  High-dimensional data, where the dimension of the feature space is much
larger than sample size, arise in a number of statistical applications. In this
context, we construct the generalized multivariate sign transformation, defined
as a vector divided by its norm. For different choices of the norm function,
the resulting transformed vector adapts to certain geometrical features of the
data distribution. Building up on this idea, we obtain one-sample and
two-sample testing procedures for mean vectors of high-dimensional data using
these generalized sign vectors. These tests are based on U-statistics using
kernel inner products, do not require prohibitive assumptions, and are amenable
to a fast randomization-based implementation. Through experiments in a number
of data settings, we show that tests using generalized signs display higher
power than existing tests, while maintaining nominal type-I error rates.
Finally, we provide example applications on the MNIST and Minnesota Twin
Studies genomic data.
