对对称张量的最佳正交逼近不能总是选择对称的
Optimal orthogonal approximations to symmetric tensors cannot always be chosen symmetric
摘要：
研究了对称张量的正交低秩逼近问题。在矩阵的情况下，近似是截断的奇异值分解，它是对称的。此外，对于任意维张量的秩一近似，Banach在1938年证明的一个经典结果表明，最优逼近总是可以选择为对称的。与这些结果相反，本文表明，相应的声明不再是正确的正交逼近的更高的秩。具体而言，对于文献中使用的四个常见的张量正交性的概念，我们表明，最佳的正交近似的秩大于1不能总是被选择为对称。
英文摘要：
We study the problem of finding orthogonal low-rank approximations of symmetric tensors. In the case of matrices, the approximation is a truncated singular value decomposition which is then symmetric. Moreover, for rank-one approximations of tensors of any dimension, a classical result proven by Banach in 1938 shows that the optimal approximation can always be chosen to be symmetric. In contrast to these results, this article shows that the corresponding statement is no longer true for orthogonal approximations of higher rank. Specifically, for any of the four common notions of tensor orthogonality used in the literature, we show that optimal orthogonal approximations of rank greater than one cannot always be chosen to be symmetric.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
