Projection pursuit in high dimensions

Significance A key challenge in analyzing high-dimensional data is to extract meaningful low-dimensional structures, which typically represent signals of interest. Standard and widely used methods include principal components analysis (PCA), independent component analysis (ICA), and projection pursuit. Limitations of PCA in high-dimensional settings have been widely studied. In this work, we show that with a limited amount of high-dimensional data, the results of projection pursuit and related ICA methods should be interpreted with great care. Specifically, even with structureless data, one may find projections with highly non-Gaussian components that have no statistical or scientific significance. In contrast, restricting to sparse projections, detecting non-Gaussian components is still possible.