The Barazilai And Borwein Method For Fitting Canonical Tensor Decompositions

Tensor decompositions are higher-order analogues of matrix decompositions and have proven to be powerful tools for data analysis. In particular, in this paper we are interested in the canonical tensor decompositions, known as CANDECOM-P/PARAFAC (CP), which expresses a tensor as the sum of component rank-one ten-sors and is used in many applications such as chemometrics, signal processing, neuro-science and web analysis. The task of computing CP, however, is difficult. The typical approach is based on alternating least-squares optimization (ALS), nonlinear least-squares method (NLS), and the nonliner conjugate gradient (NCG) method. Compu-tational experiments demonstrate that the gradient based methods are more accurate than ALS and faster than NLS. In this paper, we propose the Barzilai and Borwein gradient method for fitting canonical tensor decompositions. This method requires less storage locations and inexpensive computations. Furthermore, a nonmonotone line search strategy that guarantees global convergence is combined with the Barzilai and Borwein method. We discuss the mathematical calculation of the derivatives, pro-pose new algorithm establish the global convergence and report the numerical results which indicate that the Barzilai and Borwein gradient method is faster and more cheap than usual algorithms.