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

Ciwen Chen (), Guyan Ni () and Bo Yang ()
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Ciwen Chen: National University of Defense Technology
Guyan Ni: National University of Defense Technology
Bo Yang: National University of Defense Technology

Computational Optimization and Applications, 2023, vol. 85, issue 2, No 9, 652 pages

Abstract: 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.

Keywords: Partially symmetric tensors; Rank-R approximation; CP decomposition; BFGS method (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-023-00471-6

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