77
197.04
199.12 12
195,37
198,22
16
200k
PAIRFLOW
242.22
208.04
200.99
190.36
191.74
199.45
188.84
196.91
198.12
UDLM
263,93
192.78
167.85
167.49
155.68
150.52
152.40
151.74
154.02
32
200k
PAIRFLOW
218.48
172.27
156.35
150.53
143.83
145.77
142.54
141.04
147.57
UDLM
214.07
150.59
130.49
120.19
117.90
116.23
112.24
113.77
115.11
64
400k
PAIRFLOW
174.78
138.94
123.06
115.71
114.73
112.92
111.29
107.06
110.83
UDLM
169.61
123.48
105.13
98.94
97.89
94.92
93.75
94.12
93.59
128
600k
PAIRFLOW
167.90
121.09
102.16
96.61
93.93
91.51
90.21
89.09
89.07
Table 23: Comparison of Entropy (1) on LMIB (Chelba et al. 2014) across varying lengths (A) and training iterations (Iter.) over NFE steps to 1024. Best values are highlighted in bold.
N
Iter.
Method
4
8
16
32
64
128
256
512
1024
UDLM
2.46
2.49
2.50
2.50
2.50
2.51
2.50
2.51
2.50
16
200k
PAIRFLOW
2.48
2.49
2.51
2.52
2.52
2.53
2.52
2.53
2.52
UDLM
3.05
3.09
3.12
3.13
3.13
3.13
3.13
3.13
3.13
32
200k
PAIRFLOw
3.06
3.12
3.13
3.14
3.15
3.15
3.15
3.15
3.16
UDLM
3.57
3.63
3.67
3.68
3.69
3.70
3.70
3.69
3.70
64
400k
PAIRFLOw
3.57
3.65
3.69
3.70
3.71
3.71
3.71
3.72
3.71
UDLM
3.98
4.09
4.14
4.16
4.17
4.17
4.17
4.18
4.18
128
600k
PAIRFLOW
4.00
4.11
4.16
4.18
4.19
4.20
4.20
4.19
4.20
ANALYSIS FOR THE OVERFITTING IN IMAGE DOMAINS
In this section, evaluate method using FID computed on the test sets of two image domains:
CIFAR-10 10 (Krizhevsky et al. 2009) and MNIST-Binars (LeCun et al. 2002). For CIFAR-10, we additionally report FID scores measured with DINOv2 (Oquab et al.) 2024). The overall results summarized in Tab. 24bdd2 Tab, 25 Across all evaluation metrics, the performance trend is consistent with our main Madings-PAiRFPEE especially strong gains in the few-step generation regime.
delivers improved generation quality over the baseline, with
Table 24: FID and FID-Dino scores (L) on test dataset for CIFAR-I0 (Krizhevsky et al. comparison across extended NFE steps (1 to 1024). Best values are bolded.
2009
NPE
32
128
512
1024
Method
FID (J)
UDLM
306.45
296.77
266.64
178.11
114.04
80.40
62.70
53.83
50.96
47.61
47.16
PAIRFLOW
235.65
247.18
209.94
137.16
94.24
67.53
51.48
43.79
42.44
40.85
39.92
Method
FID-DINOV2
UDLM
2448.46
2410.65
1975.58
1344.44 44
959.39
755. 80
646.47
598.53
598.54
553.17
560.75
PAIRFLOW
2059.26
1988.21
1626.08
1127.34
828.09
623.89
530.70
486.59
484.27
470.33
470.05
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Table 25: FID scores (4) on the MNIST-Binary (LeCun al.
2002
test set across various NFE steps
(1 to 64). Best values are bolded.
Method
4
8
32
64
UDLM
129.05
42.17
11.42
6.18
5.13
5.37
5,50
PAIRFLOW
42.87
17.37
9.62
6.36
5.80
5.51
5.24
UDLM+DCD
$7.85
19.23
9.82
8,41
7.87
7.12
7.38
PAIRFLOw+DCD
19.56
13.06
10.90
8.55
7.40
7.22
7.85
UDLM+ReDi 19.08
10.79
8.77
7.01
6.89
6.57
6.61
PAIRFLOW-ReDi 13.73
9.59
8.98
7.24
7.22
6.98
7.12
We further assess potential memorization by measuring nearest-neighbor distances with respect the training set. For MNIST-BBinay (LeCun et al. 2002), we compute pixel-wise le distances, whereas for CIFAR-10 (Krizhevsky et al., 2009), evaluate both &z distance and cosine similarity between features extracted using DINOv2 (Oquab et al.) 2024). As summarized in Tab. 26 and Tab. [27] across all evaluation settings, the nearest-neighbor distances of PAIRFLOW are comparable to slightly larger than those of the baseline. These results support the conclusion that our method does not suffer from severe overfitting or excessive memorization of the training data.
