张量分析及其在高维信息处理中的应用

项目名称： 张量分析及其在高维信息处理中的应用

项目编号： No.U1536104

项目类型： 联合基金项目

立项/批准年度： 2016

项目学科： 管理科学

项目作者： 王宁

作者单位： 北京电子技术应用研究所

项目金额： 64万元

中文摘要： 在信号处理和数据分析中，张量作为向量、矩阵在组织结构上的高维扩展，可以自然地表示高维数据。以张量为视角的数据分析处理方法能够保持数据蕴含的结构特性，从而避免在降维时损失重要信息。这个优势，使得张量方法在高维数据处理中极具潜力，已经被广泛应用于心理测量、医学成像、信号处理等工程领域。张量分析一般采用的手法有特征值和张量分解两类。而计算张量的特征值一般来说是关于张量维数的NP难问题，同时张量分解也无法向矩阵奇异值分解一样同时保持核张量的对角形式和因子矩阵的正交性。基于此，本项目将针对实际工程问题的特性设计若干高效、稳定的算法来计算特定特征值和张量分解问题。本项目还将对部分模型的最优性以及所设计的算法的收敛性进行理论分析，以期达到理论创新和方法创新的统一。

中文关键词： 张量分析；特征值；张量分解；高维数据；信号处理

英文摘要： Referring to signal processing and data analysis, we consider tensors as a generalization of scalars, vectors and matrices to higher-order structures. They are natural representations of high-dimensional data, thus can describe the complex objects in reality. Tensor approach has the advantage that it can preserve the native structure of data, has demonstrated the enormous potential of high-dimensional data processing. With the successful development and application of tensor methods in psychometrics since the 1960s, interest in tensor methods recently expanded to signal and image processing, computer vision, pattern recognition and other areas. The eigenvalues of tensors can be used to characterize the nature of high-dimensional data. However, computing the eigenvalues of a tensor is NP-hard in general. How to reasonably define the tensor eigenvalues and reveal their inner meaning is also a challenging problem. Tensor decomposition, a higher-order extension of singular value decomposition, is a powerful analysis tool for exploring the composition of high-dimensional data.Unlike singular value decomposition in matrix, tensor decomposition can not keep the diagonal form of the kernel tensor and the orthogonality of the factor matrix at the same time. Based on the above observations, our program will focus on problems of practical applications, and design several efficient and stable algorithms to compute certain eigenvalue and tensor decomposition. We will also make some theoretical analysis on optimality of some models and convergency of designed algorithms, which devotes to the unity of theoretical and methodical innovations.

英文关键词： tensor analysis;eigenvalues;tensor decomposition;High-Dimensional Data;signal processing