CPSC 7430: Topics in flow-based generative modeling and optimal transport

Course Description: Generative modeling has had a major impact across many scientific disciplines in recent years. A driving force has been the development of scalable algorithms that transport, or “flow,” simple reference distributions into complex data distributions. This course will explore the mathematical foundations of generative models through the lens of flow-based transport, emphasizing how deterministic and stochastic dynamics can be used to transform probability distributions in a variety of tasks. Topics will include optimal transport maps, entropic optimal transport, flow matching and stochastic interpolants, diffusion models, Schrödinger bridges, temporal data, and data on Riemannian manifolds, as well as fine-tuning and establish connections to stochastic (optimal) control, how transport can be used for approximate inference, and in solving inverse problems. We will also touch on the basics of modern generative architectures (such as UNets, Vision Input Transformers, the role of variational autoencoders, and conditional generation). A running document acting as course notes will be available on my website that will include many of the proofs that I will likely omit from the in-class lectures. Probably by halfway through the term, we'll start reading and studying papers coming out on arxiv. Thanks to YCRC, students will have access to some computational resources, and are expected to implement many of the ideas above for the first time depending on their background.