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、全局收敛和鲁棒公式

• 批准号: 0917274
• 负责人: Chris Ding
• 金额: $ 25.08万
• 依托单位: University of Texas at Arlington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0917274&HistoricalAwards=false
• 关键词: New; Theoretical; Foundations; Tensor; Applications

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.

张量应用的新理论基础：聚类，误差分析，全局收敛和鲁棒的配方构成分解在分析高维多索引数据方面越来越重要。但是，到目前为止，张量分解的应用受到限制：（1）它们主要用于数据压缩？尚未解决至关重要的任务，例如数据聚类。 （2）重建错误没有界限吗？压缩参数是根据反复试验确定的。 （3）作为非凸优化的解决方案，张量分解并非唯一。这可能严重影响张量分析的可靠性。 （4）张量分解是通过最大程度地减少平方误差的总和而获得的，因此数据中易于噪声或离群值。对于具有较大噪音的应用，非常需要分解的强大表述。在此提案中，我们研究了张量应用的这些新的基本方面：（1）研究张量分解的聚类能力，除了既定的集群理论结果； （2）提供张量分解的全面误差分析，并得出下部和上部误差界； （3）研究张量分解的全局收敛条件，并研究全局收敛失败的情况的良好初始化。 （4）为张量分解开发可靠的配方。此外，我们将开发包含所得算法的用户友好的软件工具箱，并使公众可用。我们还将在矩阵和张量计算中教育研究生和本科生。我们将介绍教程并在这个新方向上组织研讨会。