On the Performance Bound of Sparse Estimation With Sensing Matrix Perturbation
On the Performance Bound of Sparse Estimation With Sensing Matrix Perturbation
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DOI：
10.1109/tsp.2013.2271481
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发表时间：
2012-11
影响因子：
5.4
通讯作者：
Yujie Tang;Laming Chen;Yuantao Gu
中科院分区：
文献类型：
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作者：
Yujie Tang;Laming Chen;Yuantao Gu
本文主要研究了感知矩阵和测量向量同时被加性高斯噪声破坏的稀疏估计问题。对稀疏估计的性能界进行了深入的分析和讨论。讨论了约束cram<s:1> - rao界（CCRB）和Hammersley-Chapman-Robbins界（HCRB）两种下界。结果表明，有传感矩阵摄动的情况比只有测量噪声的情况更为复杂。对于CCRB，推导了其封闭表达式。它展示了最大和非最大支持情况之间的差距。本文还揭示了CCRB与oracle伪逆估计的MSE之间存在一定的差距，但当问题维数趋于无穷时，该差距渐近于零。对于更严格的界，尽管难以获得一般感知矩阵的简单表达式，但为了定性研究性能界，推导了单元感知矩阵情况下的封闭表达式。结果表明，HCRB消除了极大值与非极大值之间的差距。数值模拟验证了本文的理论结果。
This paper focuses on the sparse estimation in the situation where both the the sensing matrix and the measurement vector are corrupted by additive Gaussian noises. The performance bound of sparse estimation is analyzed and discussed in depth. Two types of lower bounds, the constrained Cramér-Rao bound (CCRB) and the Hammersley-Chapman-Robbins bound (HCRB), are discussed. It is shown that the situation with sensing matrix perturbation is more complex than the one with only measurement noise. For the CCRB, its closed-form expression is deduced. It demonstrates a gap between the maximal and nonmaximal support cases. It is also revealed that a gap lies between the CCRB and the MSE of the oracle pseudoinverse estimator, but it approaches zero asymptotically when the problem dimensions tend to infinity. For a tighter bound, the HCRB, despite the difficulty in obtaining a simple expression for general sensing matrix, a closed-form expression in the unit sensing matrix case is derived for a qualitative study of the performance bound. It is shown that the gap between the maximal and nonmaximal cases is eliminated for the HCRB. Numerical simulations are performed to verify the theoretical results in this paper.