On norm compression inequalities for partitioned block tensors
On norm compression inequalities for partitioned block tensors
复制标题
DOI：
10.1007/s10092-020-0356-x
复制
发表时间：
2020-02
期刊：
影响因子：
1.7
通讯作者：
Zhening Li;Yun-Bin Zhao
中科院分区：
文献类型：
--
作者：
Zhening Li;Yun-Bin Zhao
当一个张量被分割成子张量时，这些子张量的一些张量范数形成一个张量，称为范数压缩张量。张量的范数压缩不等式集中在压缩张量的范数与原始张量的范数之间的关系。证明了对于张量谱范数，压缩张量的范数是原张量的范数的上界。这个结果可以推广到一般的张量谱范数。讨论了张量范数压缩不等式的各种应用。这些不等式改进了许多文献中已有的张量范数的界，特别是通过张量划分收紧了张量谱范数的一般界。研究了张量空间的谱范数与Frobenius范数的极值比，给出了估计其上界的一般方法，特别是改进了三阶非负张量和对称张量的最佳上界.我们还提出了一个更快的方法来估计一个大的张量或矩阵的谱范数通过序列范数压缩不等式的理论和数值证据。例如，我们的算法的复杂性矩阵谱范数是从0到1取决于分区和估计范围相应地从一些接近的上限到确切的谱范数。
When a tensor is partitioned into subtensors, some tensor norms of these subtensors form a tensor called a norm compression tensor. Norm compression inequalities for tensors focus on the relation of the norm of this compressed tensor to the norm of the original tensor. We prove that for the tensor spectral norm, the norm of the compressed tensor is an upper bound of the norm of the original tensor. This result can be extended to a general class of tensor spectral norms. We discuss various applications of norm compression inequalities for tensors. These inequalities improve many existing bounds of tensor norms in the literature, in particular tightening the general bound of the tensor spectral norm via tensor partitions. We study the extremal ratio between the spectral norm and the Frobenius norm of a tensor space, provide a general way to estimate its upper bound, and in particular, improve the current best upper bound for third order nonnegative tensors and symmetric tensors. We also propose a faster approach to estimate the spectral norm of a large tensor or matrix via sequential norm compression inequalities with theoretical and numerical evidence. For instance, the complexity of our algorithm for the matrix spectral norm iswhereranges from 0 to 1 depending on the partition and the estimate ranges correspondingly from some close upper bound to the exact spectral norm.