In this self-contained chapter, we revisit a fundamental problem of multivariate statistics: estimating covariance matrices from finitely many independent samples. Based on massive Multiple-Input Multiple-Output (MIMO) systems we illustrate the necessity of leveraging structure and considering quantization of samples when estimating covariance matrices in practice. We then provide a selective survey of theoretical advances of the last decade focusing on the estimation of structured covariance matrices. This review is spiced up by some yet unpublished insights on how to benefit from combined structural constraints. Finally, we summarize the findings of our recently published preprint "Covariance estimation under one-bit quantization" to show how guaranteed covariance estimation is possible even under coarse quantization of the samples.

翻译：在此自成一体的章节中,我们重新审视了多变量统计的一个根本问题:从有限的许多独立样本中估算共变矩阵。根据大规模多投入多重产出(MIMO)系统,我们说明在实际估算共变矩阵时,必须利用结构,并考虑对样本进行量化。然后我们有选择地调查过去十年的理论进展,重点是结构化共变矩阵的估计。这项审查得到一些尚未公布的关于如何从综合结构制约中获益的见解的启发的促进。最后,我们总结了我们最近出版的预印“一位数量化下的共变估计”的结论,以表明即使样品的粗略量化不足,也有可能保证共变差估计。