Towards a Theory of Non-commutative Optimization - Geodesic First and Second Order Methods for Moment Maps and Polytopes

Explore a comprehensive lecture on non-commutative optimization theory in this 32-minute talk by Peter Bürgisser from Technische Universität Berlin. Delve into the development of geodesically convex optimization problems on Riemannian manifolds arising from symmetries in complex vector spaces. Discover how this framework unifies diverse problems in computer science, mathematics, and physics. Learn about two general methods in the geodesic setting: a first-order method for minimizing moment maps and a second-order method for testing membership in null cones and moment polytopes. Understand the key parameters of underlying group actions that control convergence to the optimum in these methods. Examine the non-commutative analogues of "smoothness" and their role in efficient algorithms for null cone membership problems. Gain insights into open problems and future research directions in this field, based on joint work with Cole Franks, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson.