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: Mon/Wed 2:00 - 3:15 PM
• Location: Thornton E304 / Zoom
• Instructor: Tom Fletcher (ptf8v AT virginia DOT edu)
  • Office Hours: Tuesdays 11 AM - 12 noon, Rice 306
• TA: Nian (Nellie) Wu (bsw3ac AT virginia DOT edu)
  • Office Hours: Thursdays 5 - 6 PM, Rice 303
• 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

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