  In this paper, we address the problem of testing independence between two
high-dimensional random vectors. Our approach involves a series of max-sum
tests based on three well-known classes of rank-based correlations. These
correlation classes encompass several popular rank measures, including
Spearman's $\rho$, Kendall's $\tau$, Hoeffding's D, Blum-Kiefer-Rosenblatt's R
and Bergsma-Dassios-Yanagimoto's $\tau^*$.The key advantages of our proposed
tests are threefold: (1) they do not rely on specific assumptions about the
distribution of random vectors, which flexibility makes them available across
various scenarios; (2) they can proficiently manage non-linear dependencies
between random vectors, a critical aspect in high-dimensional contexts; (3)
they have robust performance, regardless of whether the alternative hypothesis
is sparse or dense.Notably, our proposed tests demonstrate significant
advantages in various scenarios, which is suggested by extensive numerical
results and an empirical application in RNA microarray analysis.
