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An efficient method for non-negative low-rank completion

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Abstract

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 = εUVT, 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.

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Acknowledgments

The authors thank Pratik Jawanpuria and Bamdev Mishra for providing us valuable informations about the correct use of their code (see [19]). The authors also thank the anonymous referees for their valuable remarks; in particular, for suggesting the comparison with the SVT method of [6]. The MATLAB codes implementing the methods described in this article are available upon request to the authors.

Funding

The first author thanks the INdAM GNCS (Gruppo Nazionale di Calcolo Scientifico) for financial support and Italian MIUR (PRIN 2017 project “Discontinuous dynamical systems: theory, numerics and applications.”

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Authors and Affiliations

  1. School of Advanced Studies, Gran Sasso Science Institute, via Crispi 7, I-67100, L’Aquila, Italy

    Nicola Guglielmi

  2. Dipartimento di Ingeneria e Scienze dell’Informazione e Matematica, Università degli studi dell’Aquila, via Vetoio, I-67100, L’Aquila, Italy

    Carmela Scalone

Authors
  1. Nicola Guglielmi
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  2. Carmela Scalone
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Correspondence to Nicola Guglielmi.

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Communicated by: Lothar Reichel

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Guglielmi, N., Scalone, C. An efficient method for non-negative low-rank completion. Adv Comput Math 46, 31 (2020). https://doi.org/10.1007/s10444-020-09779-x

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