This is a graduate-level course at the Courant Institute of Mathematical Sciences, NYU. By the way, Joan is fantastic and this is a very good course.
Course information
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Instructor: Joan Bruna
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Semester: Spring 2025
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Website: https://joanbruna.notion.site/ir25
- Outline: The aim of this graduate-level course is to describe the mathematical aspects of modeling high-dimensional data, with an emphasis on the computational and statistical quantitative questions. The course will describe in detail the following topics:
- Gaussian Estimation
- Gaussian PCA guarantees
- From Gaussians to Gaussian mixtures: emergence of non-linear modeling
- Probabilistic graphical models
- D-separation and conditional independence
- Gibbs models and Clifford Theorem
- Basic hardness results
- Exact inference
- Inference and Statistical Physics
- Free energies
- Approximate-Message-Passing and Belief Propagation
- Maximum-Entropy principle
- Variational Inference
- EM Algorithm
- Convex Duality
- Application: Variational Autoencoders
- Markov-Chain Monte-Carlo methods
- Metropolis-Hasting
- Importance and Rejection Sampling
- Perspective from Diffusions in continuous space-time
- Functional Inequalities, spectral gap, high-dimensional aspects of MCMC
- Measure Transportation in high dimensions
- Optimal Transport and first consequences
- Integral Probability Metrics
- Statistical and Computational aspects of IPMs
- Application: Generative Adversarial Networks
- Generative Modeling using Neural Networks
- Score-based Diffusion and Probability Flows
- Textbook: Two main good ones are Probabilistic Graphical Models - Principles and Techniques and High-Dimensional Probability - An Introduction with Applications in Data Science. The one I like a lot is Graphical models, exponential families, and variational inference which showcases extremely elegant connections between graphical models and variational inference on exponential families. See more in the course website.
Gradebook
Overall grade: A (3.00/3.00)