In this article, the rank-1 approximation of a nonnegative tensor $𝒜\in {ℝ}_{⩾0}^{n_1\times \dots \times n_m}$ is considered. Mathematically, the approximation problem can be formulated as an optimization problem. The Karush–Kuhn–Tucker (KKT) point of the optimization problem can be obtained by computing the nonnegative Z-eigenvector y of enlarged tensor $𝒢$. Therefore, we propose an iterative method with prediction and correction steps for computing nonnegative Z-eigenvector y of enlarged tensor $𝒢$, called the continuation method. In the theoretical part, we show that the computation requires only $O({\prod }_{i=1}^m{n}_i)$ flops for each iteration and the computed Z-eigenvector y has nonzero component block, and hence, the KKT point can be obtained. In addition, we show that the KKT point is a local optimizer of the optimization problem. Numerical experiments are provided to support the theoretical results.

中文翻译：

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非负张量的非负秩 1 近似的延续方法

在本文中，非负张量的 rank-1 近似 $𝒜\in {ℝ}_{⩾0}^{n_1\times \dots \times n_{米}}$被认为。在数学上，近似问题可以表述为优化问题。优化问题的 Karush-Kuhn-Tucker (KKT) 点可以通过计算放大张量的非负 Z 特征向量y来获得$𝒢$. 因此，我们提出了一种具有预测和校正步骤的迭代方法，用于计算放大张量的非负 Z 特征向量y$𝒢$，称为延续方法。在理论部分，我们表明计算只需要$哦({\prod }_{\mathrm{一世}=1}^{米}{n}_{\mathrm{一世}})$每次迭代的 flops 和计算的 Z 特征向量y具有非零分量块，因此，可以获得 KKT 点。此外，我们表明 KKT 点是优化问题的局部优化器。提供了数值实验来支持理论结果。