Resources to learn optimization on manifolds

The documentation on this website is meant to help users get to grips with Manopt. Underneath the software, there is a fair amount of math. This page provides some pointers to learn that math.

Books

The following books (both available online for free) can be helpful to learn the relevant mathematics:

An introduction to optimization on smooth manifolds
Nicolas Boumal, Cambridge University Press, 2023.

Optimization algorithms on matrix manifolds
Pierre-Antoine Absil, Robert Mahony and Rodolphe Sepulchre, Princeton University Press, 2008.

A full course

• MATH-512 at EPFL: this is a full set of video lectures, slides and exercises (with hints and solutions).
• edX course: this covers the first six weeks of MATH-512, on the edX platform.

Tutorial videos

• One of the SIAM Optimization 2023 minitutorials was about Riemannian optimization.

  • It introduce the basics of differential geometry and Riemannian geometry for optimization on smooth manifolds.
  • The two lectures of 90 minutes each are available as video 1 and video 2.
  • Here are the slides and also the Max-Cut example code.

• Earlier (and more condensed) versions of that tutorial are available as:

  • A one-hour video (Simons Institute) and
  • A two-hour video (ETHZ/EPFL summer school).

• Week 5 of MATH-512 includes a lecture about Manopt: the basics and a bit more.

Blogs and blog posts

• The Race to the bottom blog about nonconvex optimization features some Riemannian optimization.
• This blog post gives an informal overview of optimization on manifolds.

Code examples

• See the examples that ship with Manopt.