Author ORCID Identifier
https://orcid.org/0009-0008-2360-7995
Date Available
5-6-2025
Year of Publication
2025
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy (PhD)
College
Arts and Sciences
Department/School/Program
Mathematics
Faculty
Jing Qin
Faculty
Benjamin Braun
Abstract
The rapid growth of high-dimensional data has exposed the limitations of traditional vector and matrix-based methods for data analysis. These methods often struggle with computational inefficiencies, loss of critical cross-dimensional correlations, and challenges inherent in high-dimensional data. Tensors—multidimensional arrays—offer a robust framework for modeling and analyzing complex data. Tensor methods have proven valuable in tasks such as dimensionality reduction, feature extraction, and data compression, underpinning advancements in machine learning, computer vision, signal processing, and remote sensing.
This thesis focuses on two challenges in tensor analysis: tensor recovery and tensor processing. Tensor recovery addresses the reconstruction of incomplete or corrupted tensors. Leveraging low-rank or sparse structures in tensor data, this work introduces two novel regularized Kaczmarz algorithms that extend recovery methods to higher-order tensors, ensuring efficient reconstruction while maintaining computational feasibility.
In tensor processing, which involves extracting insights from high-dimensional data, this thesis explores hyperspectral band selection, proposing two methods: one utilizing spatial and spectral graph regularizations with matrix CUR decomposition, and another employing Generalized 3D Total Variation (G3DTV) with tensor CUR decomposition. These techniques enable effective classification and dimensionality reduction, improving computational efficiency and accuracy.
Throughout this thesis, we emphasize the preservation of the multidimensional structure of tensors, avoiding matricization to retain critical cross-dimensional correlations. These methodologies are supported by rigorous theoretical insights and thoroughly validated, showcasing their broad applicability and potential impact in advancing tensor-based analysis and recovery techniques.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2025.74
Recommended Citation
Henneberger, Katherine J., "Regularized Methods for Tensor Recovery and Processing" (2025). Theses and Dissertations--Mathematics. 119.
https://uknowledge.uky.edu/math_etds/119
