A Computational Perspective on Projection Pursuit in High Dimensions: Feasible or Infeasible Feature Extraction

Abstract

Finding a suitable representation of multivariate data is fundamental in many scientific disciplines. Projection pursuit ( $\text{PP}$) aims to extract interesting ‘non-Gaussian’ features from multivariate data, and tends to be computationally intensive even when applied to data of low dimension. In high-dimensional settings, a recent work (Bickel et al., 2018) on $\text{PP}$ addresses asymptotic characterization and conjectures of the feasible projections as the dimension grows with sample size. To gain practical utility of and learn theoretical insights into $\text{PP}$ in an integral way, data analytic tools needed to evaluate the behaviour of $\text{PP}$ in high dimensions become increasingly desirable but are less explored in the literature. This paper focuses on developing computationally fast and effective approaches central to finite sample studies for (i) visualizing the feasibility of $\text{PP}$ in extracting features from high-dimensional data, as compared with alternative methods like $\text{PCA}$ and $\text{ICA}$, and (ii) assessing the plausibility of $\text{PP}$ in cases where asymptotic studies are lacking or unavailable, with the goal of better understanding the practicality, limitation and challenge of $\text{PP}$ in the analysis of large data sets.