Matrix completion from any given set of observations

Abstract

In the matrix completion problem the aim is to recover an unknown real matrix from a 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 prize. A central approach to this problem is to output a matrix of lowest possible complexity (e.g. rank or trace norm) that agrees with the partially specified matrix. The performance of this approach under the assumption that the revealed entries are sampled randomly has received considerable attention (e.g. [1, 2, 3, 4, 5, 6, 7, 8]). In practice, often the set of revealed entries is not chosen at random and these results do not apply. We are therefore left with no guarantees on the performance of the algorithm we are using. We present a means to obtain performance guarantees with respect to any set of initial observations. The first step remains the same: find a matrix of lowest possible complexity that agrees with the partially specified matrix. We give a new way to interpret the output of this algorithm by next finding a probability distribution over the non-revealed entries with respect to which a bound on the generalization error can be proven. The more complex the set of revealed entries according to a certain measure, the better the bound on the generalization error.