高阶张量的低秩恢复问题研究

项目名称： 高阶张量的低秩恢复问题研究

项目编号： No.11471242

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张新珍

作者单位： 天津大学

项目金额： 72万元

中文摘要： 高阶张量的低秩恢复问题是一类极小化问题，它是信号处理、图像处理等应用学科的前沿问题,也是最优化与张量计算领域的基本问题。该问题是向量稀疏解问题与矩阵极小秩问题的高阶推广，它们都是NP-难的。由于张量与矩阵、向量的本质区别，使得我们无法套用向量稀疏解问题与矩阵极小秩问题的研究思路来研究该问题。本项目拟从以下方面对该问题展开研究。首先，研究张量的多线性秩性质，建立该问题新的等价形式及相应松弛问题，并对松弛问题提出有效数值算法。其次，研究对称张量空间上的对称CP秩极小化问题,将问题转化为带有多项式约束的向量稀疏解问题并进行松弛，建立松弛问题的优化理论并提出有效算法，且推广所建理论到一般张量的CP秩极小化问题。再次，讨论两种低秩恢复模型松弛问题的精确恢复条件及逼近解理论。最后，设计适合于大规模计算的数值算法并编制有效的实用软件。该研究不仅能够推动优化理论与张量计算的理论与算法，而且有一定的实际价值

中文关键词： 张量；张量秩；张量分解；低秩恢复；数值优化

英文摘要： The low-rank recovery of higher order tensor is a minimization problem,which is a basic problem in optimization and tensor computation, and is also frontier issue in signal processing, image processing and other disciplines in applied sciences. Generaly speaking, this problem is a generalization of the vector sparse solution and the matrix low-rank minimization, which are NP-hard to solve. However, this problem cannot be solved via the method of dealing with the problems of vector sparse solution and the matrix low-rank minimization, due to the essential difference between tensor and matrix, vector. In this project, the low-rank recovery of higher order tensor will be studied in the following aspects. Firstly, the properties of multilinear rank of higher order tensor are studied, by which a new equivalent form of the considered problem is presented as well as the relaxed one. Then a numerical algorithm for the relaxed problem is given. Secondly, the low symmetric CP-rank minimization of the symmetric tensor is transformed as a vector sparse solution problem with polynomial constraints. From this, the relaxed problem is investigated, whose optimization theory and numerical algorithms are established. Then, the obtained theories are extened to those for general CP-rank minimization. Thirdly, the exact sparse recovery conditions of two kinds of relaxed low-rank recovery problems are discussed. Then, the approximation solution theory will be studied. Finally, the numerical algorithms applicable to large-scale compulations will be designed and then the practical soft programming will be presented. In a word, this project can not only facilitate the theory on optimization theory and tensor computation, but also provide theoretical support to the technical sciences.

英文关键词： tensor;tensor rank;tensor decomposition;low-rank recovery;numerical optimization