We study a class of orbit retrieval problems that observe independent copies of unknown elements in $\mathbb{R}^p$. Each element is linearly acted upon by some group of random elements (such as $\mathbb{Z}/). p$ or $\mathrm{SO}(3)$), corrupted by additive Gaussian noise. We prove that the upper and lower bounds on the number of samples required to almost recover the group orbitals of this unknown element with high probability agree. These bounds are based on the quantitative methods of invariant theory and show a precise correspondence between the statistical difficulty of the estimation problem and the algebraic properties of the group. In addition, we provide computer-assisted procedures to prove these properties that are computationally efficient in many cases of interest.

    The model is motivated by geometric problems in signal processing, computer vision, and structural biology, and is applied to reconstruction problems in cryo-electron microscopy (cryo-EM), a problem of practical importance. Our results show that if a cryo-EM image is corrupted by noise of variance $\sigma^2$ , the number of images required to recover the molecular structure is $\sigma^6$ . We match a new (albeit computationally expensive) algorithm for ab initio reconstruction in cryo-EM based on invariant features of third order. In addition, we describe how to recover multiple molecular structures from mixed (or heterogeneous) cryo-EM samples.

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