Certifiably Correct Fr'echet Means in Orbit Spaaces

We study Fréchet means in orbit spaces of finite orthogonal group actions. Since computing these means is NP-hard in general, we take inspiration from group synchronization to develop tractable spectral and semidefinite relaxations. Pairwise alignment certifies global optimality and we develop a semidefinite certificate that can extend beyond this condition. For Gaussian data modulo sign, we prove that our certificate succeeds in a high-noise regime where alignment fails. Numerical experiments evaluate these methods in various settings, including summarizing ensembles of districting plans and clustering time series