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Locally Optimal Eigenpairs of Orthogonally Decomposable Tensors: A Generalized Proof

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Abstract

Orthogonally decomposable (odeco) tensors is a special class of symmetric tensors. Previous works have focused on investigating its E-eigenpairs problem, and made some theoretical achievements concerning the number and the local optimality of E-eigenpairs. However, concerning local optimality of each eigenpair, the existing work only analyzed the third-order tensor case. In this paper, we further exploit this issue for any higher-order tensors by checking second-order necessary condition of the related constrained optimization model and deducing an equivalent matrix formula criterion for local optimality identification. Finally, a generalized conclusion for local optimality of eigenpairs for odeco tensors is provided, and some simulated experiments are conducted for validation.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. http://www.math.msu.edu/\(\sim \)chenlipi/TenEig.html

  2. https://www.researchgate.net/publication/374976951_Code-JOTA-Odeco

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Funding

This work was partially supported by National Natural Science Fund of China (62271090).

Author information

Authors and Affiliations

  1. The School of Microelectronics and Communication Engineering, Chongqing University, Chongqing, 400044, China

    Lei Wang & Lei Zhang

  2. Aerospace Information Research Institute, Chinese Academy of Sciences, Bei**g, 100094, China

    **urui Geng

  3. School of Electronic, Electrical and Communication Engineering, University of the Chinese Academy of Sciences, Bei**g, 100049, China

    **urui Geng

  4. Key Laboratory of Technology in Geo-Spatial Information Processing and Application System, Chinese Academy of Science, Bei**g, 100190, China

    **urui Geng

Authors
  1. Lei Wang
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  2. **urui Geng
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  3. Lei Zhang
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Contributions

XG/LZ: Conceptualization, Methodology. LW: Writing - original draft.

Corresponding author

Correspondence to Lei Wang.

Additional information

Communicated by Alexander Vladimirovich Gasnikov.

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Wang, L., Geng, X. & Zhang, L. Locally Optimal Eigenpairs of Orthogonally Decomposable Tensors: A Generalized Proof. J Optim Theory Appl 201, 199–220 (2024). https://doi.org/10.1007/s10957-024-02390-w

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  • DOI: https://doi.org/10.1007/s10957-024-02390-w

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