Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a well-known, well-documented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted. We present an efficient convex optimization-based algorithm we call Outlier Pursuit, that under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the exact optimal low-dimensional subspace, and identifies the corrupted points. Such identification of corrupted points that do not conform to the low-dimensional approximation, is of paramount interest in bioinformatics and financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization, however, our results, setup, and approach, necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct column space of the uncorrupted matrix, rather than the exact matrix itself. 1

Abstract. The performance of principal component analysis (PCA) suffers badly in the presence of outliers. This paper proposes two novel approaches for robust PCA based on semidefinite programming. The first method, maximum mean absolute deviation rounding (MDR), seeks directions of large spread in the data while damping the effect of outliers. The second method produces a low-leverage decomposition (LLD) of the data that attempts to form a low-rank model for the data by separating out corrupted observations. This paper also presents efficient computational methods for solving these SDPs. Numerical experiments confirm the value of these new techniques. 1.

This paper considers the problem of matrix completion, when some number of the columns are arbitrarily corrupted. It is well-known that standard algorithms for matrix completion can return arbitrarily poor results, if even a single column is corrupted. What can be done if a large number, or even a constant fraction of columns are corrupted? In this paper, we study this very problem, and develop an robust and efficient algorithm for its solution. One direct application comes from robust collaborative filtering. Here, some number of users are so-called manipulators, and try to skew the predictions of the algorithm. Significantly, our results hold without any assumptions on the observed entries of the manipulated columns.

Abstract—Matrix factorization methods are extremely useful in many data mining tasks, yet their performances are often degraded by outliers. In this paper, we propose a novel robust matrix factorization algorithm that is insensitive to outliers. We directly formulate robust factorization as a matrix approximation problem with constraints on the rank of the matrix and the cardinality of the outlier set. Then, unlike existing methods that resort to convex relaxations, we solve this problem directly and efficiently. In addition, structural knowledge about the outliers can be incorporated to find outliers more effectively. We applied this method in anomaly detection tasks on various data sets. Empirical results show that this new algorithm is effective in robust modeling and anomaly detection, and our direct solution achieves superior performance over the state-of-the-art methods based on the L1-norm and the nuclear norm of matrices. Keywords-matrix factorization, robust, anomaly detection I.