The 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 R d into account. Using ideas for stabilizing the classical Weiszfeld algorithm at anchor points and the Kurdyka-Lojasiewicz 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.

Principal component analysis (PCA) is a powerful tool for dimensionality reduction. Unfortunately, it is sensitive to outliers, so that various robust PCA variants were proposed in the literature. One of the most frequently applied methods for high dimensional data reduction is the rotational invariant L 1 -norm PCA of Ding and coworkers. So far no convergence proof for this algorithm was available. The main topic of this paper is to fill this gap. We reinterpret this robust approach as a conditional gradient algorithm and show moreover that it coincides with a gradient descent algorithm on Grassmannian manifolds. Based on the latter point of view, we prove global convergence of the whole series of iterates to a critical point using the Kurdyka-Lojasiewicz property of the objective function, where we have to pay special attention to so-called anchor points, where the function is not differentiable.

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