Orbit recovery problems are a class of problems that often arise in practice and in various forms. In these problems, we aim to estimate an unknown function after being distorted by a group action and observed via a known operator. Typically, the observations are contaminated with a non-trivial level of noise. Two particular orbit recovery problems of interest in this paper are multireference alignment and single-particle cryo-EM modeling. In order to suppress the noise, we suggest using the method of moments approach for both problems while introducing deep neural network priors. In particular, our neural networks should output the signals and the distribution of group elements, with moments being the input. In the multireference alignment case, we demonstrate the advantage of using the NN to accelerate the convergence for the reconstruction of signals from the moments. Finally, we use our method to reconstruct simulated and biological volumes in the cryo-EM setting.

翻译：轨道恢复问题是一类在实践中常以各种形式出现的问题。在这类问题中，我们旨在估计经群作用扭曲并通过已知算子观测的未知函数。通常观测值会受到显著水平的噪声污染。本文关注的两个特定轨道恢复问题分别是多参考对齐和单颗粒冷冻电镜建模。为抑制噪声，我们建议在这两个问题中采用矩方法，同时引入深度神经网络先验。具体而言，我们的神经网络应以矩为输入，输出信号和群元素的分布。在多参考对齐案例中，我们展示了利用神经网络加速从矩中重建信号收敛的优势。最后，我们将该方法用于冷冻电镜场景中模拟体积和生物体积的重建。