Deterministic algorithms for matrix completion. Issue 2 (25th January 2013) - Heiman, Eyal; Schechtman, Gideon; Shraibman, Adi
ABSTRACT The goal of the matrix completion problem is to retrieve an unknown real matrix from a small subset of its entries. This problem comes up in many application areas, and has received a great deal of attention in the context of the Netflix challenge. This setup usually represents our partial knowledge of some information domain. Unknown entries may be due to the unavailability of some relevant experimental data. One approach to this problem starts by selecting a complexity measure of matrices, such as rank or trace norm. The corresponding algorithm outputs a matrix of lowest possible complexity that agrees with the partially specified matrix. The performance of the above algorithm under the assumption that the revealed entries are sampled randomly has received considerable attention (e.g., Refs. Srebro et al., 2005; COLT, 2005; Foygel and Srebro, 2011; Candes and Tao, 2010; Recht, 2009; Keshavan et al., 2010; Koltchinskii et al., 2010). Here we ask what can be said if the observed entries are chosen deterministically. We prove generalization error bounds for such deterministic algorithms, that resemble the results of Refs. Srebro et al. (2005); COLT (2005); Foygel and Srebro (2011) for the randomized algorithms. We still do not understand which sets of entries in a given matrix can be used to properly reconstruct it. Our hope is that the present work sheds some light on this problem. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 306–317, 2014