[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119486-en":3,"doc-seo-119486-105":30,"detail-sidebar-cat-0-en-105":91},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},119486,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Tensor-Variate Machine Learning on Graphs - Doctor of Philosophy Thesis","Traditional machine learning faces major obstacles as big data grows rapidly in both volume and application scope, expanding data sources and forms. High-dimensional and multi-way data increases computation and can lead to model over-fitting, commonly attributed to the curse of dimensionality. Tensors help address this by compressing high-dimensional data into low-rank forms while preserving structure and interpretability, reducing complexity from exponential to linear in data dimensions. The thesis further targets irregular domains and relational structure inherent in modern data by combining tensor methods with graph learning to better model complex signals. It proposes tensor-graph learning frameworks, validates them on air quality forecasting, protein classification, and financial modelling, and shows improved or comparable performance with higher interpretability and lower computational cost.","Tensor-Variate Machine Learning on Graphs  \nYao Lei Xu  \nCommunications & Signal Processing Group Department of Electrical & Electronic Engineering Imperial College London  \nA thesis submitted for the degree of  \nDoctor of Philosophy  \n2023  \nAbstract  \nTraditional machine learning algorithms are facing significant challenges as the world enters the era of big data, with a dramatic expansion in volume and range of applications and an increase in the variety of data sources. The large-and multidimensional nature of data often increases the computational costs associated with their processing and raises the risks of model over-fitting-a phenomenon known asthe curse of dimensionality. To this end, tensors have become a subject of great interest in the data analytics community, owing to their remarkable ability to super-compress high-dimensional data into a low-rank format, while retaining the original data structure and interpretability. This leads to a significant reduction in computational costs, from an exponential complexity to a linear one in the data dimensions.  \nAn additional challenge when processing modern big data is that they often reside on irregular domains and exhibit relational structures, which violates the regular grid assumptions of traditional machine learning models. To this end, there has been an increasing amount of research in generalizing traditional learning algorithms to graph data. This allows for the processing of graph signals while accounting for the underlying relational structure, such as user interactions in social networks, vehicle flows in traffic networks, transactions in supply chains, chemical bonds in proteins, and trading data in financial networks, to name a few.  \nAlthough promising results have been achieved in these fields, there is a void in literature when it comes to the conjoint treatment of tensors and graphs for data analytics. Solutions in this area are increasingly urgent, as modern big data is both large-dimensional and irregular in structure. To this end, the goal of this thesis is to explore machine learning methods that can fully exploit the advantages of both tensors and graphs. In particular, the following approaches are introduced: (i) Graphregularized tensor regression framework for modelling high-dimensional data while accounting for the underlying graph structure; (ii) Tensor-algebraic approach for computing efficient convolution on graphs; (iii) Graph tensor network framework for designing neural learning systems which is both general enough to describe most existing neural network architectures and flexible enough to model large-dimensional data on any and many irregular domains. The considered frameworks were employed in several real-world applications, including air quality forecasting, protein classification, and financial modelling. Experimental results validate the advantages of the proposed methods, which achieved better or comparable performance against stateof-the-art models. Additionally, these methods benefit from increased interpretability and reduced computational costs, which are crucial for tackling the challenges posed by the era of big data.  \nContents  \nList of Figures 1  \nList of Tables 11  \nList of Proposed Algorithms 13  \nNomenclature 14  \nStatement of Originality 17  \nCopyright Declaration 18  \nAcknowledgments 19  \nPublications 20  \n1 Introduction 23  \n1.1 Challenges of the Modern Big Data ................... 23  \n1.2 A Brief History of Tensors and Graphs .................. 27  \n1.3 Motivation and Aims of the Thesis ................... 28  \n1.4 Thesis Organization ........................... 29  \n1.5 Contributions ............................... 30  \n2 Preliminaries 37  \n2.1 Preliminaries on Tensors ......................... 37  \n2.2 Preliminaries on Graphs ......................... 46  \nI Tensor Learning Methods 50  \n3 Tensor Contraction via Tensor-Train Decomposition 51  \n3.1 Introduction ................................ 51  \n3.2 Contraction in the Tensor","cbCaivimNrzJ8W3f","https://ap.wps.com/l/cbCaivimNrzJ8W3f","pdf",20893044,1,186,"English","en",105,"# Introduction\n## Challenges of the Modern Big Data\n## A Brief History of Tensors and Graphs\n## Motivation and Aims of the Thesis\n## Thesis Organization\n## Contributions\n# Preliminaries\n## Preliminaries on Tensors\n## Preliminaries on Graphs\n# Tensor Learning Methods\n## Tensor Contraction via Tensor-Train Decomposition\n## Tensor-Train Recurrent Neural Networks for Finance\n# Graph Learning Methods\n## Graph Theory for Metro Traffic Modelling\n## Hierarchical Graph Pooling for Finance","[{\"question\":\"Why are tensors important for high-dimensional machine learning in this thesis?\",\"answer\":\"Tensors enable super-compression of high-dimensional data into low-rank formats while retaining original structure and interpretability, reducing computation complexity from exponential to linear in data dimensions.\"},{\"question\":\"What problem arises when modern big data lives on irregular domains?\",\"answer\":\"Irregular domains and relational structures violate the regular grid assumptions of traditional learning models, motivating graph-based generalizations for graph signals.\"},{\"question\":\"What major frameworks does the thesis propose and what results do they achieve?\",\"answer\":\"It introduces a graph-regularized tensor regression framework, a tensor-algebraic approach for efficient graph convolution, and a graph tensor network framework. Experiments on air quality, protein classification, and financial modelling show better or comparable performance with improved interpretability and reduced computational cost.\"}]","Tensor-Variate Machine Learning on Graphs - Doctor of Philosophy Thesis | PDF",1785724568,469,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"tensor-variate-machine-learning-on-graphs-doctor-of-philosophy-thesis","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/tensor-variate-machine-learning-on-graphs-doctor-of-philosophy-thesis/119486/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why are tensors important for high-dimensional machine learning in this thesis?","Question",{"text":75,"@type":76},"Tensors enable super-compression of high-dimensional data into low-rank formats while retaining original structure and interpretability, reducing computation complexity from exponential to linear in data dimensions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What problem arises when modern big data lives on irregular domains?",{"text":80,"@type":76},"Irregular domains and relational structures violate the regular grid assumptions of traditional learning models, motivating graph-based generalizations for graph signals.",{"name":82,"@type":73,"acceptedAnswer":83},"What major frameworks does the thesis propose and what results do they achieve?",{"text":84,"@type":76},"It introduces a graph-regularized tensor regression framework, a tensor-algebraic approach for efficient graph convolution, and a graph tensor network framework. 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