We critically review three major theories of machine learning and provide a new theory according to which machines learn a function when the machines successfully compute it. We show that this...

This lecture note is intended to prepare early-year master's and PhD students in data science or a related discipline with foundational ideas in machine learning. It starts with basic ideas in...

We currently witness the spectacular success of artificial intelligence in both science and public life. However, the development of a rigorous mathematical foundation is still at an early stage....

This book introduces the mathematical foundations and techniques that lead to the development and analysis of many of the algorithms that are used in machine learning. It starts with an...

This book aims to provide an introduction to the topic of deep learning algorithms. We review essential components of deep learning algorithms in full mathematical detail including different...

This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as...

Bayesian inference provides a methodology for parameter estimation and uncertainty quantification in machine learning and deep learning methods. Variational inference and Markov Chain Monte-Carlo...

We present an accessible first course on diffusion models and flow matching for machine learning, aimed at a technical audience with no diffusion experience. We try to simplify the mathematical...

We present an introduction to the field of statistical optimal transport, based on lectures given at École d'Été de Probabilités de Saint-Flour XLIX.

Large language models (LLMs) are remarkably efficient across a wide range of natural language processing tasks and well beyond them. However, a comprehensive theoretical analysis of the LLMs'...