Cumulative Marginal Matching for One-Step Diffusion
Paribesh Regmi, Rui Li
The sampling procedures of diffusion and flow-matching models involve hundreds of denoising steps incurring substantial inference time, which has motivated a growing body of work on training schemes for few-step and one-step generation. Existing approaches primarily seek to align the generation trajectory of few-step and base models, yet their training objectives only involve discrepancies at a single point in the trajectory, encouraging only local \emph{point-level} alignment. While recent methods such as \emph{Shortcut-CSL} introduce cumulative objective that enforce alignment over the entire trajectory, this is suboptimal as they do not directly enforce alignment between the distributions induced by the two models throughout the generation process. We propose \emph{Cumulative Marginal Matching} (CMM), a principled objective for few-step generation: by viewing flow-matching as a sequence of intermediate marginal distributions connecting the prior and data distributions, CMM directly optimizes the cumulative discrepancy between the base and few-step marginal distributions over the entire sequence. Specifically, we formulate this discrepancy using the Wasserstein-2 distance. By exploiting the pushforward representation of flow-matching, we transform an intractable cumulative Wasserstein objective into a tractable surrogate that can be optimized using samples from the intermediate marginal, which are readily generated through the conditional interpolation process of flow-matching. Furthermore, we show that the existing \emph{Shortcut-CSL} objective arises as a first-order approximation of that of the CMM, establishing a theoretical connection between the two objectives. Experiments on standard benchmarks demonstrate improvement in generation quality over \emph{Shortcut-CSL} and other state-of-the-art one- and few-step baselines under identical training budgets.
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