Certifiably Correct Fr\'echet Means in Orbit Spaces (arxiv.org)
arXiv:2610.24191v2 Announce Type: replace-cross
Abstract: We study Fr\'echet 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
Abstract: We study Fr\'echet 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
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