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Optimality conditions for Tucker low-rank tensor optimization

Author

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  • Ziyan Luo

    (Beijing Jiaotong University)

  • Liqun Qi

    (2012 Labs Huawei Technologies Investment Co., Ltd
    The Hong Kong Polytechnic University)

Abstract

Optimization problems with tensor variables are widely used in statistics, machine learning, pattern recognition, signal processing, computer vision, etc. Among these applications, the low-rankness of tensors is an intrinsic property that can help unearth potential but important structure or feature in the corresponding high-dimensional multi-way datasets, leading to the study on low-rank tensor optimization (LRTO for short). For the general framework of LRTO, little has been addressed in optimization theory. This motivates us to study the optimality conditions, with special emphasis on the Tucker low-rank constrained problems and the Tucker low-rank decomposition-based reformulations. It is noteworthy that all the involved optimization problems are nonconvex, and even discontinuous, due to the complexity of the tensor Tucker rank function or the multi-linear decomposition with the orthogonality or even group sparsity constraints imposed on factor matrices. By employing the tools in variational analysis, especially the normal cones to low-rank matrices and the properties of matrix manifolds, we propose necessary and/or sufficient optimality conditions for Tucker low-rank tensor optimization problems, which will enrich the context of the nonconvex and nonsmooth optimization.

Suggested Citation

  • Ziyan Luo & Liqun Qi, 2023. "Optimality conditions for Tucker low-rank tensor optimization," Computational Optimization and Applications, Springer, vol. 86(3), pages 1275-1298, December.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:3:d:10.1007_s10589-023-00465-4
    DOI: 10.1007/s10589-023-00465-4
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    References listed on IDEAS

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    1. Xiaoshan Li & Da Xu & Hua Zhou & Lexin Li, 2018. "Tucker Tensor Regression and Neuroimaging Analysis," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 10(3), pages 520-545, December.
    2. Bilian Chen & Zhening Li, 2020. "On the tensor spectral p-norm and its dual norm via partitions," Computational Optimization and Applications, Springer, vol. 75(3), pages 609-628, April.
    3. Hua Zhou & Lexin Li & Hongtu Zhu, 2013. "Tensor Regression with Applications in Neuroimaging Data Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 540-552, June.
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    Cited by:

    1. William W. Hager & R. Tyrrell Rockafellar & Vladimir M. Veliov, 2023. "Preface to Asen L. Dontchev Memorial Special Issue," Computational Optimization and Applications, Springer, vol. 86(3), pages 795-800, December.

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