Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery
Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery
复制标题
基于克罗内克基表示的张量稀疏性及其在张量恢复中的应用
DOI：
10.1109/tpami.2017.2734888
复制
发表时间：
2018
影响因子：
23.6
通讯作者：
Xu Zongben
中科院分区：
文献类型：
--
作者：
Xie Qi;Zhao Qian;Meng Deyu;Xu Zongben
稀疏建模作为一种很有前途的数据分析方法，在科学和工程领域取得了巨大的成功。众所周知，向量/矩阵的稀疏性/低秩可以分别由非零元素数（l0范数）/非零奇异值数（秩）来合理地度量。然而，来自真实的应用的数据通常是由多个因素的相互作用生成的，这显然不能由向量/矩阵充分表示，而高阶张量被期望提供更忠实的表示以传递这种数据集合的内在结构。与向量/矩阵的情况不同，构造张量的有理高阶稀疏性度量是一项相对困难的任务。为此，在本文中，我们提出了一种张量稀疏性的度量，称为基于Kronecker基表示的张量稀疏性度量（简称KBR），它编码了Tucker和CANDECOMP/PARAFAC（CP）低秩分解的稀疏性见解。然后，我们研究了KBR正则化最小化（KBRM）问题，并设计了一个有效的ADMM算法来解决它，其中每个涉及的参数可以更新的封闭形式的方程。这样一个高效的求解器使得KBR可以扩展到各种任务，如张量完成和张量鲁棒主成分分析。一系列的实验，包括多光谱图像（MSI）去噪，MSI完成和背景减除，证实了所提出的方法超越国家的最先进的优越性。
As a promising way for analyzing data, sparse modeling has achieved great success throughout science and engineering. It is well known that the sparsity/low-rank of a vector/matrix can be rationally measured by nonzero-entries-number (l0norm)/nonzerosingular-values-number (rank), respectively. However, data from real applications are often generated by the interaction of multiple factors, which obviously cannot be sufficiently represented by a vector/matrix, while a high order tensor is expected to provide more faithful representation to deliver the intrinsic structure underlying such data ensembles. Unlike the vector/matrix case, constructing a rational high order sparsity measure for tensor is a relatively harder task. To this aim, in this paper we propose a measure for tensor sparsity, called Kronecker-basis-representation based tensor sparsity measure (KBR briefly), which encodes both sparsity insights delivered by Tucker and CANDECOMP/PARAFAC (CP) low-rank decompositions for a general tensor. Then we study the KBR regularization minimization (KBRM) problem, and design an effective ADMM algorithm for solving it, where each involved parameter can be updated with closed-form equations. Such an efficient solver makes it possible to extend KBR to various tasks like tensor completion and tensor robust principal component analysis. A series of experiments, including multispectral image (MSI) denoising, MSI completion and background subtraction, substantiate the superiority of the proposed methods beyond state-of-the-arts.