Wang Miaoyan, Duc Khanh Dao, Fischer Jonathan, Song Yun S
Department of Mathematics, University of Pennsylvania.
Department of Statistics, University of California, Berkeley.
Linear Algebra Appl. 2017 May 1;520:44-66. doi: 10.1016/j.laa.2017.01.017. Epub 2017 Jan 17.
Interest in higher-order tensors has recently surged in data-intensive fields, with a wide range of applications including image processing, blind source separation, community detection, and feature extraction. A common paradigm in tensor-related algorithms advocates unfolding (or flattening) the tensor into a matrix and applying classical methods developed for matrices. Despite the popularity of such techniques, how the functional properties of a tensor changes upon unfolding is currently not well understood. In contrast to the body of existing work which has focused almost exclusively on matricizations, we here consider all possible unfoldings of an order- tensor, which are in one-to-one correspondence with the set of partitions of {1, …, }. We derive general inequalities between the -norms of arbitrary unfoldings defined on the partition lattice. In particular, we demonstrate how the spectral norm ( = 2) of a tensor is bounded by that of its unfoldings, and obtain an improved upper bound on the ratio of the Frobenius norm to the spectral norm of an arbitrary tensor. For specially-structured tensors satisfying a generalized definition of orthogonal decomposability, we prove that the spectral norm remains invariant under specific subsets of unfolding operations.
近年来，高阶张量在数据密集型领域的关注度激增，其应用广泛，包括图像处理、盲源分离、社区检测和特征提取等。张量相关算法中的一个常见范式主张将张量展开（或展平）为矩阵，并应用针对矩阵开发的经典方法。尽管此类技术很受欢迎，但目前对于张量展开后其功能特性如何变化尚不清楚。与几乎只专注于矩阵化的现有工作不同，我们在此考虑一个阶张量的所有可能展开，这些展开与集合{1, …, }的划分集一一对应。我们推导了在划分格上定义的任意展开的 -范数之间的一般不等式。特别地，我们展示了张量的谱范数（ = 2）如何由其展开的谱范数界定，并得到了任意张量的Frobenius范数与谱范数之比的改进上界。对于满足广义正交可分解性定义的特殊结构张量，我们证明谱范数在特定的展开操作子集中保持不变。