Table 26: Comparison of l and Dino (Oquab et al. 2024) Cosine nearest neighbor distance on the CIFAR- (Krizhevsky et al., 2009) training set across extended NFE steps (1 1024). Best values are bolded.
NEE
2
8
16
32
64
128
256
512
1024
Metric
Eg (1)
UDLM
7.97
9.13
10.06
10.06
9.40
8.94
8.63
8.41
8.29
8.29
8.22
PAIRFLOW
8.03
8.76
9.42
9.56
9.18
8.75
8.63
8.55
8.51
8.52
8.52
Metrie
Cosine(DINOv2) ())
UDLM
0.242
0.227
0.231
0.241
0.237
0.238
0.235
0.237
0.237
0.235
0.237
PAIRFLOW
0,245
0.228
0.235
0.239
0.236
0.232
0.232
0.230
0.232
0.235
0.233
Table Comparison of le nearest neighbor distance on the MNIST-Binary (LeCun et training set across extended NFE steps (I to 64). Best values are bolded.
2002
Method
2
4
8
16
32
64
UDLM
8.18
7.06
6.55
6.42
6.27
6.29
6.25
PAIRFLOW
7.36
6.95
6,63
6.42
6.26
6.34
6.30
UDLM+DCD
7.24
6.78
6.57
6.50
6.31
6.40
6.39
PAIRFLOw+DCD
7.64
7.34
7.11
6.91
6.74
6.76
6.79
UDLM+ReDi
7.14
6.81
6.49
6.24
6.10
6.14
6,08
PAIRFLOw+ReDi
6.84
6.57
6.35
6.09
5.95
5.97
5.96
K LIMITATIONS AND FUTURE WORK
We hope this work initiates broader discussion reducing training compute while still enabling fast generation in generative models. Such efficiency can have a significant impact, from reducing energy consumption in training large-scale generative models to contributing to the democratization of foundation model development.
A natural follow-up question to work is whether the same idea be applied to continuous Flow Matching (FM). We have evaluated this extension on continuous FM models, with results provided in App. Our experiments with synthetic data show that the method is effective for
27

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Published as a conference paper at ICLR 2026
relatively low-dimensional data, while its advantage bit diminishes for nigher-dimmnsional data. We will further investigate the effect of our method on continuous data, where we hypothesize that emphasize that
substantially larger number of source-target pairs will be required. Nonetheless, even in this initial exploration of accelerating flow models through well-aligned pairing, PAIR FL.Ww is particularly well-suited for low-dimensional discrete data, which includes many forms of scientific data such as molecular and protein structures.
28

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Published as a conference paper at ICLR 2026
Synthetic Data (D-2)
Synthetic Data (D=4)
CouAOT Paiiflon
Condor haidflow
10
10'
Chanferr Distance
Chanfer Distance
10"
10"
Sampling Steps
Sampling Steps
Syathetic Data (D=8)
Synthetic Data (D=16)
CanaOT PaiFFow
Conko
hid
10"
Chamfer Distance
Chamfer Distance
10'
to"
10
10'
14'
16
12
64
128
16
32
64
128
Sampling Steps
Sampling Steps
Synthetic Data (D-32)
Synthetic Data (D=64)
10"
CanAOT
Canfor
Priffow
Priffoww
In'
Chamfer Distance
Chamfer Distance
10
16
10
Sampling Steps
Sampling Steps
Synthetic Data (D-128)
Synthetic Data (D-256)
10'
Condo
Cond3
Pai:Flrn
Pai
Chanfer Distunce
10'
Chumfer Distance
10"
w/-
10"
32
32
125
Sampling Steps
Sampling Steps
Figure 7: Step-wise performance analysis on synthetic N-fold two-moons. Each plot shows Chamfer distance (log scale) vs. sampling steps for different dimensions d. Closed-form pairing (PAIRFLOW) consistently outperforms standard CondOT especialyy at few sampling steps-while the margin shrinks as d increases, indicating diminishing gains in high dimensions.
