Constructing Chebyshev approximations for matrices and tensors and their applications.

In the talk the main results of my PhD thesis will be presented. The thesis is devoted to the problems of constructing low-rank approximations of matrices and tensors in the Chebyshev norm. An important component for solving these problems is the problem of constructing the best uniform approximation by a system of vectors. We will describe the optimality criterion and an effective algorithm for solving the problem of the best uniform approximation. In addition, the alternating minimization method for constructing low-rank approximations of matrices and tensors in the Chebyshev norm for an arbitrary rank will be proposed and the theoretical properties of the method will be presented. In particular, the concept of multy-way alternance will be introduced and it will be shown that the presence of an alternance structure is a necessary condition for the optimality of the approximation, and all limit points of the alternating minimization method satisfy this property. Based on the analysis, a method for constructing optimal Chebyshev approximations of rank 1 for matrices will be proposed. All the presented results will be accompanied by a large number of numerical experiments.