Description

Modern data are high-dimensional, multi-modal, and large-scale, for example, images with millions of pixels, text corpora with millions of words, gene sequences with billions of base pairs, etc. However, these data tend to concentrate on lower-dimensional, nonlinear subspaces known as manifolds. Learning and sampling from this real distribution, hence, is of tremendous value. This class covers the mathematical theory of high-dimensional geometry and manifolds and how it applies to the latest advances in artificial intelligence.

Logistics

• Time: Tue/Thu 2:00 - 3:15 PM
• Location: Thronton E316 / Zoom
• Instructors: Tom Fletcher (ptf8v AT virginia DOT edu) and Aman Shrivastava (as3ek AT virginia DOT edu)
• Prerequisites: You should have basic (undergraduate level) knowledge of Probability, Linear Algebra, Multivariate Calculus, and be comfortable programming in Python
• Software: All homeworks will be done in Jupyter
• Office Hours: Tom: Wednesdays, 3:00 - 4:00 pm in Rice 306 & Aman: Mondays, 3:00 - 4:00 pm in Rice 342

Additional Reading

Manfredo do Carmo, Riemannian Geometry

Sigmundur Gudmundsson, Introduction to Riemannian Geometry

Example Jupyter Notebooks

For those of you who are relatively new to Jupyter, here are a few notebooks that you might find useful (from my undergraduate course Foundations of Data Analysis.)

• LaTeX math notation guide
• Data plots in Python
• Singular value decomposition
• Multiple linear regression