This is the talk I presented in the Optimization Group Seminar, titled: An Introduction to Optimization over the Space of Probability Measures: From Sampling to Wasserstein Gradient Flow.

In generative modeling, we are given a collection of training samples $\{x_i\}_{i=1}^N$ and wish to generate new samples from the underlying target distribution $\pi$. There are already many established approaches to this problem, including likelihood-based methods, implicit generative models such as GANs, and score-based diffusion models. More recently, the flow matching framework has emerged as another powerful paradigm. In what follows, we introduce the basic ideas of flow matching and explain how works.