Rates of estimation for high-dimensional multireference alignment
研究了从带噪声和随机旋转的观测中估计周期函数的问题，给出了高维情形下信号估计的极小极大速率，并分析了高低噪声区的不同表现。
We study the continuous multireference 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 σ2≳K, for signals with Fourier coefficients of roughly uniform magnitude, the rate scales as σ6 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 σ2≲K/logK, the rate scales instead as Kσ2, 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.