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

翻译：我们研究一类轨道恢复问题，其中观测到 $\mathbb{R}^p$ 中未知元素的独立副本，每个副本受到某群（例如 $\mathbb{Z}/p$ 或 $\mathrm{SO}(3)$）随机元素的线性作用，随后被加性高斯噪声破坏。我们证明了以高概率近似恢复该未知元素群轨道所需样本数量的匹配上下界。这些界基于不变理论中的定量技术，给出了估计问题的统计难度与群代数性质之间的精确对应关系。此外，我们提供了计算机辅助程序来验证这些性质，在众多感兴趣情形下这些程序在计算上是高效的。该模型源于信号处理、计算机视觉和结构生物学中的几何问题，并应用于低温电子显微镜（cryo-EM）中的重建问题——这一实际问题具有重要价值。我们的结果能够验证（针对给定问题规模）：若cryo-EM图像被方差为 $\sigma^2$ 的噪声破坏，则恢复分子结构所需的图像数量按 $\sigma^6$ 比例增长。我们通过一种新颖（尽管计算成本高昂）的算法来匹配这一界，该算法基于次数不超过3的不变特征进行cryo-EM从头重建。我们进一步讨论了如何从混合（或异质）cryo-EM样本中恢复多个分子结构。