论文标题
高维多参考对齐的估计速率
Rates of estimation for high-dimensional multi-reference alignment
论文作者
论文摘要
我们研究了从嘈杂和圆形旋转观测值估算圆上周期函数的连续多引用比对模型。由低温电子显微镜出现的类似的高维问题的促进，我们建立了最小值率，用于估计在尺寸$ k $中明确的通用信号。在具有噪声差异的高噪声状态下，对于具有大约均匀尺寸的傅立叶系数的信号，速率尺度为$σ^6 $，并且对维度无关。该速率是通过双光谱反转程序实现的，我们的分析为双光谱反演提供了新的稳定性界限，这可能具有独立的利益。在$σ^2 \ Lessim k/\ log k $的低噪声状态中，速率比例为$kσ^2 $，我们通过对最大似然估计器的最大分析来确定此速率，该速率在潜在旋转上边缘化。使用Assouad的HyperCube引理获得了这两个方案之间插值的互补下限。我们将这些分析扩展到其傅立叶系数具有缓慢功率定律衰减的信号。
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 $σ^2 \gtrsim K$, for signals with Fourier coefficients of roughly uniform magnitude, the rate scales as $σ^6$ 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 $σ^2 \lesssim K/\log K$, the rate scales instead as $Kσ^2$, 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.
