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Fan, Zhou; Lederman, Roy R; Sun, Yi; Wang, Tianhao; Xu, Sheng
Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM Technical Report
2021, (arXiv: 2107.01305).
Abstract | Links | BibTeX | Tags: Computer Science - Information Theory, Mathematics - Optimization and Control, Mathematics - Statistics Theory
@techreport{fan_maximum_2021,
title = {Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM},
author = {Zhou Fan and Roy R Lederman and Yi Sun and Tianhao Wang and Sheng Xu},
url = {http://arxiv.org/abs/2107.01305},
year = {2021},
date = {2021-07-01},
urldate = {2021-07-14},
abstract = {Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a low SNR regime, where samples are observed under random rotations of the function domain. In a general framework of group orbit estimation with linear projection, we describe a stratification of the Fisher information eigenvalues according to a sequence of transcendence degrees in the invariant algebra, and relate critical points of the log-likelihood landscape to a sequence of method-of-moments optimization problems. This extends previous results for a discrete rotation group without projection. We then compute these transcendence degrees and the forms of these moment 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, Perry, Weed, and Wein. For several of these examples, we affirmatively resolve numerical conjectures that $3textasciicircumtextbackslashtextrd$-order moments are sufficient to locally identify a generic signal up to its rotational orbit. 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 these theoretical predictions over a range of SNR, in a model of $SO(3)$ rotations without projection.},
note = {arXiv: 2107.01305},
keywords = {Computer Science - Information Theory, Mathematics - Optimization and Control, Mathematics - Statistics Theory},
pubstate = {published},
tppubtype = {techreport}
}
Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a low SNR regime, where samples are observed under random rotations of the function domain. In a general framework of group orbit estimation with linear projection, we describe a stratification of the Fisher information eigenvalues according to a sequence of transcendence degrees in the invariant algebra, and relate critical points of the log-likelihood landscape to a sequence of method-of-moments optimization problems. This extends previous results for a discrete rotation group without projection. We then compute these transcendence degrees and the forms of these moment 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, Perry, Weed, and Wein. For several of these examples, we affirmatively resolve numerical conjectures that $3textasciicircumtextbackslashtextrd$-order moments are sufficient to locally identify a generic signal up to its rotational orbit. 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 these theoretical predictions over a range of SNR, in a model of $SO(3)$ rotations without projection.