不同度量下的张量低秩近似及其扰动分析研究

The problem of the computation on the low rank approximation to a tensor and its corresponding perturbation analysis is a very important studying subject in numerical anslysis, and the research on this problem has many applications in numerical approximation, data compression, and image processing and so on. The research on the high-performance and robust algorithms and related theory for low rank approximation to a tensor under different metrics has not only significant values in expanding the research depth and breadth of the numerical linear algebra but also have very important meaning in developing the applications of numerical linear algebra in information science..Based on the algorithms and theory of the low rank approximation to a tensor under the existing Frobenius-norm, we establish the algorithms and theory of the low rank approximation to a tensor under different metrics. Specific contents are as following: (1)the 2-norm, weighted norm and nuclear norm theory of a tensor and the corresponding algorithms of low rank approximation to a tensor under the different norms; (2)the existence, uniqueness, and the perturbation invariance theory of the optimal low rank approximation to a tensor under different norms and perturbation analysis theory and robust algorithm for the low rank approximation to a tensor；(3)the simplification strategy and the order reduction method for huge tensor and the fast algorithm of the low rank tensor approximation for the applications in video compression and the denoising or repairing of the hyperspectral images.

高阶张量的低秩近似计算及其扰动分析问题是数值分析中的一个重要研究课题，该研究在数值逼近，数据压缩，图像处理等领域有着广泛的应用。对不同度量下张量低秩近似的高性能且具有稳健性算法及相关扰动分析理论的研究，不仅可以拓展数值代数理论研究的深度和广度，而且对促进数值代数在信息科学中的应用与发展具有重要作用。.本项目基于现有的Frobenius-范数框架下的张量低秩近似算法和理论，拟研究不同度量下的张量低秩近似算法和理论，具体包括如下内容：(1)张量的2-范数，加权范数，核范数等相关理论，以及相应的不同范数下的低秩近似算法；(2)不同度量下的张量最优低秩近似的存在性、唯一性、扰动不变性特征，以及张量低秩近似的扰动分析理论和鲁棒的低秩近似算法；(3)大规模张量的约简、降阶方法，以及张量低秩近似的快速算法在视频压缩和高光谱图像去噪或修复中的应用。

张量的低秩近似计算及其扰动分析问题不仅是数值分析中的一个重要研究课题，而且张量的低秩近似在信息科学中也越来越发挥着重要的作用。本项目在完善张量低秩近似的理论性质，以及探索张量低秩近似的应用方面进行了研究，所得的结果主要包括： (1)获得了张量最佳秩-1近似唯一性的刻画方式，建立了张量最佳秩-1近似的扰动不变性理论；(2)完善了张量谱范数和核范数的相关理论，给出了合理的张量谱范数、核范数，以及有限维张量空间最佳秩-1近似比上界、下界的估计；(3)得到了张量约简的可行方法，给出了一种张量稀疏性的度量方式，并且将该稀疏性度量方式成功应用在高光谱图像填充上面。