In nonparametric independence testing, we observe i.i.d.\ data $\{(X_i,Y_i)\}_{i=1}^n$, where $X \in \mathcal{X}, Y \in \mathcal{Y}$ lie in any general spaces, and we wish to test the null that $X$ is independent of $Y$. Modern test statistics such as the kernel Hilbert-Schmidt Independence Criterion (HSIC) and Distance Covariance (dCov) have intractable null distributions due to the degeneracy of the underlying U-statistics. Thus, in practice, one often resorts to using permutation testing, which provides a nonasymptotic guarantee at the expense of recalculating the quadratic-time statistics (say) a few hundred times. This paper provides a simple but nontrivial modification of HSIC and dCov (called xHSIC and xdCov, pronounced ``cross'' HSIC/dCov) so that they have a limiting Gaussian distribution under the null, and thus do not require permutations. This requires building on the newly developed theory of cross U-statistics by Kim and Ramdas (2020), and in particular developing several nontrivial extensions of the theory in Shekhar et al. (2022), which developed an analogous permutation-free kernel two-sample test. We show that our new tests, like the originals, are consistent against fixed alternatives, and minimax rate optimal against smooth local alternatives. Numerical simulations demonstrate that compared to the full dCov or HSIC, our variants have the same power up to a $\sqrt 2$ factor, giving practitioners a new option for large problems or data-analysis pipelines where computation, not sample size, could be the bottleneck.

翻译：在非对称独立测试中,我们观察 i. i. d.\\ 数据 $ {( X_i, Y_i) ⁇ i=1\ n$, 美元=x = mathcal{X}, Y = mathcal{Y} 美元存在于任何一般空间中, 我们希望测试美元是否独立于Y美元。 现代测试统计数据,如Hilbert- Schmid 独立标准( HSIC) 和远程变异( dCov) 等核心Ustat- Statistical 的变异性具有难以控制的无效分布。 因此,在实践上, 美元=xxxxxxi, Y_i= i=1\ $n$, 美元=xxxxxxxxxxxxxxxxxllmlmlmlal 和远程变异性变异性( dcov) 。 因此, 美元- caltialxxxxxalalalalalalalalalalalalalalalalalalalalalal alal al alal al al exalationalationalationalsal exal exal exal exal ex ex exmal exmal ex exm ex ex ex ex ex ex ex ex ex exm exmmmmmmm ex exm, 需要一种不进行新的但新但的变异性变换算算。