Robust Matrix Completion and Corrupted Columns

Abstract

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 {\it without any assumptions on the observed entries of the manipulated columns}.