Measuring and testing dependence between complex objects is of great importance in modern statistics. Most existing work relied on the distance between random variables, which inevitably required the moment conditions to guarantee the distance is well-defined. Based on the geometry element ``angle", we develop a novel class of nonlinear dependence measures for data in metric space that can avoid such conditions. Specifically, by making use of the reproducing kernel Hilbert space equipped with Gaussian measure, we introduce kernel angle covariances that can be applied to complex objects such as random vectors or matrices. We estimate kernel angle covariances based on $U$-statistic and establish the corresponding independence tests via gamma approximation. Our kernel angle independence tests, imposing no-moment conditions on kernels, are robust with heavy-tailed random variables. We conduct comprehensive simulation studies and apply our proposed methods to a facial recognition task. Our kernel angle covariances-based tests show remarkable performances in dealing with image data.

翻译：在现代统计学中，度量与检验复杂对象之间的依赖性具有重要意义。现有研究大多基于随机变量间的距离度量，这不可避免需要矩条件以确保距离定义良好。基于几何元素"角度"，我们开发了一类新颖的非线性依赖性度量方法，适用于度量空间中的数据，可规避此类条件。具体而言，通过利用配备高斯测度的再生核希尔伯特空间，我们引入核角度协方差，可应用于随机向量或矩阵等复杂对象。我们基于U-统计量估计核角度协方差，并借助伽马逼近建立相应的独立性检验。我们的核角度独立性检验对核函数无矩条件要求，因此能稳健处理重尾随机变量。我们开展了全面的模拟研究，并将所提方法应用于人脸识别任务。基于核角度协方差的检验在处理图像数据时展现出卓越性能。