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Published as a conference paper at ICLR 2026
MDLM
64 steps
UDLM
PAIRFLOW
MDLM
256 steps UDLM
PAIRFLON
Figure 8: Additional qualitative results of 64-step and 256-step generation on CIFAR-10 10 (32 x 32).
MDLM
L
UDLM
PAIRFLOW
C 3
UDLM DDCD
PAIRELOW DCD
0 7
3
0
UDLM + ReDs
30520997
PAIRFLOW
ReDi
Figure Additional qualitative results of 1-step generation on MNIST-Binary (28 28).
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MDLM
5 5
UDLM
PAIRFLOW
5
UDLM +DDD
0 
PAIRFLOW +DDD
9 0
UDLM ReDi
4
5
FAIRFLOW + ReDi
Figure 10: Additional qualitative results of 2-step generation MNIST-Binary (28 28).
31


Reviews:
[{"review_id": "fjgoo1U6A0", "reviewer": "Reviewer_Vm4B", "summary": "This paper proposes a new training method to accelerate discrete flow matching sampling, achieving good results on various tasks, especially CIFAR-10 generation.\n\nThe method, called PairFlow, is conceptually similar to ReFlow (Rectified Flow), which is commonly used in the continuous domain. However, PairFlow differs from ReFlow in that it starts from the true data domain and moves backward toward the noise domain using a backward velocity field.\n\nIn general, this paper aims to generalize the results obtained in the continuous domain—such as the closed-form velocity formulation and the ReFlow training approach—back to the discrete domain.", "questions": "Could you please check and rewrite the proof in Appendix A.2? I think the overall approach is sound, but there are too many careless errors. If you can make it accurate and typo-free, I will raise my score.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 4, "strengths": "- Starting from real data, back to noise. This is just a little bit change of the reflow. But I think it is a cool idea. Intuitively, if your reflow model is not trained or sampled well, then the quality of your sampled-images maybe much worse than that of real data. But starting from clean image, you can avoid this.\n- this paper has some smart ideas to get the theoretical results, for example, how to get the closed-form formula of forward velocity (A.1). Although the proof about the closed-formula of backward velocity has some flaws, but the direction of that proof (A.2) I think it's correct.", "weaknesses": "The main weakness of this paper is dued to its proof of the closed-form formular of the backward velocity.\n\n- line 770 ~ line 792, has so many typos. Those typos make the proof unreadable, though although I can grasp the approach the authors intended to use.\n- I do some calculations, and find some part results are right, but the proof process is flawed. There are numerous algebraic mistakes (e.g., signs flipped, missing ±1 terms, and other careless errors) that makes people suspect the correctness of the results.", "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": "BwHEJQeoAx", "reviewer": "Reviewer_au4X", "summary": "PairFlow introduces a pre-processing step for training discrete flow models to enable few-step sampling without the requirement of a pre-trained teacher model. The approach achieves this through around a closed-form inversion step for discrete flow models, enabling the development of source-target pairs which can be used training PairFlow.", "questions": "- Using the closed-form velocity field to generate source-target pairs may lead to the distilled model being more likely to overfit to the training data. This has been observed in continuous settings by Bertrand et al. (2025). Proving that this isn't the case would be useful across the considered datasets (my concerns around the low novelty for QM9 validate these concerns). \n- Do additional Reflow steps (deriving a new analytical vector field on reflow samples from each distilled model) applied to subsequent trained models improve performance across metrics? In other words, does the concept of straightening the paths still apply in the discrete setting with multiple iterations of distilled models? \n- How does the approach perform on more complex systems, like ImageNet? Since it's observed that the continuous approach fails for higher-dimensional systems, does this also hold for the discrete setting? \n- How does the approach perform compared to continuous flow matching methods? A contrast on the already-shown image baselines, e.g., MNIST, CIFAR10, etc., would be useful.