发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出去噪增长复杂度（DGC），利用其鞅结构开发可验证扩散采样算法，推导KL误差边界，揭示适配数据几何可带来计算收益。
AI 中文摘要
基于扩散的采样存在两个核心挑战：理论层面是理解其在高维场景下仍表现出色的根本原因，实践层面是设计具备可验证性能保证的算法。我们通过去噪增长复杂度（denoising growth complexity，$\boldsymbol{\textsf{DGC}}$）揭示这两个问题的内在关联。$\textsf{DGC}$是一种几何度量，由高斯热流过程中去噪均方误差导数的对数时间加权积分定义。我们证明$\textsf{DGC}$增量可对随机创新表示的欧拉格式的KL误差给出简单显式边界，该边界沿路径呈局部性：每一步由对应$\textsf{DGC}$增量及其相对步长控制。基于此结构，我们推导了优化步长调度的KL采样保证，涵盖更简单的单块场景与更精细的$K$-块场景。$\textsf{DGC}$函数具有自然的鞅结构，我们利用该结构开发了这些算法的完全数据可验证版本。此外，$\textsf{DGC}$可根据协方差、率失真、度量熵及庞加莱常数给出信息论上界，从而恢复并强化一系列现有扩散采样保证，同时提供新结果。在对数热时间下，精细划分极限由涉及$\textsf{DGC}$密度平方根的积分决定，而单块调度依赖其普通积分。该比较精准刻画了适配数据几何何时能带来显著计算收益，包括简单高斯混合模型中对数到常数的分离情况。
英文摘要
Two central challenges in diffusion-based sampling are the theoretical one of understanding their remarkable effectiveness even in high-dimensional settings, and the practical one of designing algorithms with certified performance guarantees. We show that these questions are intimately connected via the \emph{denoising growth complexity} ($\mathsf{DGC}$). It is a geometric measure defined by a log-time weighted integral of the derivative of the denoising mean-squared error along the Gaussian heat flow. We show how the $\mathsf{DGC}$ increments lead to a simple and explicit bound on the KL error of an Euler scheme applied to the stochastic innovations representation. The bound is local along the path: each step is controlled by the corresponding $\mathsf{DGC}$ increment and its relative stepsize. This structure allows us to derive KL sampling guarantees for optimized stepsize schedules, both in a simpler single-block setting and in a more refined $K$-block setting. The $\mathsf{DGC}$ function has a natural martingale structure, which we exploit to develop fully data-certified versions of these algorithms. It also admits information-theoretic upper bounds in terms of covariance, rate distortion, metric entropy, and the Poincar'e constant, thereby recovering and sharpening a range of existing diffusion-sampling guarantees, as well as giving new results. In log heat-time, the fine partition limit is governed by an integral involving the square root of the $\mathsf{DGC}$ density, whereas a single-block schedule depends on its ordinary integral. This comparison precisely characterizes when adaptation to data geometry yields substantial computational gains, including logarithmic-to-constant separations for simple Gaussian mixture models.
