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 $x^* \in M$. 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 $P_{\scriptscriptstyle{T}} \propto \exp(-U/T)$. In other words, instead of minimising the function $U$ itself, to minimise $\mathcal{E}_{\scriptscriptstyle{T}}(x) = \frac{1}{2}\int d^{\scriptscriptstyle 2}(x,z)P_{\scriptscriptstyle{T}}(dz)$, where $d(\cdot,\cdot)$ denotes Riemannian distance. The following original result is proved : if $U$ is invariant by geodesic symmetry about $x^*$, then for each $\delta < \frac{1}{2} r_{\scriptscriptstyle cx}$ ($r_{\scriptscriptstyle cx}$ the convexity radius of $M$), there exists $T_{\scriptscriptstyle \delta}$ such that $T \leq T_{\scriptscriptstyle \delta}$ implies $\mathcal{E}_{\scriptscriptstyle{T}}$ is strongly convex on the geodesic ball $B(x^*,\delta)\,$, and $x^*$ is the unique global minimum of $\mathcal{E}_{\scriptscriptstyle{T\,}}$. Moreover, this $T_{\scriptscriptstyle \delta}$ 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.