发表机构
University of California, Irvine（加利福尼亚大学欧文分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究从约束样本学习离散分布自然参数的问题，通过分析\(S\)几何结构改进参数估计保证，推广胖性概念并给出有效推理条件，建立理论下界，提升样本复杂度并匹配无截断极小极大率。
AI 中文摘要
从约束在子集\(S\subseteq\{0,1\}^n\)上的独立样本中学习离散分布\(\mu_z\)的自然参数\(z\in\mathbb{R}^n\)是高维统计中的一个基本挑战。现有估计截断布尔积分布的方法，如[Fotakis等人的工作]，要么对\(S\)有强局部连通性假设（称为胖性），要么有严格的反集中假设，且截断集的总质量需相对于\(n\)为常数。此外，若\(S\)的质量在\(n\)中呈指数级小，[Fotakis等人的工作]的样本复杂度为\(\Omega(2^n)\)。本文通过在测度\(\mu_z\)下分析\(S\)的几何结构规避这些限制。在胖性假设下改进了现有参数估计保证，将\(\ell_\infty\)恢复的先验样本复杂度提高到\(O(\log n / \epsilon^2)\)，与无截断的极小极大率匹配。还用布尔函数分析中使用的影响力概念推广了胖性并提供有效推理的充分条件。最后建立了理论下界，表明样本复杂度对模型宽度和集合中元素间最小距离呈现内在指数依赖。
英文摘要
Learning the natural parameters $z \in \mathbb{R}^n$ of discrete distributions $μ_z$ from independent samples constrained to a subset $S \subseteq \{0,1\}^n$ is a foundational challenge in high-dimensional statistics. Existing methods for efficiently estimating truncated Boolean product distributions, notably the work of [Fotakis et al' COLT'20, Algorithmica '22], require either strong local connectivity assumptions on $S$ -- a property denoted fatness -- or stringent anti-concentration assumptions and necessitate the total mass of the truncation set to be a constant with respect to $n$. Moreover, the results in [Fotakis et al' COLT'20, Algorithmica '22] suffer from sample complexities that scale as $Ω(2^n)$ if the mass of $S$ is exponentially small in $n$. In this work, we circumvent these limitations by analyzing the geometry of $S$ under the measure $μ_z$. We refine the existing parameter estimation guarantees under the fatness assumption, improving the prior sample complexity to $O( \log n / ε^2)$ for $\ell_\infty$-recovery, matching the untruncated minimax rate. We further generalize fatness using the notion of influence utilized in the analysis of Boolean functions and provide sufficient conditions for efficient inference. Notably, unlike previous work, our method does not require sampling at arbitrary parameterizations of the model. Lastly, we establish a theoretical lower bound demonstrating the sample complexity exhibits an intrinsic exponential dependence on the width of the model and the minimum distance between elements in the set.
