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.
Prerequisites: Advanced Probability (S&DS 400 / S&DS 600 / MATH 330), Optimization (S&DS 431 / S&DS 631, or S&DS 432 / S&DS 632), Stochastic Processes (S&DS 351 / S&DS 551), and general mathematical maturity. Prior exposure to neural networks, optimal transport, or generative modeling is not required. Enrollment is limited and requires the instructor’s permission.
Instructor: Aram-Alexandre Pooladian
When: Mondays, 9:25am to 11:20am
Location: Leet Olivier Memorial Hall (LOM) 215
Office hours: Mondays, 1–2pm or by request
Syllabus: here
Running course notes: here
Week by week content
The remaining course moves from deterministic and stochastic flows to diffusion and flow matching, Schrödinger bridges, and fast generation. The schedule is tentative; readings refer to sections of the running course notes.
| Week | Date | Content |
|---|---|---|
| Week 0 | Sept 4 | Foundations of optimal transport Sections 1.1–1.6 Transport maps, the Monge problem, couplings, Kantorovich relaxation and duality, and the Wasserstein distance. Brenier’s theorem, Gaussian transport, displacement interpolation, and one-dimensional and sliced Wasserstein methods. |
| — | Sept 7 | Labor Day — no class |
| Week 1 | Sept 14 | Stability, statistics, and algorithms for optimal transport Sections 1.7–1.9 Stability and instability of optimal transport maps, statistical estimation of Wasserstein distances, and empirical map estimation. Linear programming, sorting-based transport, Monte Carlo sliced Wasserstein distances, and Fréchet Inception Distance. |
| Week 2 | Sept 21 | Entropic optimal transport and Sinkhorn’s algorithm Sections 2.1–2.3 Entropic regularization, primal and dual formulations, recovery of the optimal coupling, and extension of dual potentials. Sinkhorn’s algorithm and its analysis through coordinate descent. |
| Week 3 | Sept 28 | Entropic maps, regularization limits, and transport regularity Sections 2.4–2.7 Entropic Brenier maps, small-regularization limits, Sinkhorn divergences, and debiased maps. Lipschitz properties of transport maps, transfer of functional inequalities, and numerical implementation of Sinkhorn and entropic map estimators. |
| Week 4 | Oct 5 | Break — no class |
| Week 5 | Oct 12 | Deterministic and stochastic transport Sections 3.1–3.3 Flow maps, continuity equation, numerical integration, Brownian motion and Itô calculus, and the Fokker–Planck equation. |
| Week 6 | Oct 19 | Reverse-time dynamics and dynamic optimal transport Sections 3.4.1–3.4.2 and 4.1 Probability flow ODEs, reverse SDEs, Benamou–Brenier, and the Hamilton–Jacobi dual. |
| Week 7 | Oct 26 | Diffusion models Chapter 5 Forward and reverse Ornstein–Uhlenbeck processes, denoising score matching, stochastic sampling, and the probability flow ODE. |
| Week 8 | Nov 2 | Flow matching and stochastic interpolants Sections 6.1–6.2 Conditional-to-marginal velocity fields, regression objectives, stochastic interpolants, and minibatch OT couplings. |
| Week 9 | Nov 9 | Schrödinger bridges: formulations and optimal control Section 3.4.3 and Sections 7.1–7.2 Girsanov’s theorem, relative entropy on path space, equivalence to entropic OT, Brownian bridge mixtures, and minimum-energy drift control. |
| Week 10 | Nov 16 | Schrödinger bridges: estimation and algorithms Sections 7.3–7.4 Markovian and reciprocal projections, Iterative Markovian Fitting, entropy descent, plug-in estimation, and Diffusion Schrödinger Bridge Matching. |
| Week 11 | Nov 23 | November recess — no class |
| Week 12 | Nov 30 | Few-step and one-step generation Chapter 8; selected background from Section 4.2 Flow-map identities, distillation, flow-map matching, and consistency models. |
| Week 13 | Dec 7 | TBD |
Project milestones
- October 26: One-page project proposal
- November 16: Preliminary results
- TBD: Final project presentations