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 sparse signals exhibit an intermediate \(\sigma ^4\) sample complexity even in the classical MRA model. Further, we characterize the dependence of the estimation rate on the support size s as \(O_p(1)\) and \(O_p(s^{3.5})\) in the dilute and moderate regimes of sparsity respectively. Our techniques have implications for the problem of crystallographic phase retrieval, indicating a certain local uniqueness for the recovery of sparse signals from their power spectrum. Our results explore and exploit connections of the MRA estimation problem with two classical topics in applied mathematics: the beltway problem from combinatorial optimization, and uniform uncertainty principles from harmonic analysis. Our techniques include a certain enhanced form of the probabilistic method, which might be of general interest in its own right.

受冷冻电子显微镜 (cryo-EM) 等尖端应用的推动，多参考对齐 (MRA) 模型需要在一组等距和添加剂的潜在作用下，从对其图像的重复测量中学习未知信号幅度噪声\(\sigma \)。尽管引起了极大的兴趣，但直到最近才出现了理解该模型中估计率的清晰图景，特别是在与应用高度相关的高噪声区域\(\sigma \gg 1\)中。最近的研究表明，对于傅里叶变换完全支持的某些信号，其阶\(\sigma ^6\)具有显着的渐近样本复杂度，这与传统的\(\sigma ^2\)形成鲜明对比。在常规模型中出现。这些结果在实践中通常大得令人望而却步，促使人们研究 MRA 模型周围的变化，在这些变化中可以实现更好的样本复杂性。在本文中，我们表明即使在经典 MRA 模型中，稀疏信号也表现出中等的\(\sigma ^4\)样本复杂度。此外，我们将估计率对支持大小s的依赖性分别描述为稀疏和中等稀疏状态下的\(O_p(1)\)和\(O_p(s^{3.5})\)。我们的技术对结晶相检索问题有影响，表明从功率谱中恢复稀疏信号具有一定的局部唯一性。我们的结果探索并利用了 MRA 估计问题与应用数学中的两个经典主题的联系：组合优化中的环城公路问题和谐波分析中的统一不确定性原理。我们的技术包括某种增强形式的概率方法，它本身可能具有普遍意义。