  We introduce sparse random projection, an important dimension-reduction tool
from machine learning, for the estimation of discrete-choice models with
high-dimensional choice sets. Initially, high-dimensional data are compressed
into a lower-dimensional Euclidean space using random projections.
Subsequently, estimation proceeds using cyclic monotonicity moment inequalities
implied by the multinomial choice model; the estimation procedure is
semi-parametric and does not require explicit distributional assumptions to be
made regarding the random utility errors. The random projection procedure is
justified via the Johnson-Lindenstrauss Lemma -- the pairwise distances between
data points are preserved during data compression, which we exploit to show
convergence of our estimator. The estimator works well in simulations and in an
application to a supermarket scanner dataset.
