Inductive Robust Principal Component Analysis

Abstract

In this paper we address the error correction problem that is to uncover the low-dimensional subspace structure from high-dimensional observations, which are possibly corrupted by errors. When the errors are of Gaussian distribution, Principal Component Analysis (PCA) can find the optimal (in terms of least-square-error) low-rank approximation to highdimensional data. However, the canonical PCA method is known to be extremely fragile to the presence of gross corruptions. Recently, Wright et al. established a so-called Robust Principal Component Analysis (RPCA) method, which can well handle grossly corrupted data [14]. However, RPCA is a transductive method and does not handle well the new samples which are not involved in the training procedure. Given a new datum, RPCA essentially needs to recalculate over all the data, resulting in high computational cost. So, RPCA is inappropriate for the applications that require fast online computation. To overcome this limitation, in this paper we propose an Inductive Robust Principal Component Analysis (IRPCA) method. Given a set of training data, unlike RPCA that targets on recovering the original data matrix, IRPCA aims at learning the underlying projection matrix, which can be used to efficiently remove the possible corruptions in any datum. The learning is done by solving a nuclear norm regularized minimization problem, which is convex and can be solved in polynomial time. Extensive experiments on a benchmark human face dataset and two video surveillance datasets show that IRPCA can not only be robust to gross corruptions, but also handle well the new data in an efficient way.