In recent years, withDeep LearningIn image recognition,Natural Language ProcessingMore and more researchers and developers are investing in this field as a breakthrough in fields such as the field. However, despite the significant success of deep learning in practice, the theoretical mechanisms behind it still confuses many people. This is why I want to recommend a book to you today - "Mathematical Theory of Deep Learning" (Mathematical Theory of Deep Learning), written by two scholars, Philipp Petersen and Jakob Zech, it explores the mathematical principles of deep learning to help readers understand whyNeural NetworkIt can solve complex problems so effectively.
1. Author background
The two authors of this book, Philipp Petersen, are from the University of Vienna and Jakob Zech, from the University of Heidelberg, have a rich academic background in mathematics and scientific computing. The content of this book also comes from the course lecture notes offered by the two authors at their respective universities. After many modifications and expansions, this highly systematic textbook was finally formed.
2. Content overview
"Mathematical Theory of Deep Learning" is divided into 16 chapters, which systematically analyze the basic concepts and principles of deep learning from a mathematical perspective. The content of the book revolves around the three pillars of deep learning—Approximate theory、Optimization theoryandStatistical learning theoryExpand and gradually reveal the working mechanism of deep learning at the mathematical level to readers.
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Approximate theorySection (Chapter 2 to 9) discusses the approximate capabilities of neural networks, especially for continuousfunctionThe approximation problem covers the classical approximation theorem (such as the general approximation theorem) and the approximation performance of ReLU activation functions.
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Optimization theoryThe section (Chapter 10 to 13) focuses on the training process of neural networks, explores gradient descent, stochastic gradient descent, backpropagation algorithms, and acceleration methods, revealing why these optimization algorithms can successfully train deep neural networks.
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Statistical learning theoryPart (Chapter 14 to Chapter 16) focuses onDeep Learning Modelgeneralization performance, especially in the case of over-parameterization, why the model can generalize well. In addition, this section discusses the confrontational sample problem and proposes coping strategies.
3. Uniqueness
Unlike other deep learning books, this book focuses onMathematical Analysis. It avoids traditionMachine LearningThe cumbersome discussion of practical applications in books focuses more on theoretical derivation and mathematical proof. This makes this book especially suitable for readers who want to delve into the mathematical principles behind deep learning. Here are some of the major features of this book:
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Focus on theoretical deduction: This book helps readers understand the approximation ability and optimization mechanism of neural networks through detailed mathematical deduction, providing a rigorous perspective to examine the working principles of deep learning.
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Easy to understand mathematical concepts: Although the content involves complex mathematical theories, the author presents concepts in a concise manner as much as possible, avoiding unnecessary abstractions, which allows readers with a certain mathematical foundation to better understand these theories.
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Comprehensive coverage: The book not only includes the basic theories of neural networks, but also covers the latest research progress, such as cutting-edge topics such as training dynamics of wide neural networks, loss landscape analysis, and generalization ability under over-parameterization.
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Application-oriented exercises: Each chapter in the book comes with a series of exercises that help readers deepen their understanding of theory from different angles, especially for researchers who want to apply the theory to practical problems.
4. Recommended objects
This book is suitable for researchers, doctoral students and senior undergraduate students in mathematics, computer science and related fields. Since it involves a lot of mathematical deduction, it is recommended that readers have basic analytics,Linear algebra, probability theory and functional analysis knowledge. If you are interested in the mathematical principles of deep learning and want to understand this field in a theoretical way, then this book will be your best choice.
V. Conclusion
As a book with strong theoretical deep learning, "Mathematical Theory of Deep Learning" answers why deep learning can be successfully applied to various complex problems through rigorous mathematical analysis. This book not only helps readers understand the core theories of deep learning, but also provides theoretical support for future research and application. If you want to lay a solid foundation in the mathematical theory of deep learning, you might as well study this book in depth, which will provide important guidance for your research path.
bibliography:
Philipp Petersen, Jakob Zech, Mathematical Theory of Deep Learning, 2024.