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 minimiseThe following original result is proved : if U is invariant by geodesic symmetry about x * , then for each δ < 1 2 rcx (rcx the convexity radius of M ), there exists Tδ such that T ≤ Tδ implies ET is strongly convex on the geodesic ball B(x * , δ) , and x * is the unique global minimum of ET . Moreover, this Tδ can be computed explicitly. This result gives rise to a general algorithm for black-box optimisation, which is briefly described, and will be further explored in future work.