低秩张量补全问题的算法研究

In processing the high dimensional data, we often use the tensors as suitable representation. However, there are so many missing values in the data. In this context, tensor completion has drawn lots of attention from reseachers in various fields in past several years and is an important topic in signal processing, computer vision, data mining and so on. Given a tensor with some elements missing, the task of tensor completion is to impute the missing values.. In this project, we mainly study the algorithms of low-rank tensor completion based on tensor TT (Tensor Train) rank minimization model. Our research content as follows. Firstly, the disadvantage of the most models at present is that there is regularization parameter, hence we will propose a new unconstraint model for tensor completion problem based on TT nuclear norm of tensor and indicator function and design a new algorithm by using the concept of proximal point mapping. Secondly, we will propose to replace the nuclear norm of the TT matricization of the tensor with a few non-convex function, such as weight TT nuclear norm and Gauss function. Then we will construct a non-convex model framework and use the methods of DC programming to solve it.. The research results of this project not only improve the shortcomings of the original methods for tensor completion, but also have extensive application prospect in image and video processing and data mining and so on.

人们在处理高维数据时通常用张量的形式来表示，并且数据中往往含有缺失值，在此背景下，低秩张量补全问题近年来受到许多领域学者的关注，已成为信号处理、计算机视觉和数据挖掘等领域炙手可热的课题，其主要任务是将低秩张量中缺失的元素补充完整。. 本项目研究基于张量TT（Tensor Train）秩极小化模型的低秩张量补全问题求解算法。研究内容分为两部分：一是针对目前求解低秩张量补全问题的大部分方法都含有正则参数的缺点，建立指示函数与张量TT核范数的极小化模型，并利用迫近映射的概念设计求解该模型的算法。二是提出用一些非凸函数（比如加权TT核范数、高斯函数等）去代替张量TT分解模式展开矩阵的核范数，构建一个非凸模型框架并采用DC规划的方法来求解。. 本项目的研究结果改进了原有低秩张量补全求解算法的不足，也为图像、视频处理，数据挖掘等领域提供了广泛的应用前景。