发表机构
Arizona State University（亚利桑那州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究从随机扫描高斯格劳伯动力学轨迹中精确恢复高斯图形模型，解决现有技术不足。提出两种无混合算法，实例化对决邻域搜索元算法，一种用最小二乘回归，另一种计特定更新模式次数，达到信息论下界依赖性，分析解决核心挑战。
AI 中文摘要
高斯图形模型选择通常在独立采样下进行研究，但在许多应用中，数据是作为相关随机过程的单个轨迹出现的。我们研究从随机扫描高斯格劳伯动力学的一个轨迹中精确恢复图形。解决此问题的现有技术要么继承链的混合时间，在无强假设下其在维度\(p\)上可能是超多项式的，要么在最小归一化边强度\(\kappa\)方面次优。我们提出两种无混合算法，达到信息论下界的\(\kappa^{-2}\)依赖性。两种算法都实例化了一个共享的对决邻域搜索元算法，使用直接从更新序列构建的局部统计量。第一种算法在每个节点的更新处拟合最小二乘回归，从\(\widetilde O(pd^{2}/\kappa^{2})\)次更新中恢复图形，其中\(d\)是最大度数。该算法的数据要求取决于一个局部条件量，但仅对数依赖，并且即使基础链混合缓慢也被证明是最优的。第二种算法基于计算特定更新模式的出现次数，需要\(\widetilde O(pd^{4}/\kappa^{2})\)次更新，不依赖于任何条件数。核心技术挑战是两个统计量都基于相关的非平稳观测构建。我们的分析通过展示如何从更新序列中提取新的高斯创新来解决此问题，从而实现对适当数量的无混合控制。算法及其分析都不调用平稳性、谱隙或混合条件，并且所有保证从任意初始化都成立。
英文摘要
Gaussian graphical model selection is usually studied under independent sampling, but in many applications the data arise as a single trajectory of a dependent stochastic process. We study exact recovery of the graph from one trajectory of random-scan Gaussian Glauber dynamics. Existing techniques for this problem either inherit the mixing time of the chain, which can be super-polynomial in the dimension $p$ without strong assumptions, or are suboptimal in the minimum normalized edge strength $κ$. We propose two algorithms that are mixing-free and attain the $κ^{-2}$ dependence of the information-theoretic lower bounds. Both instantiate a shared dueling-neighborhood search meta-algorithm with a local statistic built directly from the update sequence. For every fixed precision matrix and deterministic initialization, the first algorithm fits a least-squares regression at the updates of each node and has pointwise recovery horizon $\widetilde O(pd^{2}/κ^{2})$, where $d$ is the maximum degree. Its horizon depends logarithmically on a local conditioning quantity and on the initialization potential. The second algorithm is based on counting occurences of a specific update pattern and requires $\widetilde O(pd^{4}/κ^{2})$ updates, with no dependence on any condition number. The central technical challenge is that both statistics are built from dependent, non-stationary observations. Our analysis tackles this by demonstrating how to extract fresh Gaussian innovations from the update sequence, which yields mixing-free control of appropriate quantities. Neither the algorithms nor their analyses invoke stationarity, a spectral gap, or mixing conditions.
CommentsMinor correction to the proof of Lemma 15 (the statement and implications remain the same). For clarity and completeness: (1) separated pointwise recovery horizon guarantees from uniform ones, and (2) rewrote slow-mixing proposition (Proposition 2) over the class of problems with bounded initialization energy
