Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes

Understand

This paper initiates a systematic development of a theory of non-commutative optimization.

• It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups.
• These algorithms minimize the moment map (a non-commutative notion of the usual gradient) and test membership in null cones and moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which arise from this a priori non-convex, non-linear setting).
• This setting captures a diverse set of problems in different areas of computer science, mathematics, and physics.