[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-128221-en":3,"doc-seo-128221-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},128221,2336475104362,"Eden","https://ap-avatar.wpscdn.com/avatar/22000c4c46a41b752dd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786595829695023868",8,"Research & Report","Advanced Subspace Estimation Techniques for Statistical Analysis and Machine Learning","Rapid growth in data volume and complexity demands reliable methods for uncovering structure in high-dimensional datasets. This dissertation develops advanced subspace estimation techniques at both theoretical and practical levels. It studies Fréchet central subspace estimation for sufficient dimension reduction with non-Euclidean, metric space responses using kernel distance covariance. It also proves robust subspace recovery using an IRLS variant with dynamic smoothing regularization, establishing linear convergence under deterministic conditions and extending guarantees to affine subspaces.","University of Central Florida  \nSTARS  \nGraduate Thesis and Dissertation post-2024  \n2025  \nAdvanced Subspace Estimation Techniques for Statistical Analysis and Machine Learning  \nKang Li  \nFind similar works at: [https://stars.library.ucf.edu/etd2024](https://stars.library.ucf.edu/etd2024)  \nUniversity of Central Florida Libraries [http://library.ucf.edu](http://library.ucf.edu)  \nThis Dissertation/Thesis is brought to you for free and open access by STARS. It has been accepted for inclusion in Graduate Thesis and Dissertation post-2024 by an authorized administrator of STARS. For more information, please [contact](contact STARS@ucf.edu)[ STARS@ucf.edu](contact STARS@ucf.edu).  \nSTARS Citation  \nLi, Kang, \"Advanced Subspace Estimation Techniques for Statistical Analysis and Machine Learning\"(2025) . Graduate Thesis and Dissertation post-2024. 171.  \n[https://stars.library.ucf.edu/etd2024/171](https://stars.library.ucf.edu/etd2024/171)  \nADVANCED SUBSPACE ESTIMATION TECHNIQUES FOR STATISTICAL ANALYSIS  \nAND MACHINE LEARNING  \nby  \nKANG LI  \nB.A., Hefei University of Technology, 2012  \nM.A., Saint Louis University, 2018  \nA dissertation submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy  \nin the Department of Mathematics  \nin the College of Science  \nat the University of Central Florida  \nSpring Term  \n2025  \nMajor Professor: Teng Zhang  \n© 2025 Kang Li  \nii  \nABSTRACT  \nThe 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 highdimensional 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.  \nIn the first part, we investigate Frchet 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 Frchet 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 kernelbased transformation to map these intricate responses into a suitable feature space, our approach facilitates efficient and accurate dimension reduction while accommodating the diverse and nonEuclidean characteristics inherent in modern datasets.  \nThe 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.  \nACKNOWLEDGMENTS  \nFirst and foremost, I would like to express my deepest gratitude to my advisor, Prof. Teng Zhang, for his unwavering support, guidance, and mentorship throughout my time at UCF. I have learned a lot from them, such as independence, critical thinking, a rigorous attitude toward research, and scientific writing. I am also immensely grateful to Dr. Hsin-Hsiung Huang, a valued committee member and key collaborator, whose substantial contributions and expertise have been instrumental in shaping this work.  \nI am sincerely grateful to all the professor","cbCaikUGiA8zfqaJ","https://ap.wps.com/l/cbCaikUGiA8zfqaJ","pdf",9087214,4,1,113,"English","en",105,"# Chapter 1: Introduction\n## 1.1 Subspace Estimation\n## 1.2 Dissertation Organization\n# Chapter 2: Fr´Echet Sufficient Dimension Reduction for Metric Spacevalued Data Via Distance Covariance\n## 2.1 Introduction\n## 2.2 Methodology","[{\"question\":\"What problem does the dissertation address?\",\"answer\":\"It addresses how to extract meaningful low-dimensional structure from high-dimensional data, advancing subspace estimation for statistical analysis and machine learning.\"},{\"question\":\"How does the dissertation handle non-Euclidean responses in sufficient dimension reduction?\",\"answer\":\"It proposes a Fréchet SDR method that uses kernel distance covariance to map metric space–valued responses into a suitable feature space for efficient dimension reduction.\"},{\"question\":\"What theoretical results are established for robust subspace recovery?\",\"answer\":\"It rigorously proves that an IRLS variant with dynamic smoothing regularization converges linearly to the true underlying subspace under deterministic conditions, and it extends guarantees to affine subspace estimation.\"}]","Advanced Subspace Estimation Techniques for Statistical Analysis and Machine Learning | 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problem does the dissertation address?","Question",{"text":76,"@type":77},"It addresses how to extract meaningful low-dimensional structure from high-dimensional data, advancing subspace estimation for statistical analysis and machine learning.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the dissertation handle non-Euclidean responses in sufficient dimension reduction?",{"text":81,"@type":77},"It proposes a Fréchet SDR method that uses kernel distance covariance to map metric space–valued responses into a suitable feature space for efficient dimension reduction.",{"name":83,"@type":74,"acceptedAnswer":84},"What theoretical results are established for robust subspace recovery?",{"text":85,"@type":77},"It rigorously proves that an IRLS variant with dynamic smoothing regularization converges linearly to the true underlying subspace under deterministic conditions, and it extends guarantees to affine subspace 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