Robust PCA and subspace tracking from incomplete observations using $$\ell _0$$ ℓ 0 -surrogates

Many applications in data analysis rely on the decomposition of a data matrix into a low-rank and a sparse component. Existing methods that tackle this task use the nuclear norm and $$\ell _1$$ ℓ 1 -cost functions as convex relaxations of the rank constraint and the sparsity measure, respectively, or employ thresholding techniques. We propose a method that allows for reconstructing and tracking a subspace of upper-bounded dimension from incomplete and corrupted observations. It does not require any a priori information about the number of outliers. The core of our algorithm is an intrinsic Conjugate Gradient method on the set of orthogonal projection matrices, the so-called Grassmannian. Non-convex sparsity measures are used for outlier detection, which leads to improved performance in terms of robustly recovering and tracking the low-rank matrix. In particular, our approach can cope with more outliers and with an underlying matrix of higher rank than other state-of-the-art methods. Copyright Springer-Verlag Berlin Heidelberg 2014