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 $T \le T_{\scriptscriptstyle \delta }$ 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.