On the best rank-1 approximation of higher-order supersymmetric tensors
On the best rank-1 approximation of higher-order supersymmetric tensors
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DOI：
10.1137/s0895479801387413
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发表时间：
2002-03-06
影响因子：
1.5
通讯作者：
Regalia, PA
中科院分区：
文献类型：
--
作者：
Kofidis, E;Regalia, PA
最近的问题，确定最好的，在最小二乘意义下，秩-1逼近高阶张量进行了研究，并提出了一种迭代方法，扩展了著名的功率方法矩阵的解决方案。这种高阶幂方法也被提出用于特殊但重要的超对称张量类，没有变化。简化版本，适应于超对称问题的特殊结构，被认为是不可靠的，因为它的收敛性不能保证。本文的目的是表明，上述方法的对称版本下的凸性（或凸性）的假设下收敛的张量引起的问题，假设在实际应用中经常得到满足的功能。使用这个版本需要显着节省计算复杂性相比，无约束的高阶功率方法。此外，开发了一种新的方法，用于初始化的迭代过程中，已观察到产生的估计，位于更接近全局最优比之前建议的初始化。此外，它的接近全局最优是一个先验的定量。在分析过程中，还研究了张量的超对称性对于其方阵开折所蕴涵的一些重要性质。
Recently the problem of determining the best, in the least-squares sense, rank-1 approximation to a higher-order tensor was studied and an iterative method that extends the well-known power method for matrices was proposed for its solution. This higher-order power method is also proposed for the special but important class of supersymmetric tensors, with no change. simplified version, adapted to the special structure of the supersymmetric problem, is deemed unreliable, as its convergence is not guaranteed. The aim of this paper is to show that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications. The use of this version entails significant savings in computational complexity as compared to the unconstrained higher-order power method. Furthermore, a novel method for initializing the iterative process is developed which has been observed to yield an estimate that lies closer to the global optimum than the initialization suggested before. Moreover, its proximity to the global optimum is a priori quanti able. In the course of the analysis, some important properties that the supersymmetry of a tensor implies for its square matrix unfolding are also studied.