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Motivated by applications to single-particle cryo-electron microscopy(cryo-EM), we study several problems of function estimation in a high noiseregime, where samples are observed after random rotation and possible linearprojection of the function domain. We describe a stratification of the Fisherinformation eigenvalues according to transcendence degrees of graded pieces ofthe algebra of group invariants, and we relate critical points of thelog-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 optimizationproblems 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, Perry, Weed, and Wein. We affirmatively resolve conjecturesthat $3^\text{rd}$-order moments are sufficient to locally identify a genericsignal up to its rotational orbit in these examples.  For low-dimensional approximations of the electric potential maps of twosmall protein molecules, we empirically verify that the noise-scalings of theFisher information eigenvalues conform with our theoretical predictions over arange of SNR, in a model of $SO(3)$ rotations without projections.

Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM