Exact Matrix Completion via Convex Optimization

• Published in: Foundations of computational mathematics Vol. 9; no. 6; pp. 717 - 772
• Main Authors: Candès, Emmanuel J., Recht, Benjamin
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.12.2009; Springer Nature B.V
• Subjects: Applications of Mathematics; Computer Science; Economics; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; 15A52; Nuclear norm minimization; Random matrices; Noncommutative Khintchine inequality; Decoupling; Convex optimization; 90C25; Duality in optimization; 90C59; Low-rank matrices; Compressed sensing; Matrix completion
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-009-9045-5

We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M . Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys for some positive numerical constant C , then with very high probability, most n × n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.