On the Rotational Invariant $L_1$-Norm PCA
Provides the first convergence proof for a popular robust PCA method, addressing a theoretical gap for practitioners using robust dimensionality reduction.
The paper reinterprets rotational invariant L1-norm PCA as a conditional gradient algorithm and gradient descent on Grassmannian manifolds, proving convergence of iterates to a critical point via the Kurdyka-Łojasiewicz property.
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. Among them the so-called rotational invariant $L_1$-norm PCA is rather popular. In this paper, we reinterpret this robust method as conditional gradient algorithm and show moreover that it coincides with a gradient descent algorithm on Grassmannian manifolds. Based on this point of view, we prove for the first time convergence of the whole series of iterates to a critical point using the Kurdyka-Łojasiewicz property of the energy functional.