The Riemannian Barycentre as a Proxy for Global Optimisation

Abstract

Let M be a simply-connected compact Riemannian symmetric space, and U a twice-differentiable function on M, with unique global minimum at x∗∈ M. The idea of the present work is to replace the problem of searching for the global minimum of U, by the problem of finding the Riemannian barycentre of the Gibbs distribution PT∝ exp (- U/ T). In other words, instead of minimising the function U itself, to minimise ET(x)=12∫d2(x,z)PT(dz), where d(·, · ) denotes Riemannian distance. The following original result is proved: if U is invariant by geodesic symmetry about x∗, then for each δ<12rcx (rcx the convexity radius of M), there exists Tδ such that T≤ Tδ implies ET is strongly convex on the geodesi..