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摘要: 为了提高稀疏信号恢复的准确性, 开展了基于自适应套索算子(Least absolute shrinkage and selection operator, LASSO)先验的稀疏贝叶斯学习(Sparse Bayesian learning, SBL)算法研究. 1) 在稀疏贝叶斯模型构建阶段, 构造了一种新的多层贝叶斯框架, 赋予信号中元素独立的LASSO先验. 该先验比现有稀疏先验更有效地鼓励稀疏并且该模型中所有参数更新存在闭合解. 然后在该多层贝叶斯框架的基础上提出了一种基于自适应LASSO先验的SBL算法. 2) 为降低提出的算法的计算复杂度, 在贝叶斯推断阶段利用空间轮换变元方法对提出的算法进行改进, 避免了矩阵求逆运算, 使参数更新快速高效, 从而提出了一种基于自适应LASSO先验的快速SBL算法. 本文提出的算法的稀疏恢复性能通过实验进行了验证, 分别针对不同大小测量矩阵的稀疏信号恢复以及单快拍波达方向(Direction of arrival, DOA)估计开展了实验. 实验结果表明: 提出基于自适应LASSO先验的SBL算法比现有算法具有更高的稀疏恢复准确度; 提出的快速算法的准确度略低于提出的基于自适应LASSO先验的SBL算法, 但计算复杂度明显降低.
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关键词:
- 稀疏信号恢复 /
- 稀疏贝叶斯学习 /
- 自适应LASSO先验 /
- 贝叶斯推断
Abstract: To improve the recovery accuracy of sparse signal, we study on sparse Bayesian learning (SBL) algorithm using adaptive least absolute shrinkage and selection operator (LASSO) priors. First, a hierarchical Bayesian framework is built for Bayesian model. Each elements of the weights is assigned with hierarchical priors, resulting in adaptive LASSO priors. Compared with other priors, the proposed adaptive LASSO priors encourage sparsity more efficiently and all the variables in the proposed model can be updated using close form solution. Thus, a SBL algorithm using adaptive LASSO priors is proposed. Second, the space alternating approach is integrated into the proposed algorithm to reduce the computational complexity by avoiding matrix inverse operation. In this way, the parameters can be updated efficiently and a fast SBL algorithm using adaptive LASSO priors is proposed. The accuracy performance of the proposed algorithms are verified using numerical simulations versus different size of measurement matrix and single snapshot direction-of-arrival (DOA) estimation, respectively. The experiments show that the root mean square error (RMSE) of the proposed adaptive LASSO priors based SBL method is lower than state-of-the-art methods. Besides, the RMSE of proposed fast algorithm is slightly lower than the proposed adaptive LASSO priors based SBL method but achieves lower computational complexity performance.1) 1 Oracle特性具体包括模型选择相和性和参数估计渐进正态性. 其含义为, 在一些变量不是提前已知的情况下, 如果算法具有Oracle特性, 那么它能够筛选出正确的预测的概率为1而且能够有效而正确地估计非零估计量. -
图 1 基于自适应LASSO先验的SBL框架的因子图
Fig. 1 The factor graph of the proposed SBL framework using adaptive LASSO priors
图 2 四种算法的稀疏先验代价函数二维等高线图
Fig. 2 Two dimensional contour plots of cost functions of different sparse priors
图 3 本算法在不同参数下稀疏先验代价函数二维等高线图
Fig. 3 Two dimensional contour plots of cost functions of the proposed sparse priors versus hyperparameters
图 5 实值模型下各算法稀疏恢复准确度与测量数的关系
Fig. 5 RMSE of different algorithms with the real-value signal model versus length of measurements
图 6 复值模型下各算法稀疏恢复准确度与测量数的关系
Fig. 6 RMSE of different algorithms with the complex-value signal model versus length of measurements
图 7 高维实值信号模型下各算法稀疏恢复准确度与测量数的关系
Fig. 7 RMSE of different algorithms with the high-dimensional real-value signal model versus length of measurements
图 8 高维复值信号模型下各算法稀疏恢复准确度与测量数的关系
Fig. 8 RMSE of different algorithms with the high-dimensional complex-value signal model versus length of measurements
图 9 实值模型下各算法稀疏恢复准确度与稀疏度的关系
Fig. 9 RMSE of different algorithms with the real-value signal model versus number of non-zero elements
图 10 复值模型下各算法稀疏恢复准确度与稀疏度的关系
Fig. 10 RMSE of different algorithms with the complex-value signal model versus number of non-zero elements
图 11 高维实值信号模型下各算法稀疏恢复准确度与稀疏度的关系
Fig. 11 RMSE of different algorithms with the high-dimensional real-value signal model versus number of non-zero elements
图 12 高维复值信号模型下各算法稀疏恢复准确度与稀疏度的关系
Fig. 12 RMSE of different algorithms with the high-dimensional complex-value signal model versus number of non-zero elements
图 13 实值模型下各算法稀疏恢复准确度与信噪比的关系
Fig. 13 RMSE of different algorithms versus SNR with the real-value signal model
图 14 复值模型下各算法稀疏恢复准确度与信噪比的关系
Fig. 14 RMSE of different algorithms versus SNR with the complex-value signal model
图 15 高维实值信号模型下各算法稀疏恢复准确度与信噪比的关系
Fig. 15 RMSE of different algorithms versus SNR with the high-dimensional real-value signal model
图 16 高维复值信号模型下各算法稀疏恢复准确度与信噪比的关系
Fig. 16 RMSE of different algorithms versus SNR with the high-dimensional complex-value signal model
图 17 DOA估计的准确度与测量数的关系
Fig. 17 RMSE of DOA estimation using different algorithms versus number of measurements
图 18 DOA估计准确度与信噪比的关系
Fig. 18 RMSE of DOA estimation using different algorithms versus SNR
表 1 各算法单次运行时间
Table 1 Time consumptions of different algorithms
实值信号模型 复值信号模型 算法 用时(s) 算法 用时(s) FastLaplace 0.11 FastSBL 1.54 aLASSO 1.94 GAMP-SBL 0.51 FastSBL 0.40 MFOCUSS 0.21 GAMP-SBL 0.07 HSL-SBL 3.16 FaLASSO-SBL 0.26 FaLASSO-SBL 0.74 aLASSO-SBL 0.98 aLASSO-SBL 2.33 
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表 2 恢复高维信号时各算法单次运行时间
Table 2 Time consumptions of different algorithms when the dimension of signal is high
实值信号模型 复值信号模型 算法 用时(s) 算法 用时(s) FastLaplace 0.83 FastSBL 6.95 aLASSO 5.71 GAMP-SBL 2.17 FastSBL 3.40 MFOCUSS 2.86 GAMP-SBL 0.69 HSL-SBL 15.73 FaLASSO-SBL 1.06 FaLASSO-SBL 4.61 aLASSO-SBL 8.38 aLASSO-SBL 17.41
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表 3 单快拍DOA估计实验各算法单次运行时间
Table 3 Time consumptions of different algorithms for single snapshot DOA estimation
算法 用时(s) 算法 用时(s) SS-ESPRIT 0.37 HSL-SBL 0.85 SURE-IR 1.64 FaLASSO-SBL 0.47 L1-SR 0.91 aLASSO-SBL 0.83 OGSBL 0.69
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