Toward Theoretical Insights into Diffusion Trajectory Distillation via Operator Merging

2025-05-21

AI Summary

Diffusion model sampling is inherently slow, and while trajectory distillation accelerates inference, it lacks rigorous theoretical foundations. Method: We formulate multi-step denoising as a linear operator composition problem, revealing the geometric nature of projection and scaling—along with signal attenuation mechanisms—under arbitrary noise schedules. We identify a sharp phase transition in optimal distillation strategies, governed by the data covariance structure, and propose a dynamic programming algorithm that maximizes signal fidelity in single-step generation. Contribution/Results: Leveraging linear operator theory and covariance-driven analysis, we establish the first interpretable theoretical framework for trajectory distillation. Our framework quantitatively characterizes signal shrinkage arising from discretization errors and suboptimal optimization, yielding verifiable design principles for efficient, high-fidelity one-step synthesis.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

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.

Problem

• Understanding trade-offs in diffusion trajectory distillation strategies
• Optimizing operator merging for signal fidelity preservation
• Exploring phase transitions in optimal distillation strategies

Innovation

• Reinterprets distillation as operator merging
• Proposes dynamic programming for optimal merging
• Identifies phase transition in optimal strategy