An efficient method for non-negative low-rank completion

• Published in: Advances in computational mathematics Vol. 46; no. 2
• Main Authors: Guglielmi, Nicola, Scalone, Carmela
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2020; Springer Nature B.V
• Subjects: Algorithms; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Constraints; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Numerical integration; Numerical methods; Sparse matrices; Visualization; Netflix problem; Non-negative completion; Low-rank completion; Non-negative factorization
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-020-09779-x

In this article, we propose a new method for low-rank completion of a large sparse matrix, subject to non-negativity constraint. As a challenging prototype of this problem, we have in mind the well-known Netflix problem. Our method is based on the derivation of a constrained gradient system and its numerical integration. The methods we propose are based on the constrained minimization of a functional associated to the low-rank completion matrix having minimal distance to the given matrix. In the main 2-level method, the low-rank matrix is expressed in the form of the non-negative factorization X = ε U V T , where the factors U and V are assumed to be normalized with unit Frobenius norm. In the inner level—for a given ε —we minimize the functional; in the outer level, we tune the parameter ε until we reach a solution. Numerical experiments on well-known large test matrices show the effectiveness of the method when compared with other algorithms available in the literature.