Pierre-Emmanuel Jabin

This course aims at providing a mathematical perspective to some key elements of the so-called deep neural networks (DNNs). Much of the interest on deep learning has focused on the implementation of DNN-based algorithms. Our hope is that this compact textbook will offer a complementary point of view that emphasizes the underlying mathematical ideas. We believe that a more foundational perspective will help to answer important questions that have only received empirical answers so far. Our goal is to introduce basic concepts from deep learning in a rigorous mathematical fashion, e.g. introduce mathematical definitions of deep neural networks (DNNs), loss functions, the backpropagation algorithm, etc. We attempt to identify for each concept the simplest setting that minimizes technicalities but still contains the key mathematics. The book focuses on deep learning techniques and introduces them almost immediately. Other techniques such as regression and SVM are briefly introduced and used as a steppingstone for explaining basic ideas of deep learning. Throughout these notes, the rigorous definitions and statements are supplemented by heuristic explanations and figures. The book is organized so that each chapter introduces a key concept. When teaching this course, some chapters could be presented as a part of a single lecture whereas the others have more material and would take several lectures.

The goal of this book is to provide a mathematical perspective on some key elements of the so-called deep neural networks (DNNs). Much of the interest in deep learning has focused on the implementation of DNN-based algorithms. Our hope is that this compact textbook will offer a complementary point of view that emphasizes the underlying mathematical ideas. We believe that a more foundational perspective will help to answer important questions that have only received empirical answers so far. The material is based on a one-semester course Introduction to Mathematics of Deep Learning" for senior undergraduate mathematics majors and first year graduate students in mathematics. Our goal is to introduce basic concepts from deep learning in a rigorous mathematical fashion, e.g introduce mathematical definitions of deep neural networks (DNNs), loss functions, the backpropagation algorithm, etc. We attempt to identify for each concept the simplest setting that minimizes technicalities but still contains the key mathematics.