Multi-reference alignment entails estimating a signal in $\mathbb{R}^L$ from its circularly-shifted and noisy copies. This problem has been studied thoroughly in recent years, focusing on the finite-dimensional setting (fixed $L$). Motivated by single-particle cryo-electron microscopy, we analyze the sample complexity of the problem in the high-dimensional regime $L\to\infty$. Our analysis uncovers a phase transition phenomenon governed by the parameter $\alpha = L/(\sigma^2\log L)$, where $\sigma^2$ is the variance of the noise. When $\alpha>2$, the impact of the unknown circular shifts on the sample complexity is minor. Namely, the number of measurements required to achieve a desired accuracy $\varepsilon$ approaches $\sigma^2/\varepsilon$ for small $\varepsilon$; this is the sample complexity of estimating a signal in additive white Gaussian noise, which does not involve shifts. In sharp contrast, when $\alpha\leq 2$, the problem is significantly harder and the sample complexity grows substantially quicker with $\sigma^2$.
翻译:多参数一致要求从循环变换和吵闹的拷贝中估算一个以$mathbb{R ⁇ L$为单位的信号。近年来对这一问题进行了彻底研究,重点是有限维度设置(固定美元 美元 ) 。在单粒冷冻-电子显微镜的驱动下,我们分析了高维系统中问题的样本复杂性 $L\\ to\\\ infty$。我们的分析发现了一个由参数 $\ alpha = L/(\ sigma2\\ log L) 所规范的阶段过渡性现象,其中,$sigma2美元是噪音的差异。当 $\ pha>2美元时,未知的圆形变化对样本复杂性的影响很小。也就是说,要达到理想的精确度所需的测量数量 $\ varepsilon$ 接近$\ gmaph2\\\\\ varepsilon$\ varepsilon$\ varepsilon$;这是估算白高音中的信号的样本复杂性的样本复杂性,但不会发生急剧变化。在基比较快的年份,当 $\\\ ligh\\ lexxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx