论文标题
无排序的内核独立测试
A Permutation-Free Kernel Independence Test
论文作者
论文摘要
在非参数独立性测试中，我们观察i.i.d. \ data $ \ {（x_i，y_i）\} _ {i = 1}^n $，其中$ x \ in \ mathcal {x}，y \ y \ in \ nathcal {y} y} $ in \ mathcal in \ nathcal {y} $均为y y he y y y null y是$ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x。现代测试统计数据，例如Hilbert-Schmidt独立标准（HSIC）和距离协方差（DCOV），由于基本U统计量的退化，具有棘手的无效分布。因此，在实践中，通常有人求助于使用置换测试，这提供了非沉淀保证，而牺牲了几百次的二次时间统计（例如）。本文提供了对HSIC和DCOV（称为XHSIC和XDCOV）的简单但非平凡的修改，该作用为``Cross''HSIC/DCOV），因此它们在无效下具有限制性高斯分布，因此不需要置换。这需要建立基于Kim和Ramdas（2020）的新开发的跨U统计理论，特别是在Shekhar等人中开发了该理论的几种非平凡扩展。 （2022），它开发了类似的无排序核两样本测试。我们表明，像原始测试一样，我们的新测试与固定替代方案保持一致，而最小值速率是针对平滑局部替代方案的最佳速率。数值模拟表明，与完整的DCOV或HSIC相比，我们的变体具有相同的功率，高达$ \ sqrt 2 $ ractor，从业人员为大型问题或数据 - 分析管道提供了新的选择，而计算（而不是样本尺寸）可能是瓶颈。
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.
