[Paper Review] 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 Inference28 references4 citations

TL;DR

This paper establishes the first non-null central limit theorem (CLT) for sample generalized kernel distance covariance in high dimensions, under Gaussian assumptions. It reveals a universality phenomenon: test power is asymptotically determined solely by $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $, independent of kernel choice, and this rate is minimax optimal.

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 $Y$.

Motivation & Objective

• To close the theoretical gap in high-dimensional dependence testing by deriving non-null limiting distributions for sample distance covariance.
• To characterize the asymptotic power of generalized kernel distance correlation tests under dependence.
• To establish a universality phenomenon in test power that is independent of kernel and bandwidth choices.
• To prove minimax optimality of the separation rate $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $ for detecting dependence in high dimensions.

Proposed method

• Derives a precise high-dimensional expansion of the mean and variance of sample distance covariance, capturing the interplay between dimension-to-sample ratio and dependence.
• Applies a Hoeffding decomposition to the U-statistic form of the sample distance covariance to separate its components.
• Uses Poincaré inequalities and residual estimates to control higher-order terms in the expansion.
• Establishes a normal approximation for the truncated version of the distance covariance statistic.
• Employs a novel truncation and symmetrization strategy to handle the non-i.i.d. structure in the U-statistic under dependence.
• Combines moment expansions with concentration inequalities to derive the non-null CLT under Gaussian assumptions.

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

Experimental results

Research questions

• RQ1 What is the non-null limiting distribution of the sample generalized kernel distance covariance in high dimensions?
• RQ2 How does the power of the generalized kernel distance correlation test behave asymptotically under dependence?
• RQ3 Is the power of the test universal across different kernel choices and bandwidth parameters?
• RQ4 What is the optimal separation rate for detecting dependence in high-dimensional settings?
• RQ5 Can the non-null CLT be derived under a Gaussian assumption with diverging dimensions?

Key findings

• The first non-null central limit theorem for sample generalized kernel distance covariance is established in high dimensions under Gaussianity.
• The asymptotic power of the generalized kernel distance correlation test is completely determined by $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $, independent of kernel and bandwidth choices.
• This separation rate $ n \cdot \text{dCor}^2(X,Y)/\sqrt{2} $ is shown to be minimax optimal for detecting dependence in high-dimensional settings.
• The non-null Gaussian approximation involves a subtle interaction between the dimension-to-sample ratio and the dependence structure between $ X $ and $ Y $.
• The mean and variance of the sample distance covariance are precisely expanded in high dimensions, revealing the critical role of cross-covariance matrices.
• The power formula provides an asymptotically exact first-order approximation for the test's power, enabling accurate power analysis in high-dimensional dependence testing.

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.