Score-based diffusion models have emerged as powerful techniques for generating samples from high-dimensional data distributions. These models involve a two-phase process: first, injecting noise to transform the data distribution into a known prior distribution, and second, sampling to recover the original data distribution from noise. Among the various sampling methods, deterministic samplers stand out for their enhanced efficiency. However, analyzing these deterministic samplers presents unique challenges, as they preclude the use of established techniques such as Girsanov's theorem, which are only applicable to stochastic samplers. Furthermore, existing analysis for deterministic samplers usually focuses on specific examples, lacking a generalized approach for general forward processes and various deterministic samplers. Our paper addresses these limitations by introducing a unified convergence analysis framework. To demonstrate the power of our framework, we analyze the variance-preserving (VP) forward process with the exponential integrator (EI) scheme, achieving iteration complexity of $\tilde O(d^2/\epsilon)$. Additionally, we provide a detailed analysis of Denoising Diffusion Implicit Models (DDIM)-type samplers, which have been underexplored in previous research, achieving polynomial iteration complexity.

翻译：基于分数的扩散模型已成为从高维数据分布生成样本的强大技术。这些模型涉及两个阶段：首先注入噪声将数据分布转化为已知的先验分布，其次通过采样从噪声中恢复原始数据分布。在众多采样方法中，确定性采样器因其更高的效率而备受关注。然而，分析这些确定性采样器面临独特挑战，因为它们排除了如Girsanov定理等仅适用于随机采样器的既定技术的使用。此外，现有对确定性采样器的分析通常聚焦于特定实例，缺乏针对一般前向过程和各种确定性采样器的通用方法。本文通过引入统一的收敛性分析框架来解决这些局限性。为展示本框架的效力，我们分析了采用指数积分器（EI）方案的方差保持（VP）前向过程，实现了$\tilde O(d^2/\epsilon)$的迭代复杂度。此外，我们对去噪扩散隐式模型（DDIM）型采样器进行了详细分析，该采样器在先前研究中未得到充分探索，并实现了多项式迭代复杂度。