Introduction to optimization on smooth manifolds: first order methods

Optimization on manifolds is the result of smooth geometry and optimization merging into one elegant modern framework. We start the course at "What is a manifold?", and give the students a firm understanding of submanifolds embedded in real space. This covers numerous applications in engineering and the sciences. All definitions and theorems are motivated to build time-tested optimization algorithms. The math is precise, to promote understanding and enable computation. We build our way up to Riemannian gradient descent: the all-important first-order optimization algorithm on manifolds. This includes analysis and implementation. The lectures follow (and complement) the textbook "An introduction to optimization on smooth manifolds" written by the instructor, also available on his webpage. From there, students can explore more with numerical tools (such as the toolbox Manopt, which is the subject of the last week of the course). They will also be in a good position to tackle more advanced theoretical tools necessary for second-order optimization algorithms (e.g., Riemannian Hessians). Those are covered in further video lectures available on the instructor's textbook webpage.

• Recognize smooth manifolds and do calculus on them.
• Manipulate concepts from differential and Riemannian geometry.
• Develop geometric tools to work on new manifolds of interest.
• Recognize and formulate a Riemannian optimization problem.
• Analyze and implement first-order Riemannian optimization algorithms.
• Use toolboxes to accelerate prototyping.

• Linear algebra, Multivariable calculus, some numerical analysis.