高斯白噪声中稀疏向量的无偏估计
Unbiased Estimation of a Sparse Vector in White Gaussian Noise
摘要：
本文研究了被加性高斯白噪声污染的稀疏非随机向量的无偏估计。我们表明，虽然有无限多的无偏估计这个问题，他们都没有一致的最小方差。因此，我们专注于局部最小方差无偏（LMVU）估计。我们得到简单的封闭形式的下限和上界的方差LMVU估计，或等价地，Barankin界（BB）。我们的界限允许分离低SNR和高SNR制度的阈值区域的估计，并且它们指示BB在高SNR下的渐近行为。我们还开发了数值下限和上界，这是严格的封闭形式的界限，从而更准确地描述BB。数值研究比较我们的特性的BB与建立有偏估计方案，并表明，而无偏估计在低信噪比表现不佳，他们可能会表现得更好，在高信噪比有偏估计。我们的分析的一个有趣的结论是，BB的高SNR行为仅取决于稀疏向量的最小非零分量的值，并且这种类型的依赖性也表现出某些实际估计的性能。
英文摘要：
We consider unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise. We show that while there are infinitely many unbiased estimators for this problem, none of them has uniformly minimum variance. Therefore, we focus on locally minimum variance unbiased (LMVU) estimators. We derive simple closed-form lower and upper bounds on the variance of LMVU estimators or, equivalently, on the Barankin bound (BB). Our bounds allow an estimation of the threshold region separating the low-SNR and high-SNR regimes, and they indicate the asymptotic behavior of the BB at high SNR. We also develop numerical lower and upper bounds which are tighter than the closed-form bounds and thus characterize the BB more accurately. Numerical studies compare our characterization of the BB with established biased estimation schemes, and demonstrate that while unbiased estimators perform poorly at low SNR, they may perform better than biased estimators at high SNR. An interesting conclusion of our analysis is that the high-SNR behavior of the BB depends solely on the value of the smallest nonzero component of the sparse vector, and that this type of dependence is also exhibited by the performance of certain practical estimators.
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