In recent years, there is a growing need for processing methods aimed at extracting useful information from large datasets. In many cases, the challenge is to discover a low-dimensional structure in the data, often concealed by the existence of nuisance parameters and noise. Motivated by such challenges, we consider the problem of estimating a signal from its scaled, cyclically shifted and noisy observations. We focus on the particularly challenging regime of low signal-to-noise ratio (SNR), where different observations cannot be shift-aligned. We show that an accurate estimation of the signal from its noisy observations is possible, and derive a procedure which is proved to consistently estimate the signal. The asymptotic sample complexity (the number of observations required to recover the signal) of the procedure is \(1{/}{\text {SNR}}^4\). Additionally, we propose a procedure which is experimentally shown to improve the sample complexity by a factor equal to the signal’s length. Finally, we present numerical experiments which demonstrate the performance of our algorithms and corroborate our theoretical findings.

近年来，对旨在从大型数据集中提取有用信息的处理方法的需求不断增长。在许多情况下，挑战在于在数据中发现一个低维结构，而该结构通常被讨厌的参数和噪声的存在所掩盖。受此类挑战的驱使，我们考虑了根据缩放后的，周期性移位的和嘈杂的观测值来估计信号的问题。我们专注于低信噪比（SNR）的特别具有挑战性的方案，在该方案中不同的观察结果无法进行移位对齐。我们表明，从其嘈杂的观察中准确估计信号是可能的，并推导了证明可以一致估计信号的过程。该过程的渐进样本复杂度（恢复信号所需的观察数）为\（1 {/} {\ text {SNR}} ^ 4 \）。此外，我们提出了一种程序，该程序已通过实验证明可以将采样复杂度提高一个等于信号长度的因子。最后，我们提出了数值实验，这些实验证明了我们算法的性能并证实了我们的理论发现。