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The approach is mathematically interesting, and offers significant practical utility, as it eliminates the need for a pre-trained teacher for distillation while accelerating inference through few-step sampling in discrete settings.", "weaknesses": "Deriving the closed-form velocity field although possible requires summing over all the training data, which can become significant for large systems. Further, it is possible that using the analytical vector field to determine source-target pairs leads to a distilled model that overfits to the data due to the way the analytical field is obtained. It would be useful to demonstrate that this in fact does not occur. There is some evidence to suggest that this may be happening (novelty of molecules on the molecular datasets is the worst compared to other baselines).", "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": "TsexBRTj5P", "reviewer": "Reviewer_vdPJ", "summary": "This paper presents PairFlow, a novel and highly efficient preprocessing method for training Discrete Flow Models (DFMs) that enables high-quality generation with very few sampling steps. Inspired by ReFlow and ReDi, the authors uses coupled source and target distributions for training. The core innovation is the derivation of closed-form forward and backward velocity fields for DFMs, which allows for the direct construction of optimized source-target data pairs from a collection of samples from the target distribution, without needing a pretrained teacher model.", "questions": "PairFlow now focus on uniform source distribution. What's the difficulties of (or reasons of not) considering a more general class of source distributions?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-motivated and easy to follow.\n2. The derivation of the closed-form velocity field of DFM is a significant theoretical contribution. \n3. The proposed method has a huge improvement in computation complexity.", "weaknesses": "1. The continuous flow experiments in Appendix E.3 suggest that the advantage of closed-form pairing may diminish with increasing data dimensionality. A brief discussion and further experiments on the scalability of the discrete PairFlow method to very large vocabularies and sequence lengths (Image datasets of higher resolution or language modeling) would be beneficial for setting expectations for future applications.\n2. Typos in Line 127-128: the codomain of $p_t(\\cdot)$ and $v_t(\\cdot)$.", "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": "lCw247fg5e", "reviewer": "Reviewer_PZ3N", "summary": "This paper introduces PAIRFLOW, a lightweight preprocessing method that accelerates sampling in Discrete Flow Models (DFMs) without relying on pretrained teachers or costly finetuning. DFMs, though powerful for modeling discrete data such as molecules and images, typically require many iterative steps for generation. PAIRFLOW addresses this by deriving closed-form forward and backward velocity fields that allow efficient pairing of source (prior) and target (data) samples directly in discrete spaces, in terms of the Hamming distance. This pairing enables the model to learn “straighter” probability paths, drastically reducing the number of sampling steps needed. Despite its minimal computational overhead (≤1.7% of total training cost), PAIRFLOW achieves or surpasses the performance of state-of-the-art distillation-based methods like ReDi and Discrete Consistency Distillation, across both molecular (QM9, ZINC-250k) and image (MNIST-Binary, CIFAR-10) benchmarks. Furthermore, models trained with PAIRFLOW serve as stronger bases for later distillation, achieving additional speed and quality gains, suggesting a general strategy for efficient few-step generation in discrete generative models.", "questions": "- Can the proposed method be extended to masked priors? It doesn’t look like it can. Masked priors are relevant because they are the best performing methods for some applications.\n\n- The proposed method is close in spirit to the works in continuous diffusion that use empirical estimates of the score function (with different goals). This kind of approaches are fundamentally cursed by dimension, and although they may provide some gains, are hard and expensive to scale to very high dimensions and very large training datasets. Can the authors comment on this?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The authors show that the method works well, especially in training small text-to-image models. It can also be combined successfully with distillation methods such as DCD and ReDI.\n\n- Despite the weaknesses I point out below, the method can be a cheap way to boost pre-training of DFMs, and is relevant to the community.", "weaknesses": "- The concept of straightness of probability paths is central to the paper, but the explanation in Section 3.2 is lacking and leaves the questions with many questions as to what it means to straighten probability paths. This explains my low presentation score. Happy to raise my score if this is addressed.\n\n- As the authors point out in Section 6, the methods seems to perform better for relatively low dimensional data. This is not surprising, because the construction of the backward trajectories relies on the training dataset having good coverage over the distribution measured in the Hamming distance, which can be hard when the dimension is very high. \n\n- The authors try their method on datasets of tens to low hundreds of thousands of samples. Since evaluating the backward velocity field involves a sum over all training samples, I wonder how scalable the method is when the training set is much larger, which may be needed for good performance in higher dimensional settings (related to the previous point). Can the authors comment on this?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)