【作者】 贾晶晶；

【导师】 杨庆之；

【作者基本信息】 南开大学 ， 计算数学， 2017， 硕士

【摘要】 张量是高维数据的一种排列,可以看成矩阵的一种高阶推广。它在心理测量学,信号处理,神经科学,数值线性代数,数据挖掘,图分析等领域有着越来越重要的应用。近年来,关于张量理论的研究主要从张量分解和张量特征值两方面入手。论文中介绍了张量补全的一个可行模型和非负张量谱半径的一些性质。论文内容共分为四个部分。第一部分简要介绍了张量分解和张量特征值的发展现状。第二部分主要介绍了张量补全的优化模型,包括已被多数学者接受的一个模型以及我们提出来的基于张量的平衡展开形式的一个模型,并说明了该模型的可行性。第三部分给出了正张量的H-谱半径的Birkhof f-Hopf定理以及非负张量的Z-谱半径的上界。第四部分总结了论文的主要内容并指出了下一步的研究方向。更多还原

【Abstract】 A tensor is an array of high-dimensional data and can be regarded as a high-order generalization of a matrix.Tensors have important applications in some fields,such as psychometrics,signal processing,neuroscience,numerical linear algebra,data mining,graph analysis and so on.Recently,the study of tensor theories focuses on tensor decompositions and tensor eigenvalues.This paper mainly introduces a feasible model for tensor completion and some results on the spectral radii of nonnegative tensors.This paper falls into four parts.In the first part,we briefly introduce the development of tensor decompositions and tensor eigenvalues.In the second part,we mainly introduce optimization models for tensor completion,including a general model which has been accepted by many researchers and a new feasible model based on balanced unfoldings of tensors.In the third part,the Birkhoff-Hopf Theorem of H-spectral radius for positive tensors is given and upper bounds of Z-spectral radius for nonnegative tensors are also presented.In the last part,we carry on a summary of tensor completion and eigenvalues of nonnegative tensors and point out the further research work.更多还原