Determining causal relationship between high dimensional observations are among the most important tasks in scientific discoveries. In this paper, we revisited the \emph{linear trace method}, a technique proposed in~\citep{janzing2009telling,zscheischler2011testing} to infer the causal direction between two random variables of high dimensions. We strengthen the existing results significantly by providing an improved tail analysis in addition to extending the results to nonlinear trace functionals with sharper confidence bounds under certain distributional assumptions. We obtain our results by interpreting the trace estimator in the causal regime as a function over random orthogonal matrices, where the concentration of Lipschitz functions over such space could be applied. We additionally propose a novel ridge-regularized variant of the estimator in \cite{zscheischler2011testing}, and give provable bounds relating the ridge-estimated terms to their ground-truth counterparts. We support our theoretical results with encouraging experiments on synthetic datasets, more prominently, under high-dimension low sample size regime.
翻译:确定高维观测之间的因果关系是科学发现中最重要的任务之一。 在本文中,我们重新审视了\ emph{ 线性跟踪方法} 。 这是在“citep{janzing2009telling,zscheischler2011测试”中提议的一种技术,用以推断两个高维随机变量之间的因果关系方向。我们大大加强了现有结果,除了根据某些分布假设将结果扩大到非线性微量功能外,还改进了尾料分析,在某些分布式假设中将结果扩大到具有更清晰的置信界限的非线性微量功能。我们通过将因果系统中的微量估量器解释为随机或线性矩阵的函数,将利普西茨在这种空间上的功能集中应用。我们还提议在\ cite{zscheischler2011测试} 中建立一个新的脊柱定型的脊柱变体,并将估计值值的界限与地面图比相联系起来。我们支持我们的理论结果,在高二分层低采样系统下鼓励合成数据集的实验。</s>