TY - JOUR
T1 - Projection pursuit in high dimensions
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 9151
LP - 9156
M3 - 10.1073/pnas.1801177115
VL - 115
IS - 37
AU - Bickel, Peter J.
AU - Kur, Gil
AU - Nadler, Boaz
Y1 - 2018/09/11
UR - http://www.pnas.org/content/115/37/9151.abstract
N2 - 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.Projection pursuit is a classical exploratory data analysis method to detect interesting low-dimensional structures in multivariate data. Originally, projection pursuit was applied mostly to data of moderately low dimension. Motivated by contemporary applications, we here study its properties in high-dimensional settings. Specifically, we analyze the asymptotic properties of projection pursuit on structureless multivariate Gaussian data with an identity covariance, as both dimension p and sample size n tend to infinity, with p/n→γ∈[0,∞]. Our main results are that (i) if γ=∞, then there exist projections whose corresponding empirical cumulative distribution function can approximate any arbitrary distribution; and (ii) if γ∈(0,∞), not all limiting distributions are possible. However, depending on the value of γ, various non-Gaussian distributions may still be approximated. In contrast, if we restrict to sparse projections, involving only a few of the p variables, then asymptotically all empirical cumulative distribution functions are Gaussian. And (iii) if γ=0, then asymptotically all projections are Gaussian. Some of these results extend to mean-centered sub-Gaussian data and to projections into k dimensions. Hence, in the “small n, large p” setting, unless sparsity is enforced, and regardless of the chosen projection index, projection pursuit may detect an apparent structure that has no statistical significance. Furthermore, our work reveals fundamental limitations on the ability to detect non-Gaussian signals in high-dimensional data, in particular through independent component analysis and related non-Gaussian component analysis.
ER -