Toward Theoretical Insights into Diffusion Trajectory Distillation via Operator Merging

Diffusion trajectory distillation methods aim to accelerate sampling in diffusion models, which produce high-quality outputs but suffer from slow sampling speeds. These methods train a student model to approximate the multi-step denoising process of a pretrained teacher model in a single step, enabling one-shot generation. However, theoretical insights into the trade-off between different distillation strategies and generative quality remain limited, complicating their optimization and selection. In this work, we take a first step toward addressing this gap. Specifically, we reinterpret trajectory distillation as an operator merging problem in the linear regime, where each step of the teacher model is represented as a linear operator acting on noisy data. These operators admit a clear geometric interpretation as projections and rescalings corresponding to the noise schedule. During merging, signal shrinkage occurs as a convex combination of operators, arising from both discretization and limited optimization time of the student model. We propose a dynamic programming algorithm to compute the optimal merging strategy that maximally preserves signal fidelity. Additionally, we demonstrate the existence of a sharp phase transition in the optimal strategy, governed by data covariance structures. Our findings enhance the theoretical understanding of diffusion trajectory distillation and offer practical insights for improving distillation strategies.