Kernel Angle Dependence Measures in Metric Spaces

Measuring and testing dependence between data in separable metric spaces 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 various types of data, including low dimensional vector, high dimensional vectors, non-Euclidean data like symmetric positive definite matrices, and compositional data. 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. All the codes and proofs are included in the Supplementary Materials.