On the convergence of higher-order orthogonal iteration

作     者：Xu, Yangyang 

作者机构：Rensselaer Polytech Inst Dept Math Sci Troy NY 12180 USA 

出 版 物：《LINEAR & MULTILINEAR ALGEBRA》 (线性与多重线性代数)

年 卷 期：2018年第66卷第11期

页      面：2247-2265页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：National Science Foundation, NSF, (1719549, DMS-1719549) National Science Foundation, NSF 

主　　题：Higher-order orthogonal iteration (HOOI) global convergence Kurdyka-lojasiewicz (KL) property greedy algorithm block coordinate descent 

摘      要：The higher-order orthogonal iteration (HOOI) has been popularly used for finding a best low-multilinear rank approximation of a tensor. However, its convergence is still an open question. In this paper, we first analyse a greedy HOOI, which updates each factor matrix by selecting from the best candidates one that is closest to the current iterate. Assuming the existence of a block-nondegenerate cluster point, we establish its global iterate sequence convergence through the so-called Kurdyka-?ojasiewicz property. In addition, we show that if the starting point is sufficiently close to any block-nondegenerate globally optimal solution, the greedy HOOI produces an iterate sequence convergent to a globally optimal solution. Relating the iterate sequence by the original HOOI to that by the greedy HOOI, we then show that the original HOOI has global convergence on the multilinear subspace sequence and thus positively address the open question.