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We describe a stratification of the Fisher information eigenvalues according to transcendence degrees of graded pieces of the algebra of group invariants, and we relate critical points of the log-likelihood landscape to a sequence of moment optimization problems, extending previous results for a discrete rotation group without projections. We then compute the transcendence degrees and forms of these optimization problems for several examples of function estimation under SO(2) and SO(3) rotations, including a simplified model of cryo-EM as introduced by Bandeira, Blum-Smith, Kileel, Niles-Weed, Perry and Wein. We affirmatively resolve conjectures that third-order moments are sufficient to locally identify a generic signal up to its rotational orbit in these examples. For low-dimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noise scalings of the Fisher information eigenvalues conform with our theoretical predictions over a range of SNR, in a model of SO(3) rotations without projections.<\/jats:p>","DOI":"10.1214\/23-aos2292","type":"journal-article","created":{"date-parts":[[2024,3,8]],"date-time":"2024-03-08T01:31:23Z","timestamp":1709861483000},"page":"52-77","source":"Crossref","is-referenced-by-count":8,"title":["Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM"],"prefix":"10.1214","volume":"52","author":[{"given":"Zhou","family":"Fan","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, Yale University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roy R.","family":"Lederman","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, Yale University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Chicago"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tianhao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, Yale University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sheng","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, Yale University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"20","doi-asserted-by":"crossref","unstructured":"<small>GOWER, J. 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