标题： 高维多参考对齐的估计速率 显示英文标题

标题： Rates of estimation for high-dimensional multi-reference alignment

摘要： 我们研究了从带有噪声和圆周旋转观测中估计圆周上的周期函数的连续多参考对齐模型。 受冷冻电子显微镜中出现的类似高维问题的启发，我们建立了对于维度$K$明确的通用信号的极小极大率。 在噪声方差为$\sigma^2 \gtrsim K$的高噪声情况下，对于傅里叶系数大致均匀幅度的信号，该速率按$\sigma^6$缩放，并且不再依赖于维度。 该速率通过双谱反演过程实现，我们的分析为双谱反演提供了新的稳定性界限，这可能具有独立的兴趣。 在噪声较低的情况下，当$\sigma^2 \lesssim K/\log K$时，该速率则按$K\sigma^2$缩放，我们通过对外部旋转进行边缘化的最大似然估计器的精确分析来建立该速率。 使用 Assouad 超立方体引理得到了一个在两种情况之间插值的互补下界。 我们将这些分析也扩展到傅里叶系数具有缓慢幂律衰减的信号。 显示英文摘要

摘要： We study the continuous multi-reference alignment model of estimating a periodic function on the circle from noisy and circularly-rotated observations. Motivated by analogous high-dimensional problems that arise in cryo-electron microscopy, we establish minimax rates for estimating generic signals that are explicit in the dimension $K$. In a high-noise regime with noise variance $\sigma^2 \gtrsim K$, for signals with Fourier coefficients of roughly uniform magnitude, the rate scales as $\sigma^6$ and has no further dependence on the dimension. This rate is achieved by a bispectrum inversion procedure, and our analyses provide new stability bounds for bispectrum inversion that may be of independent interest. In a low-noise regime where $\sigma^2 \lesssim K/\log K$, the rate scales instead as $K\sigma^2$, and we establish this rate by a sharp analysis of the maximum likelihood estimator that marginalizes over latent rotations. A complementary lower bound that interpolates between these two regimes is obtained using Assouad's hypercube lemma. We extend these analyses also to signals whose Fourier coefficients have a slow power law decay.