Bounded Matrix Low Rank Approximation

  • Ramakrishnan Kannan
  • Mariya Ishteva
  • Barry Drake
  • Haesun Park
Chapter
Part of the Signals and Communication Technology book series (SCT)

Abstract

Low rank approximation is the problem of finding two matrices \(\mathbf {P} \in \mathbb {R}^{m \times k}\) and \(\mathbf {Q} \in \mathbb {R}^ {k \times n}\) for input matrix \(\mathbf {R} \in \mathbb {R}^{m \times n}\), such that \(\mathbf {R} \approx \mathbf {PQ} \). It is common in recommender systems rating matrix, where the input matrix \(\mathbf {R}\) is bounded in the closed interval \([r_{min},r_{max}]\) such as [1, 5]. In this chapter, we propose a new improved scalable low rank approximation algorithm for such bounded matrices called bounded matrix low rank approximation (BMA) that bounds every element of the approximation \(\mathbf {PQ}\). We also present an alternate formulation to bound existing recommender systems algorithms called BALS and discuss its convergence. Our experiments on real-world datasets illustrate that the proposed method BMA outperforms the state-of-the-art algorithms for recommender system such as stochastic gradient descent, alternating least squares with regularization, SVD++ and bias-SVD on real-world datasets such as Jester, Movielens, Book crossing, Online dating, and Netflix.

Notes

Acknowledgments

This work was supported in part by the NSF Grant CCF-1348152, the Defense Advanced Research Projects Agency (DARPA) XDATA program grant FA8750-12-2-0309, Research Foundation Flanders (FWO-Vlaanderen), the Flemish Government (Methusalem Fund, METH1), the Belgian Federal Government (Interuniversity Attraction Poles—IAP VII), the ERC grant 320378 (SNLSID), and the ERC grant 258581 (SLRA). Mariya Ishteva is an FWO Pegasus Marie Curie Fellow. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF or the DARPA.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ramakrishnan Kannan
    • 1
  • Mariya Ishteva
    • 2
  • Barry Drake
    • 1
  • Haesun Park
    • 1
  1. 1.Georgia Institute of TechnologyAtlantaUSA
  2. 2.Vrije Universiteit Brussel (VUB)BrusselsBelgium

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