[论文解读] Generalized kernel distance covariance in high dimensions: non-null CLTs and power universality

Qiyang Han, Yandi Shen|arXiv (Cornell University)|Jun 14, 2021

Statistical Methods and Bayesian Inference参考文献 28被引用 4

一句话总结

该论文在高维条件下，于高斯假设下首次建立了样本广义核距离协方差的非零均值中心极限定理（CLT）。其揭示了普遍性现象：检验功效在渐近意义上仅由 $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $ 决定，与核函数选择无关，且该速率达到极小极大最优。

ABSTRACT

Distance covariance is a popular dependence measure for two random vectors $X$ and $Y$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that $X$ and $Y$ are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert-Schmidt) kernel distance covariance in high dimensions, primarily in the Gaussian case. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between $X$ and $Y$. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by $n\cdot ext{dcor}^2(X,Y)/\sqrt{2}$ in the high dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between $X$ and $Y$.

研究动机与目标

• 通过推导样本距离协方差的非零极限分布，填补高维依赖性检验的理论空白。
• 刻画广义核距离相关检验在依赖性下的渐近功效。
• 建立检验功效的普遍性现象，该现象与核函数和带宽选择无关。
• 证明在高维中检测依赖性的分离速率 $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $ 的极小极大最优性。

提出的方法

• 精确推导了样本距离协方差的均值与方差在高维下的展开式，捕捉了维度与样本量之比与依赖性之间的相互作用。
• 对样本距离协方差的U-统计量形式应用Hoeffding分解，以分离其各组成部分。
• 利用Poincaré不等式与余项估计，控制展开式中的高阶项。
• 为截断版本的距离协方差统计量建立了正态近似。
• 采用一种新颖的截断与对称化策略，以处理在依赖性下U-统计量中的非独立同分布结构。
• 结合矩展开与集中不等式，在高斯假设下推导出非零均值CLT。

Figure 1 . Verification of CLTs. Solid lines correspond to the standard normal quantiles, and dashed lines correspond to sample quantiles with the identity, Gaussian, and Laplace kernels, respectively. Simulation parameters: $(n,p,q)=(1000,100,100)$ , $B=200$ replications, bandwidth choices $\rho_{X

实验结果

研究问题

• RQ1 在高维条件下，样本广义核距离协方差的非零极限分布是什么？
• RQ2 在依赖性下，广义核距离相关检验的功效如何渐近表现？
• RQ3 该检验的功效是否在不同核函数选择与带宽参数下具有普遍性？
• RQ4 在高维设定下检测依赖性的最优分离速率是什么？
• RQ5 在维度发散的高斯假设下，能否推导出非零均值CLT？

主要发现

• 在高斯性假设下，首次建立了高维样本广义核距离协方差的非零中心极限定理。
• 广义核距离相关检验的渐近功效完全由 $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $ 决定，与核函数和带宽选择无关。
• 该分离速率 $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $ 被证明在高维设定下检测依赖性时达到极小极大最优。
• 非零均值高斯近似涉及维度与样本量之比和 $ X $ 与 $ Y $ 之间依赖结构之间的微妙相互作用。
• 精确展开了高维下样本距离协方差的均值与方差，揭示了交叉协方差矩阵的关键作用。
• 功效公式为检验功效提供了渐近精确的一阶近似，使得高维依赖性检验中的功效分析更加准确。

Figure 2 . Verification of power universality in choice of bandwidth parameter (left and middle) and choice of kernel (right). Solid lines correspond to the standard normal quantiles, and dashed lines correspond to sample quantiles.