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.

翻译：我们研究了连续多基准对齐模型，该模型旨在从含噪且经圆周旋转的观测中估计圆上的周期函数。受冷冻电镜中类似高维问题的启发，我们建立了在维度$K$上显式表达的一般信号的最小最大估计速率。在噪声方差$\sigma^2 \gtrsim K$的高噪声条件下，对于傅里叶系数大致均匀幅值的信号，估计速率按$\sigma^6$标度且不再依赖维度。该速率通过双谱反演程序实现，我们的分析为双谱反演提供了新的稳定性界，这可能具有独立的研究意义。在$\sigma^2 \lesssim K/\log K$的低噪声条件下，估计速率转而按$K\sigma^2$标度，我们通过对潜旋转进行边缘化的极大似然估计的精细分析确立了该速率。利用阿苏阿德超立方体引理，我们得到了在这两种条件之间插值的互补下界。此外，我们将这些分析推广到傅里叶系数呈缓慢幂律衰减的信号。