  The paper presents new metrics to quantify and test for (i) the equality of
distributions and (ii) the independence between two high-dimensional random
vectors. We show that the energy distance based on the usual Euclidean distance
cannot completely characterize the homogeneity of two high-dimensional
distributions in the sense that it only detects the equality of means and the
traces of covariance matrices in the high-dimensional setup. We propose a new
class of metrics which inherits the desirable properties of the energy distance
and maximum mean discrepancy/(generalized) distance covariance and the
Hilbert-Schmidt Independence Criterion in the low-dimensional setting and is
capable of detecting the homogeneity of/completely characterizing independence
between the low-dimensional marginal distributions in the high dimensional
setup. We further propose t-tests based on the new metrics to perform
high-dimensional two-sample testing/independence testing and study their
asymptotic behavior under both high dimension low sample size (HDLSS) and high
dimension medium sample size (HDMSS) setups. The computational complexity of
the t-tests only grows linearly with the dimension and thus is scalable to very
high dimensional data. We demonstrate the superior power behavior of the
proposed tests for homogeneity of distributions and independence via both
simulated and real datasets.
