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摘要: 基于泛函深度作用的思想, 通过将两种非凸稀疏泛函进行复合, 构造了一种新的稀疏信号重构模型, 实现了对0范数的深度逼近. 综合运用MM (Majorize minimization)技术、外点罚函数法和共轭梯度法, 提出一种求解该模型的算法, 称为NCCS (Non-convex composite sparse)算法. 为降低重构信号陷入局部极值的可能性, 提出在算法的每步迭代中以BP (Basis pursuit)模型的解作为初始迭代值. 为验证所建模型和所提算法的有效性, 进行了多项数值实验. 实验结果表明, 相较于SL0 (Smoothed
$L_0$ )算法、IRLS (Iterative reweighed least squares)算法、SCSA (Successive concave sparsity approximation)算法以及BP 算法等经典算法, 提出的算法在重构误差、信噪比、归一化均方差、支撑集恢复成功率等方面都有更优的表现.Abstract: Based on the idea of deep composition of functionals, this paper constructs a new sparse signal reconstruction model by combining two non-convex sparse functionals to achieve the depth of 0 norm approaching. Using the MM (Majorize minimization) technology, the exterior penalty function method and the conjugate gradient method, an algorithm for solving the model is proposed, which is called the NCCS (Non-convex composite sparse) algorithm. In order to reduce the possibility of the reconstructed signal falling into a local extreme value, it is proposed to use the solution of the BP (Basis pursuit) model as the initial iterative value in each iteration of the algorithm. To verify the effectiveness of the model and the proposed algorithm, a number of numerical experiments have been carried out in this paper. The experimental results show that compared with classic algorithms such as the SL0 (Smoothed$L_0 $ ) algorithm, IRLS (Iterative reweighed least squares) algorithm, SCSA (Successive concave sparsity approximation) algorithm and BP algorithm, the algorithm proposed in this paper has better performance in reconstruction error, signal-to-noise ratio, normalized mean square error, and support set recovery success rate. -
图 1 4种函数在
$\sigma = 0.1$ 时的一元函数分布Fig. 1 The unary distribution of the four functions at
$\sigma = 0.1$ 
图 2
${p_\sigma }({\boldsymbol{x}})$ 、${h_\sigma }({\boldsymbol{x}})$ 、${g_\sigma }({\boldsymbol{x}})$ 和函数${f_\sigma }({\boldsymbol{x}})$ 在$\sigma = 0.1$ 时的二元函数分布Fig. 2 The bivariate distribution of
${p_\sigma }({\boldsymbol{x}}),$ ${h_\sigma }({\boldsymbol{x}}),$ ${g_\sigma }({\boldsymbol{x}})$ and the function${f_\sigma }({\boldsymbol{x}})$ at$\sigma = 0.1$ 
图 3 待定数
$\alpha $ 对NCCS算法运行时间的影响Fig. 3 The influence of undetermined number
$\alpha $ on the running time of NCCS algorithm
图 4 NCCS算法的一维信号重构仿真图, 信号大小为
$500 \times 1,$ 稀疏度为$65$ Fig. 4 One-dimensional signal reconstruction simulation diagram of NCCS algorithm, the signal size is 500 × 1, the sparsity is 65
图 5 SL0、IRLS、BP、SCSA、NCCS 5种算法的重构误差和稀疏度的变化关系
Fig. 5 The relationship between the reconstruction error and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS
图 6 SL0、IRLS、BP、SCSA、NCCS 5种算法的重构信噪比和稀疏度的变化关系
Fig. 6 The relationship between the reconstructed signal-to-noise ratio and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS
图 7 SL0、IRLS、BP、SCSA、NCCS 5种算法的运行时间和稀疏度的变化关系
Fig. 7 The relationship between the running time and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS
图 8 SL0、IRLS、BP、SCSA、NCCS 5种算法的支撑集恢复成功率和稀疏度的变化关系
Fig. 8 The relationship between the recovery success rate of the support set and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS
图 9 SL0、IRLS、BP、SCSA、NCCS 5种算法的归一化均方差和稀疏度的变化关系
Fig. 9 The relationship between the normalized mean square error and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS
表 1 5种算法的归一化均方差的数值记录
Table 1 Numerical records of the normalized mean square error of five algorithms
算法 归一化均方差$(NMSE)$ k = 20 k = 30 k = 40 k = 50 k = 60 k = 70 k = 80 k = 90 k = 100 k = 110 SL0 4.11×10−10 2.86×10−10 2.06×10−10 1.55×10−10 1.16×10−10 9.38×10−11 1.46×10−4 1.28×10−3 9.22×10−3 3.13×10−2 IRLS 2.02×10−4 4.08×10−4 1.15×10−3 4.23×10−3 1.07×10−2 2.79×10−2 5.04×10−2 7.20×10−2 1.14×10−1 1.38×10−1 BP 1.65×10−22 4.49×10−22 5.68×10−22 4.95×10−22 4.19×10−22 4.38×10−15 1.18×10−5 9.52×10−4 1.03×10−2 3.57×10−2 SCSA 2.05×10−23 1.94×10−22 1.94×10−22 2.45×10−22 2.50×10−22 2.96×10−22 3.18×10−22 1.27×10−16 1.27×10−13 3.57×10−12 NCCS 5.36×10−27 2.10×10−26 4.81×10−24 4.98×10−24 5.87×10−15 1.85×10−17 3.35×10−16 2.62×10−17 3.58×10−16 1.00×10−16 
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