Intrinsic Grassmann Averages for Online Linear and Robust Subspace Learning

Principal Component Analysis (PCA) is a fundamental method for estimating a linear subspace approximation to high-dimensional data. Many algorithms exist in literature to achieve a statistically robust version of PCA called RPCA. In this paper, we present a geometric framework for computing the principal linear subspaces in both situations that amounts to computing the intrinsic average on the space of all subspaces (the Grassmann manifold). Points on this manifold are defined as the subspaces spanned by K-tuples of observations. We show that the intrinsic Grassmann average of these subspaces coincide with the principal components of the observations when they are drawn from a Gaussian distribution. Similar results are also shown to hold for the RPCA. Further, we propose an efficient online algorithm to do subspace averaging which is of linear complexity in terms of number of samples and has a linear convergence rate. When the data has outliers, our proposed online robust subspace averaging algorithm shows significant performance (accuracy and computation time) gain over a recently published RPCA methods with publicly accessible code. We have demonstrated competitive performance of our proposed online subspace algorithm method on one synthetic and two real data sets. Experimental results depicting stability of our proposed method are also presented. Furthermore, on two real outlier corrupted datasets, we present comparison experiments showing lower reconstruction error using our online RPCA algorithm. In terms of reconstruction error and time required, both our algorithms outperform the competition.