[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119421-en":3,"doc-seo-119421-105":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},119421,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Metric magnitude and topological methods for machine learning and biomedical data analysis - Doctor of Philosophy thesis","Methods based on geometry and topology enable analysis of high-dimensional data with complex structure by studying distances and connectivity relations. Magnitude, a recently introduced geometric invariant, captures intrinsic geometric properties and can quantify geometric quantities such as curvature, volume, and diameter for machine learning. This thesis presents the first applications of magnitude to theoretical deep learning, representation learning, and biomedical data analysis, while comparing magnitude’s geometric insights with topological views from persistent homology. It also introduces efficient algorithms to address the computational cost of magnitude.","This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e. g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use:  \n• This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated.  \n• A copy can be downloaded for personal non-commercial research or study, without prior permission or charge.  \n• This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author.  \n• The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author.  \n• When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given.  \nMetric magnitude and topological methods for machine learning and biomedical data analysis  \nRayna Andreeva  \nU  \nR  \nG  \nH  \nO  \nF  \nE  \nD  \nDoctor of Philosophy  \nInstitute for Adaptive and Neural Computation School of Informatics University of Edinburgh 2025  \nAbstract  \nWe live in a world which generates vast amounts of data with highly complex structure. Methods based on geometry and topology are suited to analyse the shape of high-dimensional data and thus can provide unique insights. While geometry is concerned with studying distances, topology focuses on connectivity relations. The main advantage of these methods is that they can generate compact summaries of the data to highlight and unravel distinct patterns and relationships. Magnitude is a recently introduced geometric invariant, capable of capturing important properties of the intrinsic geometry of a space. It has potential for applications in machine learning as it can measure a number of geometric quantities such as curvature, volume and diameter. In this thesis, we provide the first applications of magnitude to theoretical deep learning, representation learning and biomedical data analysis. In addition, we compare the geometric insights from magnitude with the topological insights from persistent homology. This thesis contains three parts, the first addresses one of the main difficulties in the application of magnitude, which is the computational cost. To compute magnitude, one needs to invert a matrix, which is an expensive procedure, particularly for large datasets. We provide new faster algorithms for speeding up this computation and approximate magnitude well. These new algorithms enable the applicability of magnitude to data analysis, providing a solid foundation for its wider adoption. The second part examines the intrinsic geometric aspect of machine learning. Here we show the unique uses of magnitude to generalization and the space of latent representations. In the third part, we demonstrate novel biomedical applications of magnitude to the surface of the human tongue and brain artery trees.  \nLay Summary  \nIn today’s world, we generate large amounts of complex data. To understand this data, we can use methods from geometry (which focuses on distances) and topology (which focuses on connectivity) . These methods help us summarize and uncover important patterns in the data. A new geometric concept, called ”magnitude,”can measure properties like curvature and volume, making it potentially useful for machine learning. This thesis presents the first applications of magnitude to areas such as deep learning and biomedical data analysis. It also compares magnitude’s geometric insights with those from a topological method called persistent homology. The thesis is divided into three parts: first, it addresses the high computational cost of calculating magnitude by introducing faster algorithms. These improvements make magnitude more practical for data analysis. The second part explores how magnitude can be used to understand key geometric aspects in machine learning, like how models generalize an","cbCait9KVukDl6nU","https://ap.wps.com/l/cbCait9KVukDl6nU","pdf",19531105,1,236,"English","en",105,"# Abstract\n# Lay Summary\n# Acknowledgements","[{\"question\":\"What are the key geometry and topology concepts used in this thesis?\",\"answer\":\"Geometry focuses on studying distances, while topology focuses on connectivity relations in high-dimensional data. The thesis uses both perspectives to summarize and reveal distinct patterns.\"},{\"question\":\"What is magnitude, and how is it used for machine learning and biomedical analysis?\",\"answer\":\"Magnitude is a geometric invariant designed to capture intrinsic geometric properties of a space, enabling measurement of quantities like curvature, volume, and diameter. The thesis applies magnitude to theoretical deep learning, representation learning, and biomedical data analysis.\"},{\"question\":\"How does the thesis address the computational challenge of calculating magnitude?\",\"answer\":\"Computing magnitude requires expensive matrix inversion, especially for large datasets. The thesis develops faster algorithms that approximate magnitude effectively, improving practical applicability for data analysis.\"}]","Metric magnitude and topological methods for machine learning and biomedical data analysis - Doctor of Philosophy thesis | PDF",1785724218,595,{"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},"metric-magnitude-and-topological-methods-for-machine-learning-and-biomedical-data-analysis-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/metric-magnitude-and-topological-methods-for-machine-learning-and-biomedical-data-analysis-doctor-of-philosophy-thesis/119421/",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},"What are the key geometry and topology concepts used in this thesis?","Question",{"text":75,"@type":76},"Geometry focuses on studying distances, while topology focuses on connectivity relations in high-dimensional data. The thesis uses both perspectives to summarize and reveal distinct patterns.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is magnitude, and how is it used for machine learning and biomedical analysis?",{"text":80,"@type":76},"Magnitude is a geometric invariant designed to capture intrinsic geometric properties of a space, enabling measurement of quantities like curvature, volume, and diameter. The thesis applies magnitude to theoretical deep learning, representation learning, and biomedical data analysis.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the thesis address the computational challenge of calculating magnitude?",{"text":84,"@type":76},"Computing magnitude requires expensive matrix inversion, especially for large datasets. 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