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Orthogonal Low Rank Tensor Approximation

2016年06月21日 09:11  点击:[]

海天学者学术报告

报告人简介:

Moody T. Chu(朱天照)教授的研究兴趣包括数值代数、特征值反问题以及常微方程的数值解法等,共发表SCI检索论文100余篇,其中包括数学类最顶级期刊SIAM Review3篇), Acta Numerica2篇)等,在牛津大学出版社出版专著一本,担任计算数学顶级期刊SIAM Journal on Matrix and Applications的编辑及数十个期刊的审稿人,曾受邀近百次在国际会议做邀请报告。

报告安排:

时间:2016.06.22(周三)下午13:30-15:30    地点:研教楼508

报告题目及摘要:

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.

 

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数学科学学院

2016.06.20

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