基于张量结构和lq范数的低秩张量恢复和补全

项目名称： 基于张量结构和lq范数的低秩张量恢复和补全

项目编号： No.61501300

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 无线电电子学、电信技术

项目作者： 孙维泽

作者单位： 深圳大学

项目金额： 21万元

中文摘要： 低秩张量恢复和补全是一个在数据挖掘、视频数据和生物信息数据处理等多个领域有着广泛应用的科学与工程问题。精确、高效和鲁棒性张量恢复及补全算法研究是数据处理中需要解决的一个核心问题。本项目通过深入研究数据的张量特性、张量分解所得元素的特征，在矩阵恢复和补全的算法的基础上，提出基于张量及其分解所得元素的结构和特征的算法，从而取得更好的恢复和补全结果。同时，通过将矩阵子空间的分析方法拓展至张量子空间，进一步的发掘张量子空间和核心张量的应用机理，从而从理论上对张量恢复和补全的方法进行深入的分析和探讨。更进一步地，通过结合lq范数的思想与张量恢复的模型，我们给出基于lq范数的张量恢复的公式，及应用迭代、l2范数平滑、凸优化等数学方法求解，提出相应的恢复算法，并将其拓展至张量补全中。本项目的研究将拓展高维张量恢复和补全的算法尤其是鲁棒性算法在多个领域中的应用，并为其提供充分的数学和理论依据。

中文关键词： 张量恢复；张量补全；高维信号；稀疏表示；低秩张量

英文摘要： Low-rank tensor recovery and completion are important topics in science and engineering because they are widely used in many real-world applications such as data mining, video signal and biological information data processing. Derivation of Accurate, efficient and robust tensor recovery and completion algorithms is a key problem in data processing. Based on the researches on matrix recovery and completion, and a deep study on the structure and characteristics of tensor as well as the components from its decompositions, we propose to derive new tensor recovery and completion algorithms to obtain better recovery and completion results. At the same time, by extending the analysis methods of recovery and completion algorithms from matrix subspace to tensor subspace, we explore the application mechanism of the tensor subspaces and the core tensor, and then perform the theoretical analysis of the tensor recovery and completion algorithms. Furthermore, by combing the idea of lq norm and the model of tensor recover, we propose the tensor recovery equations based on lq norm, and then apply the mathematical methods such as iteration procedure, l2 norm smoothing and convex optimization to derive corresponding algorithms, as well as extend it to the area of tensor completion. These researches will expand the application of high-dimensional tensor recovery and completion algorithms, especially robust algorithms, in many applications, and provide sufficient mathematical and theoretical analysis for the algorithms.

英文关键词： Tensor Recovery;Tensor Completion;Multidimentional Signal;Sparse Representation;Low-rank Tensors