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Low-rank factorization for rank minimization with nonconvex regularizers

Author

Listed:
  • April Sagan

    (Rensselaer Polytechnic Institute)

  • John E. Mitchell

    (Rensselaer Polytechnic Institute)

Abstract

Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective technique to solve the problem with strong performance guarantees. However, nonconvex relaxations have less estimation bias than the nuclear norm and can more accurately reduce the effect of noise on the measurements. We develop efficient algorithms based on iteratively reweighted nuclear norm schemes, while also utilizing the low rank factorization for semidefinite programs put forth by Burer and Monteiro. We prove convergence and computationally show the advantages over convex relaxations and alternating minimization methods. Additionally, the computational complexity of each iteration of our algorithm is on par with other state of the art algorithms, allowing us to quickly find solutions to the rank minimization problem for large matrices.

Suggested Citation

  • April Sagan & John E. Mitchell, 2021. "Low-rank factorization for rank minimization with nonconvex regularizers," Computational Optimization and Applications, Springer, vol. 79(2), pages 273-300, June.
    • Handle: RePEc:spr:coopap:v:79:y:2021:i:2:d:10.1007_s10589-021-00276-5
    DOI: 10.1007/s10589-021-00276-5
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    References listed on IDEAS

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    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    2. Magnus, Jan R., 1985. "On Differentiating Eigenvalues and Eigenvectors," Econometric Theory, Cambridge University Press, vol. 1(2), pages 179-191, August.
    3. Xin Shen & John E. Mitchell, 2018. "A penalty method for rank minimization problems in symmetric matrices," Computational Optimization and Applications, Springer, vol. 71(2), pages 353-380, November.
    4. April Sagan & Xin Shen & John E. Mitchell, 2020. "Two Relaxation Methods for Rank Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 806-825, September.
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