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

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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.