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We consider the general dimensionality reduction problem of locating in ahigh-dimensional data cloud, a $k$-dimensional non-Gaussian subspace ofinteresting features. We use a projection pursuit approach -- we search formutually orthogonal unit directions which maximise the 2-Wasserstein distanceof the empirical distribution of data-projections along these directions from astandard Gaussian. Under a generative model, where there is a underlying(unknown) low-dimensional non-Gaussian subspace, we prove rigorous statisticalguarantees on the accuracy of approximating this unknown subspace by thedirections found by our projection pursuit approach. Our results operate in theregime where the data dimensionality is comparable to the sample size, and thussupplement the recent literature on the non-feasibility of locating interestingdirections via projection pursuit in the complementary regime where the datadimensionality is much larger than the sample size.

Wasserstein Projection Pursuit of Non-Gaussian Signals