The Limits of Closed-Form Couplings for Few-Step Generation

Few-step generation remains an open challenge for flow models. Rectified flows address it by distilling a pretrained teacher into a few-step student using the noise–data couplings induced by the teacher. Obtaining the pretrained model and sampling the couplings is expensive. However, for a finite dataset and a Gaussian probability path, the probability-flow vector field of the empirical distribution is available in closed form, so one could use this field to construct the couplings directly and skip pretraining altogether. Surprisingly, this fails, even though the pretrained coupling is almost always compatible with the data-point assignment induced by the closed-form flow. Students trained on closed-form couplings perform much closer to those trained on random noise–data pairings than to those trained on teacher-generated couplings. We trace this failure to an exact endpoint identity for the closed-form flow. The integration returns the noise vector drawn at initialisation, plus a drift that is empirically a faint copy of the data point, almost independent of the noise. The coupling therefore differs from an independent pairing only through this drift. By bounding the drift and combining the bound with the known collapse time of empirical diffusion dynamics, we show that the informative share of the coupling grows with the dataset size $N$ and shrinks with the data dimension $d$, scaling as $\frac{\log N}{d}$. Growing the dataset therefore helps only logarithmically, while lowering the dimension helps linearly. We confirm this empirically by decreasing the dimension of the data itself, downsampling images to a range of resolutions. The drift share grows as $\frac{1}{d}$, and students trained on closed-form couplings recover an increasing fraction of the gap between random and teacher couplings. However, even at the lowest resolutions the recovery remains incomplete, suggesting that closed-form couplings become competitive teachers only at values of $\frac{\log N}{d}$ far beyond those seen in natural image data.