Generative Modeling by Transport I: Stochastic Interpolants and Continuous Probability Paths
Introduction
Modern generative modeling can be understood as the problem of transporting probability measures through continuous stochastic dynamics. This series develops a unified mathematical framework for diffusion models, flow matching, Schrödinger bridges, and related methods through the theory of stochastic interpolants. Beginning from rigorous probabilistic and analytical foundations—including stochastic calculus, transport equations, and measure-valued dynamics—we derive the governing objectives, likelihood formulations, and geometric principles underlying contemporary generative models from first principles. Rather than treating these systems as isolated architectures or engineering heuristics, the course presents them as different realizations of a single transport-theoretic formalism over probability paths.
The latter portion of the series studies compositional and ratio-based probability paths via the Feynman–Kac framework, sequential Monte Carlo transport, and recent FKC/ACE-style samplers. These topics motivate the broader perspective that generation itself is fundamentally a problem of constructing, composing, and controlling probability flows. By the end of the series, students will possess a rigorous understanding of continuous-time generative modeling theory and the mathematical tools necessary to derive and analyze new transport-based generative systems.
Study Notes
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Mathematical Preliminaries, Background, Motivation, and Related Work [pdf]
Homework 1 [pdf] -
Stochastic Interpolant Definitions, Transport Equations, and Quadratic Objectives [pdf]
Homework 2 [pdf] -
Generative Models, Likelihood Control, Density Estimation [pdf]
Homework 3 [pdf] -
Instantiations of Stochastic Interpolants: Diffusive, One-sided, Mirror, and Schrodinger Bridge Interpolants [pdf]
Homework 4 [pdf] -
Spatially Linear Interpolants [pdf]
Homework 5 [pdf] -
Connections, Algorithms, Implementations, and Numerical Experiments [pdf]
Homework 6 [pdf] -
Composing Generative Paths: Feynman-Kac Formula, Correctors, and SMC [pdf]
Homework 7 [pdf] -
ACE: Path Existence, Marginal Path Collapse, Adaptive Exponents, and Generator Composition [pdf]
Homework 8 [pdf]
References
- Albergo, M., Boffi, N. M., & Vanden-Eijnden, E. (2025). Stochastic interpolants: A unifying framework for flows and diffusions. Journal of Machine Learning Research, 26(209), 1-80.
- Lee, Z., Hwang, M., Jo, S., Lee, W., Ko, J., Park, Y. B., ... & Kim, K. (2026). On the Collapse of Generative Paths: A Criterion and Correction for Diffusion Steering. International Conference on Machine Learning (ICML), 2026.
- Skreta, M., Akhound-Sadegh, T., Ohanesian, V., Bondesan, R., Aspuru-Guzik, A., Doucet, A., ... & Neklyudov, K. (2025). Feynman-kac correctors in diffusion: Annealing, guidance, and product of experts. International Conference on Machine Learning (ICML), 2025.
- Holderrieth, P., Havasi, M., Yim, J., Shaul, N., Gat, I., Jaakkola, T., ... & Lipman, Y. (2025, May). Generator matching: Generative modeling with arbitrary markov processes. In International Conference on Learning Representations (Vol. 2025, pp. 52153-52219).
- Pauline, V., Höppe, T., Neklyudov, K., Tong, A., Bauer, S., & Dittadi, A. (2025). Foundations of diffusion models in general state spaces: A self-contained introduction. arXiv preprint arXiv:2512.05092.