通过凸优化从二次采样中获得精确且稳定的协方差估计
Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming
AI总结：
本文研究了通过凸优化从二次采样中提取高维数据协方差结构的方法，探讨了低秩、Toeplitz低秩、稀疏性等结构假设，并展示了在无噪声情况下能实现准确的协方差估计。
摘要：
高维数据的统计推断和信息处理往往需要高效且准确的二阶统计量估计。在数据快速变化、处理能力和存储有限的情况下，提取协方差结构需要单次数据遍历和少量存储测量。本文探讨了一种二次（或秩一）测量模型，该模型在采样过程中具有最小的内存需求和低计算复杂性，并在保持各种低维协方差结构方面被证明是最优的。具体而言，研究了四种流行的协方差矩阵结构假设，即低秩、Toeplitz低秩、稀疏性以及联合秩一和稀疏结构，通过针对相应结构的凸松弛范式实现恢复。所提出的二次采样框架具有多种潜在应用，包括流数据处理、高频无线通信、相空间断层扫描和光学相位恢复，以及非相干子空间检测。我们的方法在无噪声情况下，只要测量数量超过信息论极限即可实现普遍准确的协方差估计。我们还展示了该方法在噪声和不完美结构假设下的鲁棒性。我们的分析基于一种新的概念，称为混合范数限制等距性质（RIP-ℓ₂/ℓ₁），以及传统的RIP-ℓ₂/ℓ₂用于近各向同性和有界测量。此外，我们的结果在使用PhaseLift进行相位恢复（包括密集和稀疏信号）的已知最佳保证方面，采用了一种显著更简单的方法。
英文摘要：
Statistical inference and information processing of high-dimensional data often require efficient and accurate estimation of their second-order statistics. With rapidly changing data, limited processing power and storage at the acquisition devices, it is desirable to extract the covariance structure from a single pass over the data and a small number of stored measurements. In this paper, we explore a quadratic (or rank-one) measurement model which imposes minimal memory requirements and low computational complexity during the sampling process, and is shown to be optimal in preserving various low-dimensional covariance structures. Specifically, four popular structural assumptions of covariance matrices, namely low rank, Toeplitz low rank, sparsity, jointly rank-one and sparse structure, are investigated, while recovery is achieved via convex relaxation paradigms for the respective structure. The proposed quadratic sampling framework has a variety of potential applications including streaming data processing, high-frequency wireless communication, phase space tomography and phase retrieval in optics, and non-coherent subspace detection. Our method admits universally accurate covariance estimation in the absence of noise, as soon as the number of measurements exceeds the information theoretic limits. We also demonstrate the robustness of this approach against noise and imperfect structural assumptions. Our analysis is established upon a novel notion called the mixed-norm restricted isometry property (RIP-$\ell_{2}/\ell_{1}$), as well as the conventional RIP-$\ell_{2}/\ell_{2}$ for near-isotropic and bounded measurements. In addition, our results improve upon the best-known phase retrieval (including both dense and sparse signals) guarantees using PhaseLift with a significantly simpler approach.
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