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Best sparse rank-1 approximation to higher-order tensors via a truncated exponential induced regularizer

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  • Mao, Xianpeng
  • Yang, Yuning

Abstract

Best sparse tensor rank-1 approximation consists of finding a projection of a given data tensor onto the set of sparse rank-1 tensors, which is important in sparse tensor decomposition and related problems. Existing models used ℓ0 or ℓ1 norms to pursue sparsity. In this work, we first construct a truncated exponential induced regularizer to encourage sparsity, and prove that this regularizer admits a reweighted property. Lower bounds for nonzero entries and upper bounds for the number of nonzero entries of the stationary points of the associated optimization problem are studied. By using the reweighted property of the regularizer, we develop an iteratively reweighted algorithm for solving the problem, and establish its convergence to a stationary point without any assumption. In particular, we show that if the parameter of the regularizer is small enough, then the support of the iterative points will be fixed after finitely many steps. Numerical experiments illustrate the effectiveness of the proposed model and algorithm.

Suggested Citation

  • Mao, Xianpeng & Yang, Yuning, 2022. "Best sparse rank-1 approximation to higher-order tensors via a truncated exponential induced regularizer," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    • Handle: RePEc:eee:apmaco:v:433:y:2022:i:c:s0096300322005070
    DOI: 10.1016/j.amc.2022.127433
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    References listed on IDEAS

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    1. Xianpeng Mao & Yuning Yang, 2022. "Several approximation algorithms for sparse best rank-1 approximation to higher-order tensors," Journal of Global Optimization, Springer, vol. 84(1), pages 229-253, September.
    2. 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.
    3. Will Wei Sun & Lexin Li, 2019. "Dynamic Tensor Clustering," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(528), pages 1894-1907, October.
    4. Junhui Wang, 2010. "Consistent selection of the number of clusters via crossvalidation," Biometrika, Biometrika Trust, vol. 97(4), pages 893-904.
    5. Will Wei Sun & Junwei Lu & Han Liu & Guang Cheng, 2017. "Provable sparse tensor decomposition," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 899-916, June.
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    Cited by:

    1. Cao, Baiheng & Wu, Xuedong & Wang, Yaonan & Zhu, Zhiyu, 2024. "Modified hybrid B-spline estimation based on spatial regulator tensor network for burger equation with nonlinear fractional calculus," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 253-275.

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