The lecture presents the state-of-the-art in the generation of images, sounds and scientific data by deep neural networks. The focus is on sampling probability distributions obtained by transporting white Gaussian noise. After a review of the state of the art, we study score diffusion transport, which performs progressive denoising to generate data (images, sounds, etc.). This involves estimating the probability density score, using a deep neural network. The lecture introduces the mathematical and algorithmic foundations with their applications. The following topics will be covered:
- Probability transport in deep learning. Transport generation and sampling;
- Fokker Plank equation giving the evolution of the probability density of a dynamic system. Langevin equation for probability sampling;
- Diffusion score data generation. Score estimation by denoising with the Tweety-Myasawa formula;
- Score learning with deep neural networks. Generalization of learning. Applications to image and sound generation;
- Analysis of neural network computation. Denoising and parsimony in orthogonal bases;
- Data generation conditional on additional information;
- Stochastic interpolation for prediction. Applications to the prediction of chaotic physical systems such as meteorology.