乘积参考离散扩散的信息几何：交互增长复杂度与最优调度

中文摘要 AI 辅助

我们研究一类用于从离散分布中采样的乘积参考扩散算法。研究表明，其采样性能可通过一种基于路径的数据几何度量来刻画，该度量被命名为交互增长复杂度（IGC）。双变量IGC核可精确表示KL离散化误差及一个简单的单步上界；而更简单的单变量IGC密度可用于研究步长选择对获取KL散度下ε-精确样本所需迭代复杂度的影响。以对数平方可靠性赔率的等距步长遍历路径的采样器，其性能取决于总IGC质量；而精细的步长选择则能获得更低的复杂度，取决于平方根泛函。在细网格极限下，这两种刻画均变得精确。我们还考虑了一般乘积参考分布，表明参考分布可显著重塑IGC轮廓及由此产生的采样复杂度；特别地，既远离均匀分布又远离数据边际分布的参考分布可产生依赖于维度的改进。最后，总IGC质量可通过总相关和对偶总相关获得界，从而将路径几何与经典的多元依赖度量联系起来。

英文摘要

We study a class of product-reference diffusion algorithms for sampling from a discrete distribution. We show that their sampling performance can be characterized using a path-based measure of data geometry that we call the interaction growth complexity (IGC). We show that a bivariate IGC kernel gives an exact representation of both the KL discretization error and a simple one-step upper bound. The simpler univariate IGC density can be used to study the effect of stepsize choices on the iteration complexity required to obtain $ε$-accurate samples in KL divergence. Samplers that traverse the path with equi-spaced steps in log-squared-reliability-odds have performance that depends on the aggregate IGC mass, whereas refined choices of stepsizes have a lower complexity depending on a square-root functional. In the fine-grid limit, both of these characterizations become sharp. We also allow general product reference distributions and show that the reference law can substantially reshape the IGC profile and the resulting sampling complexity; in particular, references far from both the uniform and the data marginals can yield dimension-dependent improvements. Finally, the aggregate IGC mass admits bounds in terms of total correlation and dual total correlation, thereby connecting the pathwise geometry to classical measures of multivariate dependence.

发表机构

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

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