Regularization methods for learning incomplete matrices

AbstractWe use convex relaxation techniques to provide a sequence of solutions to the matrix completionproblem. Using the nuclear norm as a regularizer, we provide simple and very efficient algorithms forminimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm iterativelyreplaces the missing elements with those obtained from a thresholded SVD. With warm starts this allowsus to efficiently compute an entire regularization path of solutions. 1 Introduction In many applications measured data can be represented in a matrix X m×n , for which only a relativelysmall number of entries are observed. The problem is to “complete” the matrix based on the observedentries, and has been dubbed the matrix completion problem [CCS08, CR08, RFP07, CT09, KOM09]. The“Netflix” competition is a primary example, where the data is the basis for a recommender system. Therows correspond to viewers and the columns to movies, with the entry X ij being the rating ∈{1,...,5}byviewer i for movie j. There are 480K viewers and 18K movies, and hence 8.6 billion (8.6 ×10