非负张量分解的算法研究及其应用

In this project, we are mainly concerned with three kinds of nonnegative tensor factorizations: the Tucker decomposition, the CANDECOMP/PARAFAC (CP) decomposition and the Tensor-train (TT) decomposition. The nonnegative Tucker decomposition with the semi-orthogonal constraint is concerned and reformulated into its equivalent Riemannian optimization problem. We wish to construct some stable and efficient algorithms from Riemannian optimization viewpoint. Analysis on the convergence and stability properties of these algorithms will be investigated. The new algorithms will be used to pattern recognition, cluster analysis, data mining, and so on. For the nonnegative CP decomposition, we will try to propose an algorithm for sparse nonnegative CP decomposition. In fact, since the data in many applications are usually sparse, it is reasonable to add the sparsity in the constraint conditions. Besides, developing Riemannian optimization methods for the orthogonal nonnegative CP decomposition is also an important content of this project. For the nonnegative TT decomposition, we also consider adding the semi-orthogonal constraint to the problem. Then we will try to solve the problem by reformulating it into its equivalent optimization problem on the product of manifolds.

在本项目中，我们将从优化的角度考虑三类非负张量分解问题：Tucker分解、CANDECOMP/PARAFAC (CP)分解与Tensor-train (TT)分解。对于非负Tucker分解，我们将考虑带有半正交约束条件的非负Tucker分解，并将问题等价地转化为相应的黎曼优化问题，然后从黎曼优化的角度设计稳定有效的算法。我们还将分析相关算法的收敛性和稳定性，并将算法应用于模式识别、聚类分析、数据挖掘等领域中。对于非负CP分解，我们将考虑稀疏的非负CP分解算法。许多实际应用问题的数据往往带有稀疏性，在非负张量分解中添加稀疏性约束条件是合理的。将黎曼优化方法应用于正交非负CP分解也是本项目的一个主要内容。对于非负TT分解，我们也将考虑在约束中加入半正交的约束条件，从而利用相应的流形优化方法，把问题等价地转化为相应地乘积流形上的优化问题。

在本项目中，我们原计划从优化的角度考虑三类非负张量分解：Tucker分解、CANDECOMP/PARAFAC（CP）分解与Tensor-train（TT）分解的算法设计及应用问题。在项目进行中，我们针对一类基于Einstein乘积的Toeplitz类的张量方程设计了最优预处理子（optimal preconditioner），并应用于图像的恢复问题中。我们还研究了基于张量缩并乘积与张量T-乘积的交换子（李乘积）的Frobenius范数的最优上界问题。为了构建新的张量分解模型，我们还将矩阵的半张量积推广到张量领域，分别给出两个张量的半张量积定义和张量与矩阵的新模式乘积定义，并研究这些新乘积的性质与应用。此外，基于张量与多项式问题的联系，项目成员还研究了有限域上的若干类置换多项式的构造问题。

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