GrIP-PCA: Grassmann Iterative P-Norm Principal Component Analysis

Abstract

Principal component analysis is one of the most commonly used methods for dimensionality reduction in signal processing. However, the most commonly used PCA formulation is based on the L2-norm, which can be highly influenced by outlier data. In recent years, there has been growing interest in the development of more robust PCA methods. Recent works explore alternative norms, such as the L1-norm or the more general Lp-norms, which significantly improve robustness over the L2-norm. In this work, we present the Grassmann Iterative P-norm PCA (GrIP-PCA) method, which uses an iterative Grassmann manifold optimization approach to find the solution to the highly non-convex Lp-norm PCA problem. The increased flexibility of this iterative optimization approach allows for the first ever direct comparison between the projection maximization and reprojection minimization objective functions for general Lp-PCA. Our results demonstrate that the underutilized reprojection formulation leads to improved robustness of PCA in multiple experiments.

Keywords

• Dimensionality reduction
• Grassmann manifold
• $L_{p}$ -PCA%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX">$L_{p}$</tex-math> </inline-formula>-PCA
• principal component analysis