New Theoretical Foundations of Tensor Applications: Clustering, Error Analysis, Global Convergence, and Robust Formulations

张量应用的新理论基础：聚类、误差分析、全局收敛和稳健公式张量分解在分析高维多指标数据中变得越来越重要。然而，到目前为止，张量分解的应用受到了限制：(1)它们主要用于数据压缩？数据集群等至关重要的任务尚未得到解决。(2)不存在重建误差的界限？压缩参数是在反复试验的基础上确定的。(3)作为非凸优化问题的解，张量分解不是唯一的。这可能会严重影响张量分析的可靠性。(4)张量分解是通过最小化误差平方和得到的，因此容易受到噪声或离群点的影响。对于具有大噪声的应用，稳健的分解公式是非常理想的。在这个方案中，我们研究了张量应用的这些新的基本方面：(1)除了已有的关于聚类的理论结果之外，还研究了张量分解的聚簇能力；(2)提供了张量分解的全面误差分析，并得到了误差的上下界；(3)研究了张量分解的全局收敛的条件，并研究了对于全局收敛失败的情况下良好的初始化。(4)开发稳健的张量分解公式。此外，我们将开发用户友好的软件工具箱，其中包含所得到的算法，并将其提供给公众。我们还将教育研究生和本科生学习矩阵和张量计算的基础知识。我们将就这一新方向提供教程和组织研讨会。

New Theoretical Foundations of Tensor Applications: Clustering, Error Analysis, Global Convergence, and Robust FormulationsTensor decompositions become increasingly important in analyzing high-dimensional multi-index data. However, applications of tensor decompositions are so far restricted: (1) they are mainly used for data compression ? critically important tasks such as data clustering have not been addressed. (2) No bounds on reconstruction error exist ? the compression parameters are determined on a trial-and-error basis. (3) As solutions to non-convex optimizations, tensor decompositions are not unique. This could severely affect the reliability of tensor analysis. (4) Tensor decompositions are obtained via minimizing the sum of squared errors, thus are prone to noise or outliers in the data. A robust formulation of decomposition is highly desirable for applications with large noises. In this proposal, we investigate these new fundamental aspects of tensor applications: (1) Investigate the clustering capabilities of tensor decompositions, in addition to the established theoretical results on clustering; (2) Provide comprehensive error analysis of tensor decompositions and derive lower and upper error bounds; (3) Investigate conditions for global convergence for tensor decompositions and investigate good initializations for the cases where global convergence fails. (4) Develop robust formulations for tensor decompositions.In addition, we will develop user-friendly software toolbox that contains the resulting algorithms and make it available to the public. We will also educate graduate and undergraduate students with fundamentals in matrix and tensor computations. We will present tutorials and organize workshops on this new direction.