Robust Principal Component Analysis as a Nonlinear Eigenproblem

Principal Component Analysis (PCA) is a widely used tool for, e.g., exploratory data analysis, dimensionality reduction and clustering. However, it is well known that PCA is strongly aected by the presence of outliers and, thus, is vulnerable to both gross measurement error and adversarial manipulation of the data. This phenomenon motivates the development of robust PCA as the problem of recovering the principal components of the uncontaminated data. In this thesis, we propose two new algorithms, QRPCA and MDRPCA, for robust PCA components based on the projection-pursuit approach of Huber. While the resulting optimization problems are non-convex and non-smooth, we show that they can be eciently minimized via the RatioDCA using bundle methods/accelerated proximal methods for the interior problem. The key ingredient for the most promising algorithm (QRPCA) is a robust, location invariant scale measure with breakdown point 0.5. Extensive experiments show that our QRPCA is competitive with current state-of-the-art methods and outperforms other methods in particular for a large number of outliers