OPERATOR NORM INEQUALITIES BETWEEN TENSOR UNFOLDINGS ON THE PARTITION LATTICE.
OPERATOR NORM INEQUALITIES BETWEEN TENSOR UNFOLDINGS ON THE PARTITION LATTICE.
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DOI：
10.1016/j.laa.2017.01.017
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发表时间：
2017-05-01
影响因子：
1.1
通讯作者：
Song YS
中科院分区：
文献类型：
--
作者：
Wang M;Duc KD;Fischer J;Song YS
高阶张量的兴趣最近在数据密集型领域激增，具有广泛的应用，包括图像处理，盲源分离，社区检测和特征提取。张量相关算法中的一个常见范例主张将张量展开（或展平）为矩阵，并应用为矩阵开发的经典方法。尽管这种技术很受欢迎，但张量的功能特性如何在展开时变化目前还没有很好的理解。与现有的几乎只关注矩阵化的工作相反，我们在这里考虑k阶张量的所有可能的开折，它们与{1，.，k}的划分集一一对应。我们得到了一般的LP-范数之间的定义在划分格上的任意开折不等式。特别地，我们证明了张量的谱范数（p = 2）是如何被其开折的谱范数所约束的，并且得到了任意张量的Frobenius范数与谱范数之比的一个改进的上界.对于满足广义正交可分解性定义的特殊结构张量，我们证明了谱范数在展开操作的特定子集下保持不变。
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-k tensor, which are in one-to-one correspondence with the set of partitions of {1, …, k}. We derive general inequalities between the lp-norms of arbitrary unfoldings defined on the partition lattice. In particular, we demonstrate how the spectral norm (p = 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.