Toward theoretical insights into diffusion trajectory distillation via operator merging.

Area of Science:

• Artificial Intelligence
• Machine Learning
• Computational Mathematics

Background:

• Diffusion models generate high-quality data but require many sampling steps.
• Diffusion trajectory distillation (DTD) trains a student model to mimic a teacher model's denoising process in fewer steps.
• The theoretical underpinnings of DTD's impact on generative quality are not well understood.

Purpose of the Study:

• To theoretically characterize the trade-offs in diffusion trajectory distillation.
• To identify bottlenecks and optimal strategies for distillation in different regimes.
• To provide principled guidance for selecting distillation methods.

Main Methods:

• Reinterpreting trajectory distillation as an operator merging problem.
• Analyzing distillation in the linear Gaussian regime, focusing on optimization error.
• Analyzing distillation in the nonlinear Gaussian mixture regime, focusing on approximation error.
• Deriving an optimal merging strategy using Pareto dynamic programming.

Main Results:

• In the linear Gaussian regime, optimization error (signal shrinkage) is the main bottleneck, with a theoretically optimal merging strategy identified.
• In the nonlinear Gaussian mixture regime, approximation error increases with distillation steps due to component growth.
• The study quantifies error amplification across merges in nonlinear settings.

Conclusions:

• The theoretical framework clarifies distinct error mechanisms in linear and nonlinear distillation regimes.
• Principled guidance is provided for selecting distillation strategies based on model characteristics.
• Understanding these trade-offs is crucial for improving the efficiency and quality of diffusion model sampling.