Estimation of third-order statistics relies on the availability of a huge amount of data records, which can pose severe challenges on the data collecting hardware in terms of considerable storage costs, overwhelming energy consumption, and unaffordably high sampling rate especially when dealing with high-dimensional data sources such as wideband signals. To overcome these challenges, this paper focuses on the reconstruction of the third-order cumulants under the compressive sensing framework. We derive a transformed linear system that directly connects the cross-cumulants of compressive measurements to the desired third-order statistics. We provide sufficient conditions for lossless third-order cumulant reconstruction via solving simple least-squares, and quantify the strongest achievable compression ratio. To reduce the computational burden, we also propose an approach to recover diagonal cumulant slices directly from compressive measurements, which is useful when the cumulant slices are sufficient for the inference task at hand. As testified by extensive simulations, the developed joint sampling and reconstruction approaches to third-order statistics estimation are able to reduce the required sampling rates significantly by exploiting the cumulant structure resulting from signal stationarity, even in the absence of any sparsity constraints on the signal or cumulants.

中文翻译：

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压缩测量的三阶统计重建

三阶统计量的估计依赖于海量数据记录的可用性，这对数据采集硬件提出了严峻的挑战，包括可观的存储成本、巨大的能源消耗和难以承受的高采样率，尤其是在处理高数据时。维数据源，如宽带信号。为了克服这些挑战，本文重点研究了压缩感知框架下三阶累积量的重建。我们推导出一个转换的线性系统，该系统将压缩测量的交叉累积量直接连接到所需的三阶统计量。我们通过求解简单的最小二乘法为无损三阶累积量重建提供了充分条件，并量化了可实现的最强压缩比。为了减少计算负担，我们还提出了一种直接从压缩测量中恢复对角累积量切片的方法，当累积量切片足以完成手头的推理任务时，这很有用。正如大量模拟所证明的那样，开发的三阶统计估计联合采样和重建方法能够通过利用信号平稳性产生的累积量结构显着降低所需的采样率，即使在对信号或累积量。