Matrix completion

Article summary

Matrix completion is the task of filling in the missing entries of a partially observed matrix, which is equivalent to performing data imputation in statistics. A wide range of datasets are naturally organized in matrix form. One example is the movie-ratings matrix, as appears in the Netflix problem: Given a ratings matrix in which each entry (i,j) represents the rating of movie j by customer i , if customer i has watched movie j and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. Another example is the document-term matrix: The frequencies of words used in a collection of documents can be represented as a matrix, where each entry corresponds to the number of times the associated term appears in the indicated document. Without any restrictions on the number of degrees of freedom in the completed matrix, this problem is underdetermined since the hidden entries could be assigned arbitrary values. Thus, we require some assumption on the matrix to create a well-posed problem, such as assuming it has maximal determinant, is positive definite, or is low-rank. For example, one may assume the matrix has low-rank structure, and then seek to find the lowest rank matrix or, if the rank of the completed matrix is known, a matrix of rank r that matches the known entries.…

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