Abstract: Various metrics have been developed to test for statistical independence and measure the degree of nonlinear dependence between two random objects. Most of these metrics achieve their lower bound if and only if the two random objects are independent. However, it is often unclear how the two random objects are dependent if they attain their upper bound. Moreover, how to implement these metrics when one of the objects is matrix-valued is rarely touched in the literature. To address these issues, we introduce a new metric called trace correlation, which ranges from zero to one. It equals zero only if the two random objects are independent and attains one only if one random object is functionally dependent on the other. In addition, trace correlation allows one of the random objects to be matrix-valued. We estimate trace correlation using standard U-statistic theory and thoroughly study the asymptotic properties of resultant estimates. Furthermore, we adapt trace correlation in the reproducing kernel Hilbert space. Extensive simulations and an application to the MNIST dataset demonstrate the effectiveness and usefulness of trace correlation.

Key words and phrases: Complete dependence, independence test, matrix-valued object, trace correlation.