Tensor Computations Using Optimization Methods And Their Applications

Tensor computation theory and methods have more and more applications in numerous areas of science and engineering, such as Chemometrics, Signal Processing, Data Mining and Medical Engineering. This thesis is devoted to exploit the inherent structure of real problems involving tensors and develop corresponding optimization methods.First, we study an issue from Medical Engineering in Chapter2. The positive definiteness of diffusion tensors is important in magnetic resonance imaging because it reflects the phenomenon of water molecular diffusion in a complicated biological tissues environment. To preserve this property, we represent it as an explicit pos-itive semidefinite (PSD) matrix constraint and some linear matrix equalities. The objective function is the regularized linear least squares fitting for the log-linearized Stejskal-Tanner equation. The regularization term is the heuristic nuclear norm of the PSD matrix, since we expect it to be of low-rank. In this way, we establish a con-vex quadratic semidefinite programming (SDP) model, whose global solution exists. For such a model, we design a new augmented Lagrangian based alternating direc-tion method (ADM) for the dual problem. Sensitivity analyses on the coefficients of the optimal diffusion tensor and the optimal objective function value with respect to noise-corrupted signals are presented. Experiments on synthetic data with multiply fibers show that the new method is robust to the Rician noise and it outperforms sev-eral existing methods. Furthermore, when the fibre orientation distribution function is considered, the new method is competitive to the Q-ball imaging. Using the human brain data, we illustrate that the fibre reconstruction produced by the new method ex-presses the true configuration of nervous fibre bundles. Additionally, the new method generates positive definite generalized diffusion tensors in all voxels while the uncon-strained least squares fitting fails. Finally, we confirm that the ADM solver is more efficient than SDPT3and QSDP for this special problem.Second, we consider the problem of tensor decompositions in Chapter3. In signal processing, data analysis and scientific computing, one often encounters the problem of decomposing a tensor into a sum of contributions. To solve such prob- lems, both the search direction and the step size are two crucial elements in numerical algorithms, such as the alternating least squares algorithm (ALS). Owing to the non-linearity of the problem, the often used linear search direction is not always powerful enough. In this paper, we propose two higher-order search directions. The first one, geometric search direction, is constructed via a combination of two successive linear search directions. The second one, algebraic search direction, is constructed via a quadratic approximation of three successive iterates. Then, in an enhanced line search along these directions, the optimal complex step size contains two arguments:modu-lus and phase. A current strategy is ELSCS that finds these two arguments alternately. So it may suffer from a local optimum. We broach a direct method, which determines these two arguments simultaneously, so as to obtain the global optimum. Finally, nu-merical comparisons on various search directions and step size schemes are reported in the context of blind separation-equalization of convolutive DS-CDMA mixtures. The results show that the new search directions greatly improve the efficiency of ALS and the new step size strategy is competitive.In Chapter4, a successive unconstrained dual optimization (SUDO) algorithm is proposed for best rank-one approximation problems of high order tensors, in the least-squares sense. The main idea is to transform the constrained dual program of the best rank-one approximation problems of tensors into a successive unconstrained optimization problems. Then the unconstrained optimization problems are solved by a fast gradient based method which makes up of the steepest ascent direction, a novel initial step length strategy and the backtracking line search rule at each iteration. The global convergence theorems of the SUDO algorithm to a Karush-Kuhn-Tucker point of the dual program is proved. Preliminary numerical experiments show that the new SUDO method outperforms the popular alternating least squares method.