On the convergence of higher-order orthogonal iteration
On the convergence of higher-order orthogonal iteration
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DOI：
10.1080/03081087.2017.1391743
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发表时间：
2015-04
影响因子：
1.1
通讯作者：
Yangyang Xu
中科院分区：
文献类型：
--
作者：
Yangyang Xu
摘要高阶正交迭代法（HOOI）已被广泛用于寻找张量的最佳低秩多线性逼近。然而，它的收敛性仍然是一个悬而未决的问题。在本文中，我们首先分析了一个贪婪的HOOI，它更新每个因子矩阵，从最佳候选人中选择一个最接近当前的因子。假设块非退化簇点的存在性，通过Kurdyka-Schojasiewicz性质证明了其全局收敛性.此外，我们表明，如果起点是足够接近任何块非退化的全局最优解，贪婪HOOI产生一个收敛到全局最优解的序列。通过将原始HOOI生成的序列与贪婪HOOI生成的序列相联系，证明了原始HOOI在多线性子空间序列上具有全局收敛性，从而肯定地解决了这个问题。
Abstract 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–ᴌojasiewicz 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.