Principal subspace estimation for low-rank Toeplitz covariance matrices with binary sensing

In this paper, we consider the problem of estimating the principal subspace of data in decentralized sensing systems with resource constraints, where the sensors only transmit a single bit to the fusion center to minimize communication costs. In particular, the data covariance matrix is modeled as a low-rank Toeplitz positive semidefinite (PSD) matrix, which arises in applications such as array signal processing and power spectrum estimation in cognitive radios. Our algorithm is based on convex regularization using carefully designed one-bit measurements based on comparing the energy projections of the data seen at each sensor onto pairs of randomly selected Gaussian vectors. We provide finite sample performance guarantees for the accuracy of the subspace estimate in the rank-one case, even when a constant fraction of the 1-bit measurements are randomly flipped. Numerical experiments demonstrate the method continues to work well in the low-rank case, and is capable of separating close located modes from fewer bits in direction-of-arrival estimation, outperforming existing approaches that do not consider the Toeplitz structure.

在本文中，我们考虑在具有资源约束的分布式传感系统中估计数据主子空间的问题，其中传感器仅向融合中心传输单个比特以最小化通信成本。特别地，数据协方差矩阵被建模为低秩托普利兹半正定（PSD）矩阵，这在诸如阵列信号处理和认知无线电中的功率谱估计等应用中会出现。我们的算法基于凸正则化，它使用精心设计的单比特测量，这些测量是通过比较在每个传感器处看到的数据在成对随机选择的高斯向量上的能量投影得到的。我们为秩为1的情况下子空间估计的准确性提供了有限样本性能保证，即使当一部分单比特测量被随机翻转时也是如此。数值实验表明，该方法在低秩情况下仍然能很好地工作，并且在到达方向估计中能够从较少的比特中分离出位置接近的模式，优于不考虑托普利兹结构的现有方法。