掩码扩散的数据几何：通过去掩蔽增长复杂度实现可验证最优调度

中文摘要 AI 辅助

我们研究用于离散采样的掩码扩散，并引入一种名为去掩蔽增长复杂度（Unmasking Growth Complexity，UGC）的路径解析数据几何度量。其局部增量直接控制Kullback-Leibler（KL）离散化误差，从而对伯努利子集和固定基数去掩蔽方案进行统一分析。在对数揭示几率坐标中，该结构可生成优化的单块和多块调度，并量化将计算工作量适配数据几何所带来的收益。关键是，我们展示了如何通过耦合揭示轨迹上的KL增量从样本中估计UGC增量，这进而得到可验证最优采样器，其能以高概率达到规定的KL误差，且迭代复杂度在对应神谕过程的常数因子范围内。将UGC路径折叠得到聚合UGC质量，其与经典多元相关性度量及离散扩散先前分析中的复杂度度量相关联。在精细划分极限下，UGC密度平方根的平方积分确定了尖锐的主导阶最优欧拉离散化误差。示例表明，相较于粗调度存在显著的维度相关收益，包括使用恒定数量自适应放置的块可实现的$\tilde{\u03A9}(\u221A{d})$改进。

英文摘要

We study masking diffusion for discrete sampling and introduce a path-resolved measure of data geometry called the \emph{unmasking growth complexity} ({\textsf{UGC}\xspace}). Its local increments directly control Kullback--Leibler (KL) discretization error, yielding a unified analysis of Bernoulli-subset and fixed-cardinality unmasking schemes. In log-reveal-odds coordinates, this structure yields optimized single-block and multi-block schedules, and quantifies the gains from adapting computational effort to data geometry. Crucially, we show how {\textsf{UGC}\xspace} increments can be estimated from samples via KL increments along coupled reveal trajectories. This leads to \emph{certified-optimal} samplers that achieve a prescribed KL error with high probability and iteration complexity within a constant factor of the corresponding oracle procedure. Collapsing the \ugc path yields the aggregate {\textsf{UGC}\xspace} mass, which connects to classical multivariate dependence measures and complexity measures from previous analyses of discrete diffusion. In the fine-partition limit, the squared integral of the square-root {\textsf{UGC}\xspace} density determines the sharp leading-order optimal Euler discretization error. Examples exhibit substantial dimension-dependent gains over coarse schedules, including $\widetildeΩ(\sqrt{d})$ improvements achievable with a constant number of adaptively placed blocks.

发表机构

• Massachusetts Institute of Technology（麻省理工学院）

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