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PreprintCC BY 4.0arXiv

Certifiably Correct Fréchet Means in Orbit Spaces

  • Dustin G. Mixon
  • Tom Needham
  • Nathan Willey
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Abstract

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

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Cite

BibTeXRIS
BibTeX
@misc{mixon2026certifiably,
  title = {{Certifiably Correct Fréchet Means in Orbit Spaces}},
  author = {Dustin G. Mixon and Tom Needham and Nathan Willey},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24191},
  eprint = {2610.24191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24191},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Certifiably Correct Fréchet Means in Orbit Spaces
AU  - Dustin G. Mixon
AU  - Tom Needham
AU  - Nathan Willey
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.24191
UR  - https://arxiv.org/abs/2610.24191
AB  - 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
ER  -