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We introduce principal differences analysis (PDA) for analyzing differencesbetween high-dimensional distributions. The method operates by finding theprojection that maximizes the Wasserstein divergence between the resultingunivariate populations. Relying on the Cramer-Wold device, it requires noassumptions about the form of the underlying distributions, nor the nature oftheir inter-class differences. A sparse variant of the method is introduced toidentify features responsible for the differences. We provide algorithms forboth the original minimax formulation as well as its semidefinite relaxation.In addition to deriving some convergence results, we illustrate how theapproach may be applied to identify differences between cell populations in thesomatosensory cortex and hippocampus as manifested by single cell RNA-seq. Ourbroader framework extends beyond the specific choice of Wasserstein divergence.

Principal Differences Analysis: Interpretable Characterization of Differences between Distributions