Semidefinite and Second Order Cone Programming Seminar Fall 2012 Low Rank Matrix Completion

Imagine we have an n1×n2 matrix from which we only get to see a small number of the entries. Is it possible from the available entries to guess the many entries that are missing? In general it is an impossible task because the unknown entries could be anything. However, if one knows that the matrix is low rank and makes a few reasonable assumptions, then the matrix can indeed be reconstructed and often from a surprisingly low number of entries. This field of research, matrix completion, was started with the results in [1] and [2]. There, it was shown, that under some conditions, recovering a rank−r matrix from randomly selected matrix elements, can be done efficiently by minimizing the nuclear norm of the matrix, which can be converted in a semi-definite program. In this work we review in an intuitive way the main results of two seminal papers and some of the well-known applications of the matrix completion problem.