  We introduce a new method for two-sample testing of high-dimensional linear
regression coefficients without assuming that those coefficients are
individually estimable. The procedure works by first projecting the matrices of
covariates and response vectors along directions that are complementary in sign
in a subset of the coordinates, a process which we call 'complementary
sketching'. The resulting projected covariates and responses are aggregated to
form two test statistics, which are shown to have essentially optimal
asymptotic power under a Gaussian design when the difference between the two
regression coefficients is sparse and dense respectively. Simulations confirm
that our methods perform well in a broad class of settings and an application
to a large single-cell RNA sequencing dataset demonstrates its utility in the
real world.
