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Orthogonal Low Rank Tensor Approximation


Academic report of Haitian scholars


Reporter: Prof. Moody T. Chu


Report time: June 22, 2016 PM 13:30-15:30


Location: 508#room, Research and education building


Contact: DONG Bo (tel: 84708351-8313)


Report title and abstract: 

1.Title: Orthogonal Low Rank Tensor Approximation

Abstract: With the notable exceptions of two cases-that tensors of order 2, namely, matrices, always have best approximations of arbitrary low ranks and that tensors of any order always have the best rank-one approximation, it is known that high-order tensors may fail to have best low rank approximations. When the condition of orthogonality is imposed, even under the modest assumption that only one set of components in the decomposed rank-one tensors is required to be mutually perpendicular, the situation is changed completely-orthogonal low rank approximations always exist. The purpose of this paper is to discuss the best low rank approximation subject to orthogonality. The conventional high-order power method is modified to address the orthogonality via the polar decomposition. Algebraic geometry technique is employed to show that for almost all tensors the orthogonal alternating least squares method converges globally.


2.Title: Fej\'{e}r-Riesz Factorization: Challenges and Applications

Abstract: Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fej\'{e}r-Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. Such a notion includes the Cholesky decomposition as a trivial case and generalizes the notion of sums-of-squares. The challenge is at its numerical realization which has been long sought after in engineering applications. In this work, the simplest form of degree one factorization finds its application to the nonlinear matrix equation X+A^{*}X^{-1}A = Q. A suitable parametrization recasts the nonlinear equation as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired.


Brief introductio to the reporter: Prof. Moody T. Chu's research directions include numerical algebra, eigenvalue inverse problem, numerical solution methods of ODE and so on. Prof. CHU has already published more than one hundred SCI retrival papers, which include the top journals such as SIAM Review (three papers), Acta Numerica (two papers). As an editor of the top journal of computational mathematics, SIAM Journal on Matrix and Applications and reviewers in dozens of journals, he has published a monograph in Oxford University press, and been invited to make speeches in international conferences many times.



School of Mathematical Sciences, Dalian University of Technology 

June 20, 2016