Gradient flows for the minimum distance to the sum of adjoint orbits

Abstract

Let G be a connected semisimple Lie group and its Lie algebra. Let be the Cartan decomposition corresponding to a Cartan involution θ of . The Killing form B induces a positive definite symmetric bilinear form B θ on defined by B θ(X, Y) = − B(X, θY). Given , we consider the optimization problem where the norm ‖ · ‖ is induced by B θ and K is the connected subgroup of the G with the Lie algebra . We obtain the gradient flow of the corresponding smooth function on the manifold K × ··· × K. Our results give unified extensions of several results of Li, Poon and Schulte-Herbrüggen [C.K. Li, Y.T. Poon, and T. Schulte-Herbruggen, Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact Lie groups, Math. Comp. 80 (2011), 1601–1621]. They are also true for reductive Lie groups.