Population-Based Optimization on Riemannian Manifolds

Manifold optimization is an emerging field of contemporary optimization thatconstructs efficient and robust algorithms by exploiting the specific geometricalstructure of the search space. In our case the search space takes the form of amanifold. Manifold optimization methods mainly focus on adapting existing optimizationmethods from the usual "easy-to-deal-with" Euclidean search spaces to manifoldswhose local geometry can be defined e.g. by a Riemannian structure. In this waythe form of the adapted algorithms can stay unchanged. However, to accommodatethe adaptation process, assumptions on the search space manifold often have tobe made. In addition, the computations and estimations are confined by the localgeometry. This book presents a framework for population-based optimization on Riemannianmanifolds that overcomes both the constraints of locality and additional assumptions.Multi-modal, black-box manifold optimization problems on Riemannian manifoldscan be tackled using zero-order stochastic optimization methods from a geometricalperspective, utilizing both the statistical geometry of the decision spaceand Riemannian geometry of the search space. This monograph presents in a self-contained manner both theoretical and empiricalaspects ofstochastic population-based optimization on abstract Riemannianmanifolds.