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Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical. David Hilbert The goal of this book is to explain the principles that made support vector machines (SVMs) a successful modeling and prediction tool for a variety of applications. We try to achieve this by presenting the basic ideas of SVMs together with the latest developments and current research questions in a uni?ed style. In a nutshell, we identify at least three reasons for the success of SVMs: their ability to learn well with only a very small number of free parameters, their robustness against several types of model violations and outliers, and last but not least their computational e?ciency compared with several other methods. Although there are several roots and precursors of SVMs, these methods gained particular momentum during the last 15 years since Vapnik (1995, 1998) published his well-known textbooks on statistical learning theory with aspecialemphasisonsupportvectormachines. Sincethen,the?eldofmachine learninghaswitnessedintenseactivityinthestudyofSVMs,whichhasspread moreandmoretootherdisciplinessuchasstatisticsandmathematics. Thusit seems fair to say that several communities are currently working on support vector machines and on related kernel-based methods. Although there are many interactions between these communities, we think that there is still roomforadditionalfruitfulinteractionandwouldbegladifthistextbookwere found helpful in stimulating further research. Many of the results presented in this book have previously been scattered in the journal literature or are still under review. As a consequence, these results have been accessible only to a relativelysmallnumberofspecialists,sometimesprobablyonlytopeoplefrom one community but not the others.
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In a nutshell, we identify at least three reasons for the success of SVMs: their ability to learn well with only a very small number of free parameters, their robustness against several types of model violations and outliers, and last but not least their computational e?ciency compared with several other methods.
2 646 kr
Kommande
This book offers a comprehensive exploration of the theoretical foundations of deep learning, bridging the gap between its ever increasing empirical success and rigorous mathematical understanding. With a focus on interdisciplinary approaches, it illuminates the interplay between mathematics, statistics, and computer science to address pressing challenges in deep learning research. While deep learning continues to revolutionize industries such as healthcare, gaming, and autonomous systems, its applications in solving mathematical problems like inverse problems and partial differential equations mark a paradigm shift in scientific inquiry. Yet, this rapid progress often outpaces our theoretical understanding, leaving critical questions about expressivity, optimization, robustness, and fairness unanswered.This book aims to close this gap by presenting perspectives from mathematics, statistics, and applications. This interdisciplinary endeavor draws upon diverse mathematical frameworks, including algebraic geometry, differential geometry, stochastics, functional analysis, and topology, alongside fundamental tools from statistics and theoretical computer science. By synthesizing these fields, the book not only advances the theoretical foundation of deep learning but also sets the stage for novel applications across scientific domains. Targeted at researchers, advanced students, and professionals in mathematics, computer science, and related fields, this book serves as both an introduction to emerging theoretical insights and a roadmap for interdisciplinary collaboration. It is will be of value to anyone seeking to understand or to contribute to the future of deep learning.