Motivated by cutting-edge applications like cryo-electron microscopy (cryo-EM), the Multi-Reference Alignment (MRA) model entails the learning of an unknown signal from repeated measurements of its images under the latent action of a group of isometries and additive noise of magnitude $\sigma$. Despite significant interest, a clear picture for understanding rates of estimation in this model has emerged only recently, particularly in the high-noise regime $\sigma \gg 1$ that is highly relevant in applications. Recent investigations have revealed a remarkable asymptotic sample complexity of order $\sigma^6$ for certain signals whose Fourier transforms have full support, in stark contrast to the traditional $\sigma^2$ that arise in regular models. Often prohibitively large in practice, these results have prompted the investigation of variations around the MRA model where better sample complexity may be achieved. In this paper, we show that \emph{sparse} signals exhibit an intermediate $\sigma^4$ sample complexity even in the classical MRA model. Our results explore and exploit connections of the MRA estimation problem with two classical topics in applied mathematics: the \textit{beltway problem} from combinatorial optimization, and \textit{uniform uncertainty principles} from harmonic analysis.

翻译：多参考协调模式(MRA)模型受到冷冻-电子显微镜(cryo-EM)等尖端应用的激励,它需要从一组异地美食家的潜伏动作和规模为$gma$的添加性噪声中反复测量其图像,从而了解一个未知信号。尽管人们对此模型的估算率有极大的兴趣,但直到最近才出现了一个清晰的了解该模型估算率的图象,特别是在高新制度$\sigma \gg 1美元的应用中,这在应用中具有高度相关性。最近的调查显示,对于Fourier变换的具有充分支持力的某些信号来说,其订单的杂质复杂性为$\sigma_6美元,这与常规模型中出现的传统的$sgma_2美元相鲜明相反。在实践中,这些结果往往令人无法接受地大,促使对可实现更高样本复杂性的MRA模型的变异性进行调查。在本文件中,显示,即使在古典的MSA模型模型模型模型模型中,我们探索并探索了MRA模型模型模型与两个模型的精确性分析问题的链接。