Keywords
Robust subspace recovery, iteratively reweighted least squares, dynamic smoothing, nonconvex optimization, data visualization, kernel distance covariance
Abstract
The rapid expansion of data volume and complexity across diverse domains has underscored the need for robust and efficient techniques capable of extracting meaningful information from high-dimensional datasets. Subspace estimation—identifying the underlying low-dimensional structures within high-dimensional data—has emerged as a cornerstone method in modern data analysis and machine learning. This dissertation aims to advance both the theoretical and practical aspects of subspace estimation problems.
In the first part, we investigate Fréchet central subspace estimation, a critical component of sufficient dimension reduction (SDR) challenges. Traditional SDR methods often fall short when dealing with non-Euclidean responses. To address this, we propose a novel Fréchet SDR method that leverages kernel distance covariance, specifically designed for metric space-valued responses such as count data, probability densities, and other complex structures. By employing a kernel-based transformation to map these intricate responses into a suitable feature space, our approach facilitates efficient and accurate dimension reduction while accommodating the diverse and non-Euclidean characteristics inherent in modern datasets.
The second part of the dissertation focuses on robust subspace recovery, a fundamental task with applications in clustering, anomaly detection, and image processing, among others. Although Iteratively Reweighted Least Squares (IRLS) has demonstrated strong empirical performance, its theoretical foundations have remained largely unexplored. We rigorously establish that, under a set of deterministic conditions, a variant of IRLS augmented with dynamic smoothing regularization converges linearly to the true underlying subspace from any initialization. Additionally, we extend our theoretical guarantees to the more general setting of affine subspace estimation, offering novel insights and recovery guarantees in an area where existing theory is notably sparse.
Completion Date
2025
Semester
Spring
Committee Chair
Zhang, Teng
Degree
Doctor of Philosophy (Ph.D.)
College
College of Sciences
Department
Mathematics
Identifier
DP0029340
Document Type
Dissertation/Thesis
Campus Location
Orlando (Main) Campus
STARS Citation
Li, Kang, "Advanced Subspace Estimation Techniques for Statistical Analysis and Machine Learning" (2025). Graduate Thesis and Dissertation post-2024. 171.
https://stars.library.ucf.edu/etd2024/171