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A nonconvex formulation for low rank subspace clustering: algorithms and convergence analysis

Author

Listed:
  • Hao Jiang

    (Johns Hopkins University)

  • Daniel P. Robinson

    (Johns Hopkins University)

  • René Vidal

    (Johns Hopkins University)

  • Chong You

    (Johns Hopkins University)

Abstract

We consider the problem of subspace clustering with data that is potentially corrupted by both dense noise and sparse gross errors. In particular, we study a recently proposed low rank subspace clustering approach based on a nonconvex modeling formulation. This formulation includes a nonconvex spectral function in the objective function that makes the optimization task challenging, e.g., it is unknown whether the alternating direction method of multipliers (ADMM) framework proposed to solve the nonconvex model formulation is provably convergent. In this paper, we establish that the spectral function is differentiable and give a formula for computing the derivative. Moreover, we show that the derivative of the spectral function is Lipschitz continuous and provide an explicit value for the Lipschitz constant. These facts are then used to provide a lower bound for how the penalty parameter in the ADMM method should be chosen. As long as the penalty parameter is chosen according to this bound, we show that the ADMM algorithm computes iterates that have a limit point satisfying first-order optimality conditions. We also present a second strategy for solving the nonconvex problem that is based on proximal gradient calculations. The convergence and performance of the algorithms is verified through experiments on real data from face and digit clustering and motion segmentation.

Suggested Citation

  • Hao Jiang & Daniel P. Robinson & René Vidal & Chong You, 2018. "A nonconvex formulation for low rank subspace clustering: algorithms and convergence analysis," Computational Optimization and Applications, Springer, vol. 70(2), pages 395-418, June.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:2:d:10.1007_s10589-018-0002-6
    DOI: 10.1007/s10589-018-0002-6
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    References listed on IDEAS

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    1. Kristian Bredies & Dirk A. Lorenz & Stefan Reiterer, 2015. "Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 78-112, April.
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