Estimation under group actions: recovering orbits from invariants

We study a class of orbit recovery problems in which we observe independent copies an unknown element $\mathbb{R}^p$, each linearly acted upon by random some group (such as $\mathbb{Z}/p$ or $\mathrm{SO}(3)$) and then corrupted additive Gaussian noise. prove matching upper lower bounds on the number samples required to approximately recover this with high probability. These bounds, based quantitative techniques invariant theory, give precise correspondence between statistical difficulty estimation problem algebraic properties group. Furthermore, computer-assisted procedures certify these that are computationally efficient many cases interest. The model is motivated geometric signal processing, computer vision, structural biology, applies reconstruction cryo-electron microscopy (cryo-EM), significant practical Our results allow us verify (for given size) if cryo-EM images noise variance $\sigma^2$, molecule structure scales $\sigma^6$. match bound novel (albeit expensive) algorithm for ab initio cryo-EM, features degree at most 3. further discuss how multiple molecular structures from mixed (or heterogeneous) samples.