Study of Low Rank Approximation of Tensorial Data Set via Non-convex Regularization

高阶张量的低秩近似在各种实际应用中是非常理想的，并且正在成为高效处理多维阵列的主要主题。在许多应用中，由于高度多维数组的某些局部相似性或某些周期性，例如在图像处理中，这种张量的秩通常看起来明显低于其大小。当多维数组由低秩结构控制时，处理这样的数据集变得更加容易，更重要的是，低秩属性表明数据集可以以有意义的方式进行显着压缩。这解释了为什么低秩特性在各种应用中如此有吸引力和实际有用。该项目的广泛意义和重要性主要体现在两个方面：第一，该项目旨在促进下一代数学理论/工具的创建和发展，以更有效和更高效地处理和处理高维数据集，从而扩展现有的方法论;第二，该项目将加强研究生/本科生培训的多学科课程，并提高本地K-12学生的数学学习兴趣，伊利诺斯州南部的一个农村地区，有许多低收入家庭。在过去的十年中，张量的低秩近似主要集中在凸正则化，而这种方法似乎是不够的，由于凸形式的限制。在这个提议中，我们将通过非凸正则化来开发张量的低秩近似，目前还没有很好地建立多维数据集的研究。在这一点上，很少有研究张量下的非凸公式。在这个建议中，我们提出了一个框架，在这个框架中，等价的问题可以在傅立叶域中制定，其中张量秩可以以更接近的方式来表征。在此框架下，非凸公式可以提供更有效的方法，张量相关problems than the existing methods.This award reflects NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Low rank approximation of higher-order tensors is highly desirable in various practical applications and is becoming a main theme in processing multi-dimensional arrays efficiently and effectively. Quite often in many applications, due to some local similarity or certain periodicity of a highly multidimensional array such as in image processing, the rank of such tensor usually appears to be significantly lower than its size. When a multi-dimensional array is governed by a low-rank structure, the handling of such a dataset becomes much more approachable, and more importantly, the low rank property indicates that the dataset can be significantly compressed in a meaningful way. This explains why the low rank characteristic is so attractive and practically useful in various applications. The broader significance and importance of this project are mainly reflected by two aspects: firstly, the project aims to promote the creation and development of the next generation of mathematical theory/tools for handling and processing high dimensional data sets more effectively and more efficiently, leading to expand the existing methodology; secondly, the project will enhance the multidisciplinary program for graduate/undergraduate student training and promote the mathematical learning interests for K-12 students in the local community, a rural area at Southern Illinois with many low-income families. During the last decade, the low rank approximation of tensors mainly focuses on convex regularization, and the approach appears to be insufficient due to the limitations of convex formulation. In this proposal, we will develop the low rank approximation of tensors via non-convex regularization, which currently is not well established yet for the study of multi-dimensional datasets. There exists very few study for tensors under the non-convex formulation at this point. In this proposal, we propose a framework in which equivalent problems can be formulated in the Fourier domain, where tensor ranks can be characterized in a more approachable way. In this framework, the non-convex formulation can provide more effective approach for tensor related problems than the existing methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.