部分谱估计问题的精确简化
Exact Dimensionality Reduction for Partial Line Spectra Estimation Problems
摘要：
线性谱估计理论的目标是以最佳精度估计时间信号的离网谱分量。最近的研究结果表明，它是可能的，通过使用凸规划恢复信号稀疏线频谱从几个时间观测。然而，这种方法的计算成本仍然是其应用于实际系统的主要缺陷。本文研究了从低维部分测量中恢复谱稀疏信号的问题。本文的第一部分证明，在对子采样矩阵作较轻的假设下，部分谱估计问题可化为一个低维半定规划。证明技术依赖于一个新的扩展的革兰参数化的三角多项式的子空间。本文的第二部分着重分析了两种特殊的子采样模式：多速率采样和随机选择采样。结果表明，这些采样模式保证完美的恢复线频谱，并重建可以实现在一个多对数时间相对于完整的观察情况。此外，这种采样模式的次奈奎斯特恢复能力突出。在此框架下，采用原子软阈值方法估计噪声环境下的稀疏谱，并提出了一种可扩展的分辨率算法。
英文摘要：
Line spectral estimation theory aims to estimate the off-the-grid spectral components of a time signal with optimal precision. Recent results have shown that it is possible to recover signals having sparse line spectra from few temporal observations via the use of convex programming. However, the computational cost of such approaches remains the major flaw to their application to practical systems. This work investigates the recovery of spectrally sparse signal from low-dimensional partial measurements. It is shown in the first part of this paper that, under a light assumption on the sub-sampling matrix, the partial line spectral estimation problems can be relaxed into a low-dimensional semidefinite program. The proof technique relies on a novel extension of the Gram parametrization to subspaces of trigonometric polynomials. The second part of this work focuses on the analysis of two particular sub-sampling patterns: multirate sampling and random selection sampling. It is shown that those sampling patterns guarantee perfect recovery of the line spectra, and that the reconstruction can be achieved in a poly-logarithmic time with respect to the full observation case. Moreover, the sub-Nyquist recovery capabilities of such sampling patterns are highlighted. The atomic soft thresholding method is adapted in the presented framework to estimate sparse spectra in noisy environments, and a scalable algorithm for its resolution is proposed.
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