  We consider the general dimensionality reduction problem of locating in a
high-dimensional data cloud, a $k$-dimensional non-Gaussian subspace of
interesting features. We use a projection pursuit approach -- we search for
mutually orthogonal unit directions which maximise the 2-Wasserstein distance
of the empirical distribution of data-projections along these directions from a
standard Gaussian. Under a generative model, where there is a underlying
(unknown) low-dimensional non-Gaussian subspace, we prove rigorous statistical
guarantees on the accuracy of approximating this unknown subspace by the
directions found by our projection pursuit approach. Our results operate in the
regime where the data dimensionality is comparable to the sample size, and thus
supplement the recent literature on the non-feasibility of locating interesting
directions via projection pursuit in the complementary regime where the data
dimensionality is much larger than the sample size.
