Result: 一个新的零模非凸代理函数及其应用研究.
Further Information
Based on the SCAD (Smoothly Clipped Absolute Deviation) function and the elastic net function, a new zero-norm nonconvex surrogate function(EN-SCAD function) is proposed, which is the difference between the elastic net function and a continuous differentiable convex function, thus is a DC (difference of convex) function. Then, the EN-SCAD function is applied to the sparse linear regression problem, and the EN-SCAD nonconvex surrogate model is established, and the statistical error bound between the stationary point of the model and the true sparse vector is obtained under appropriate restricted strong convex condition. Secondly, a multi-stage convex relaxation algorithm is designed according to the EN-SCAD nonconvex surrogate model, and obtain the statistical error bounds between the columns of iterative points generated by the algorithm and the true sparse vector. Finally, comparing the numerical effects of the algorithm designed based on the EN-SCAD nonconvex surrogate model with a convex relaxation method for adaptive elastic net. Numerical experimental results show that when the column vectors of the sampling matrix are strongly correlated, the estimation error produced by the algorithm based on the EN-SCAD nonconvex surrogate model is smaller than that produced by the convex relaxation method for adaptive elastic net. [ABSTRACT FROM AUTHOR]
基于带平滑削边绝对偏离(Smoothly Clipped Absolute Deviation, SCAD)函数和弹性网络(Elastic Net)函数，提出了一个零模非凸代理函数(EN-SCAD函数)，该代理函数是弹性网络函数与一个连续可微凸函数的差，因此是一个凸差(Difference of Convex，DC)函数；然后，将EN-SCAD函数应用于稀疏线性回归问题，建立了EN-SCAD非凸代理模型，在适当的限制强凸条件下得到该模型的稳定点与真实稀疏向量之间的统计误差界；其次，根据EN-SCAD非凸代理模型设计了一个多阶段凸松弛算法，并得到了该算法产生的迭代点列与真实稀疏向量之间的统计误差界；最后，将基于EN-SCAD非凸代理模型设计的算法与自适应弹性网络凸松弛方法的数值效果进行比较，数值实验结果表明：当采样矩阵的列向量具有强相关性时，基于EN-SCAD非凸代理模型的算法产生的估计误差小于自适应弹性网络凸松弛方法产生的估计误差。 [ABSTRACT FROM AUTHOR]
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