  We introduce principal differences analysis (PDA) for analyzing differences
between high-dimensional distributions. The method operates by finding the
projection that maximizes the Wasserstein divergence between the resulting
univariate populations. Relying on the Cramer-Wold device, it requires no
assumptions about the form of the underlying distributions, nor the nature of
their inter-class differences. A sparse variant of the method is introduced to
identify features responsible for the differences. We provide algorithms for
both the original minimax formulation as well as its semidefinite relaxation.
In addition to deriving some convergence results, we illustrate how the
approach may be applied to identify differences between cell populations in the
somatosensory cortex and hippocampus as manifested by single cell RNA-seq. Our
broader framework extends beyond the specific choice of Wasserstein divergence.
