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

基于非凸正则化的张量数据集低秩逼近研究

基本信息

批准号：
1854638
负责人：
MingQing Xiao
金额：
$ 15.11万
依托单位：
Southern Illinois University at Carbondale
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854638&HistoricalAwards=false
关键词：
Study Low Rank Approximation Tensorial

项目摘要

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.

高阶张量的低秩逼近在各种实际应用中是非常需要的，并且正在成为高效处理多维数组的主要主题。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，此类张量的秩通常显着低于其大小。当多维数组由低秩结构管理时，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著压缩。这就解释了为什么低阶特性在各种应用中如此有吸引力和实际用途。本项目更广泛的意义和重要性主要体现在两个方面：第一，本项目旨在促进更有效和更高效地处理和处理高维数据集的下一代数学理论/工具的创建和发展，从而扩展现有的方法；其次，该项目将加强研究生/本科生多学科培训计划，并促进当地社区K-12学生的数学学习兴趣，该社区位于伊利诺伊州南部的农村地区，有许多低收入家庭。近十年来，张量的低秩逼近主要集中在凸正则化上，由于凸公式的限制，这种方法显得不够充分。在本提案中，我们将通过非凸正则化发展张量的低秩逼近，该方法目前尚未很好地建立用于多维数据集的研究。在这一点上，关于非凸形式下张量的研究很少。在这个建议中，我们提出了一个框架，其中等效问题可以在傅里叶域中表述，其中张量秩可以以更接近的方式表征。在此框架下，非凸公式可以提供比现有方法更有效的张量相关问题的解决方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。