Mathematical Aspects of Deep Learning

doi:10.1017/9781009025096

In recent years the development of new classification and regression algorithms based on deep learning has led to a revolution in the fields of artificial intelligence, machine learning, and data analysis. The development of a theoretical foundation to guarantee the success of these algorithms constitutes one of the most active and exciting research topics in applied mathematics. This book presents the current mathematical understanding of deep learning methods from the point of view of the leading experts in the field. It serves both as a starting point for researchers and graduate students in computer science, mathematics, and statistics trying to get into the field and as an invaluable reference for future research.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-xii
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Contributors
pp xiii-xiv
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Preface
pp xv-xviii
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1 - The Modern Mathematics of Deep Learning
pp 1-111
- By Julius Berner, Philipp Grohs, Gitta Kutyniok, Philipp Petersen
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2 - Generalization in Deep Learning
pp 112-148
- By K. Kawaguchi, Y. Bengio, L. Kaelbling
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3 - Expressivity of Deep Neural Networks
pp 149-199
- By Ingo Gühring, Mones Raslan, Gitta Kutyniok
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4 - Optimization Landscape of Neural Networks
pp 200-228
- By René Vidal, Zhihui Zhu, Benjamin D. Haeffele
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5 - Explaining the Decisions of Convolutional and Recurrent Neural Networks
pp 229-266
- By Wojciech Samek, Leila Arras, Ahmed Osman, Grégoire Montavon, Klaus-Robert Müller
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6 - Stochastic Feedforward Neural Networks: Universal Approximation
pp 267-313
- By Thomas Merkh, Guido Montúfar
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7 - Deep Learning as Sparsity-Enforcing Algorithms
pp 314-337
- By A. Aberdam, J. Sulam
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8 - The Scattering Transform
pp 338-399
- By Joan Bruna
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9 - Deep Generative Models and Inverse Problems
pp 400-421
- By Alexandros G. Dimakis
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10 - Dynamical Systems andOptimal Control Approach to Deep Learning
pp 422-438
- By Weinan E, Jiequn Han, Qianxiao Li
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11 - Bridging Many-Body Quantum Physics and Deep Learning via Tensor Networks
pp 439-474
- By Yoav Levine, Or Sharir, Nadav Cohen, Amnon Shashua
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Mathematical Aspects of Deep Learning

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Contents

Frontmatter
pp i-iv

Contents
pp v-xii

Contributors
pp xiii-xiv

Preface
pp xv-xviii

1 - The Modern Mathematics of Deep Learning
pp 1-111

2 - Generalization in Deep Learning
pp 112-148

3 - Expressivity of Deep Neural Networks
pp 149-199

4 - Optimization Landscape of Neural Networks
pp 200-228

5 - Explaining the Decisions of Convolutional and Recurrent Neural Networks
pp 229-266

6 - Stochastic Feedforward Neural Networks: Universal Approximation
pp 267-313

7 - Deep Learning as Sparsity-Enforcing Algorithms
pp 314-337

8 - The Scattering Transform
pp 338-399

9 - Deep Generative Models and Inverse Problems
pp 400-421

10 - Dynamical Systems andOptimal Control Approach to Deep Learning
pp 422-438

11 - Bridging Many-Body Quantum Physics and Deep Learning via Tensor Networks
pp 439-474

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Mathematical Aspects of Deep Learning

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