Notes and Comments on S. Mallat’s Lectures at Collège de France (2025): Learning and generation using random sampling

In this course, Stéphane Mallat explores generative models, which are essential to recent advances in artificial intelligence. He focuses on understanding the mathematical foundations of these models by highlighting the regularities present in data that help overcome the curse of dimensionality. The goal is to grasp the principles underlying the generation of realistic data from simple rules. Among the approaches studied are Generative Adversarial Networks (GANs), where a generator learns to produce data that can fool a discriminator. While effective, GANs present limitations such as mode collapse and unstable training. The course then introduces a more theoretical perspective: data generation is modeled as a probability transport from a latent distribution to the distribution of real data. Normalizing Flows implement this transport through differentiable bijections, enabling explicit density evaluation but remaining constrained in high dimensions. Diffusion models, based on stochastic differential equations (SDEs), offer a more flexible alternative with strong theoretical guarantees. In this context, key concepts of denoising are also covered in the course.