Title: A Quantitative Framework for Approximating Flow Distillation in Diffusion Models

Abstract: This study introduces a quantitative approximation framework for diffusion distillation, interpreting few-step sampling as the propagation of errors through compositions of learned flow maps. By concentrating on trajectory distillation within the context of the probability-flow ordinary differential equation (ODE), we demonstrate that local approximation errors can be significantly magnified in low-noise, multimodal environments where underlying dynamics exhibit stiffness. Using a Gaussian-mixture Ornstein–Uhlenbeck model that allows for analytical tractability, we isolate two primary challenges: approximating the time-varying score field and managing dynamical amplification dictated by the time-integrated Jacobian bound of the probability-flow ODE. Regarding approximation capabilities, we establish constructive L^p(p_t) guarantees, proving that ReLU–ReQU networks can uniformly approximate the Gaussian-mixture score across all time steps. These networks achieve this with depth and width scaling polylogarithmically with respect to target accuracy and explicitly dependent on mixture geometry. In terms of stability, we derive an explicit bound, L(t), for the spatial Lipschitz constant of the probability-flow velocity. This bound is translated into a stability estimate for the flow map, controlled by the integral \int_s^t L(u)\,du, thereby rendering late-time amplification in stiff regimes computable. Leveraging these estimates, we show that deep residual compositions effectively approximate long-horizon transport, with global error managed by the stability amplification factor. Additionally, we identify a Lipschitz-mismatch regime where one-step distillation is inherently suboptimal. The theoretical findings lead to a stability-balanced, non-uniform time grid created by uniformly partitioning the cumulative stability coordinate. Experimental results validate these predictions, demonstrating a reduction in end-to-end relative mean squared error (MSE) of up to 51.9% when using eight segments, compared to uniform grids.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC