Wasserstein Projection Pursuit of Non-Gaussian Signals
This addresses dimensionality reduction for non-Gaussian signals, providing theoretical support in a regime where previous methods were infeasible, though it appears incremental as it supplements existing literature.
The paper tackles the problem of finding non-Gaussian subspaces in high-dimensional data using a projection pursuit method based on 2-Wasserstein distance, proving statistical guarantees for accurate subspace approximation under a generative model when data dimensionality is comparable to sample size.
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.