  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.
