The Optimization Algorithms For Canonical Tensor Decomposition And The Generalized Eigenvalue Of The Symmetric Tensor

In recent years, the tensor decomposition and tensor eigenvalue have been wide-ly applied to many fields, such as chemometrics, image processing, hypergraph the-ory and high order Markov chain, etc, and researchments on their theories and algorithms has attracted many scholars to study.In this paper, we first consider the problem of the canonical decomposition of real valued tensor, a modified Barzilai-Borwein method is proposed for fitting canonical decomposition, and the convergence of this algorithm is discussed. In order to verify the effectiveness of the algorithm, we apply the new algorithm to the amino acid fluorescence data and artificially generated tensors decomposition,and compare it with CPOPT and CPBBOPT. In many cases, the numerical results show that the new algorithm is superior to CPOPT and CPBBOPT in the number of iterations and CPU times. Then we study the generalized eigenvalue problem of real valued symmetric tensor. The unconstrained optimization method for the gen-eralized eigenvalue problem of real valued symmetric matrix is extended to the real valued symmetric tensor. Two unconstrained optimization models for solving gen-eralized eigenvalue of real valued symmetric tensors are presented. The properties of these models axe analyzed and a numerical example is given, the limited memory quasi-newton method is used for solving these two models and the numerical results show the validity of these models.Finally, we make a summary for this paper and put forward some problems to be solved.