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pairflow closed form source target coupling for few step generation in discrete flow models pairflow closed form source target coupling for few step generation in discrete flow models iclr 2026 mingue park jisung hwang seungwoo yoo kyeongmin yeo minhyuk sung kaist paper arxiv code tl dr pairflow is a teacher free acceleration framework for discrete flow models that builds source target training pairs via a closed form inversion backward velocity so the model learns straighter few step paths it s cheap 0 2 1 7 of full training compute yet improves few step sampling and can even strengthen the base model for later distillation abstract we introduce 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 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 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 method pairflow is a training framework for dfms that enables few step sampling by constructing paired source target samples using closed form velocities while inspired by redi style coupling driven training our approach eliminates the need for a pretrained teacher by using closed form formulations and achieves acceleration without finetuning the algorithm for computing source target pairs is fully parallelizable and requires at most 1 7 of compute needed for full model training figure 1 illustrations of data inversion in pairflow and the standard corruption process in udlm pairflow achieves a lower average hamming distance 6 47 vs 9 0 promoting straighter paths during training algorithm 1 procedure for generating source target pairs using the closed form backward velocity closed form velocity fields we derive the closed form forward velocity field hat v _t x i z for discrete domains for the empirical target distribution tilde q x the closed form denoiser p_ 1 t x i z and its associated velocity field hat v _t x i z are given by begin align p_ 1 t x i z cfrac sum_ m 1 m delta_ d_ m i x i gamma h d_ m z sum_ m 1 m gamma h d_ m z quad longrightarrow quad hat v _t x i z frac dot kappa_t 1 kappa_t left p_ 1 t x i z delta_ z x i right tag 9 end align to effectively identify suitable source target pairs we derive the closed form backward velocity specifically we first derive the closed form noise predictor p_ 0 t x i z begin align p_ 0 t x i z delta_ z x i frac kappa_t k delta_ x i z i 1 1 k 1 kappa_t frac sum_ m 1 m delta_ d_m i z i gamma h d_m z sum_ m 1 m gamma h d_m z tag 11 end align substituting this into the definition of the backward velocity field we obtain the desired closed form expression begin align check v _t x i z frac dot kappa _t k delta_ x i z i 1 1 k 1 kappa_t frac sum_ m 1 m delta_ d_m i z i gamma h d_m z sum_ m 1 m gamma h d_m z tag 12 end align pairflow algorithm using the derived closed form backward velocity we can efficiently compute source target pairs as outlined in algorithm 1 starting from a given target data sample x_1 we iteratively apply the backward update rule using check v _t x i z to trace back the probability path the final state x_0 reached by this iterative process serves as the source noise this directly constructs a well aligned source target pair x_0 x_1 purely from data effectively eliminating the need for generating samples from a pretrained teacher model quantitative 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 figure 2 step wise performance analysis on the qm9 dataset figure 3 step wise performance analysis on the zinc 250k dataset figure 4 step wise performance analysis on discretized image datasets qualitative results mnist binary 1 step mdlm udlm pairflow udlm dcd pairflow dcd udlm redi pairflow redi cifar 10 64 steps mdlm udlm pairflow cifar 10 256 steps mdlm udlm pairflow citation inproceedings park2026 pairflow title pairflow closed form source target coupling for few step generation in discrete flow models author park mingue and hwang jisung and yoo seungwoo and yeo kyeongmin and sung minhyuk booktitle iclr year 2026