On the Robust PCA and Weiszfeld's Algorithm

Abstract

Principal component analysis (PCA) is a powerful standard tool for reducing the dimensionality of data. Unfortunately, it is sensitive to outliers so that various robust PCA variants were proposed in the literature. This paper addresses the robust PCA by successively determining the directions of lines having minimal Euclidean distances from the data points. The corresponding energy functional is not differentiable at a finite number of directions which we call anchor directions. We derive a Weiszfeld-like algorithm for minimizing the energy functional which has several advantages over existing algorithms. Special attention is paid to the careful handling of the anchor directions, where we take the relation between local minima and one-sided derivatives of Lipschitz continuous functions on submanifolds of $\mathbb R^d$ into account. Using ideas for stabilizing the classical Weiszfeld algorithm at anchor points and the Kurdyka-Łojasiewicz property of the energy functional, we prove global convergence of the whole sequence of iterates generated by the algorithm to a critical point of the energy functional. Numerical examples demonstrate the very good performance of our algorithm.