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Partially symmetric tensor structure preserving rank-R approximation via BFGS algorithm

Author

Listed:
  • Ciwen Chen

    (National University of Defense Technology)

  • Guyan Ni

    (National University of Defense Technology)

  • Bo Yang

    (National University of Defense Technology)

Abstract

It is known that many tensor data have symmetric or partially symmetric structure and structural tensors have structure preserving Candecomp/Parafac (CP) decompositions. However, the well-known alternating least squares (ALS) method cannot realize structure preserving CP decompositions of tensors. Hence, in this paper, we consider numerical problems of structure preserving rank-R approximation and structure preserving CP decomposition of partially symmetric tensors. For the problem of structure preserving rank-R approximation, we derive the gradient formula of the objective function, obtain BFGS iterative formulas, propose a BFGS algorithm for positive partially symmetric rank-R approximation, and discuss the convergence of the algorithm. For the problem of structure preserving CP decomposition, we give a necessary condition for partially symmetric tensors with even orders to have positive partially symmetric CP decompositions, and design a general partially symmetric rank-R approximation algorithm. Finally, some numerical examples are given. Through numerical examples, we find that if a tensor has a positive partially symmetric CP decomposition then its partially symmetric rank CP decomposition must be a positive CP decomposition. In addition, we compare the BFGS algorithm proposed in this paper with the standard CP-ALS method. Numerical examples show that the BFGS algorithm has better stability and faster computing speed than CP-ALS algorithm.

Suggested Citation

  • Ciwen Chen & Guyan Ni & Bo Yang, 2023. "Partially symmetric tensor structure preserving rank-R approximation via BFGS algorithm," Computational Optimization and Applications, Springer, vol. 85(2), pages 621-652, June.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:2:d:10.1007_s10589-023-00471-6
    DOI: 10.1007/s10589-023-00471-6
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    References listed on IDEAS

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