This article discusses a generalization of the 1-dimensional multi-reference alignment problem. The goal is to recover a hidden signal from many noisy observations, where each noisy observation includes a random translation and random dilation of the hidden signal, as well as high additive noise. We propose a method that recovers the power spectrum of the hidden signal by applying a data-driven, nonlinear unbiasing procedure, and thus the hidden signal is obtained up to an unknown phase. An unbiased estimator of the power spectrum is defined, whose error depends on the sample size and noise levels, and we precisely quantify the convergence rate of the proposed estimator. The unbiasing procedure relies on knowledge of the dilation distribution, and we implement an optimization procedure to learn the dilation variance when this parameter is unknown. Our theoretical work is supported by extensive numerical experiments on a wide range of signals.
翻译:文章讨论了一维多参考对齐问题的一般化。 目标是从许多吵闹的观测中恢复隐藏的信号, 每一个吵闹的观测都包括隐藏信号的随机翻译和随机放大, 以及高添加噪音。 我们提出一种方法, 通过应用数据驱动的非线性无偏移程序来恢复隐藏信号的能量谱, 从而获得隐藏信号到一个未知的阶段。 确定了一个不偏倚的电谱测算器, 其错误取决于样本大小和噪声水平, 我们精确量化提议的测算器的汇合率。 不偏差程序依赖于对放大分布的了解, 我们实施一个优化程序, 以在未知参数时学习乘积差。 我们的理论工作得到了广泛关于广泛信号的数字实验的支持 。