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目的压缩感知信号重构过程是求解不定线性系统稀疏解的过程。针对不定线性系统稀疏解3种求解方法不够鲁棒的问题:最小化l_0-范数属于NP问题,最小化l_1-范数的无解情况以及最小化l_p-范数的非凸问题,提出一种基于光滑正则凸优化的方法进行求解。方法为了获得全局最优解并保证算法的鲁棒性,首先,设计了全空间信号l_0-范数凸拟合函数作为优化的目标函数;其次,将n元函数优化问题转变为n个一元函数优化问题;最后,求解过程中利用快速收缩算法进行求解,使收敛速度达到二阶收敛。结果该算法无论在仿真数据集还是在真实数据集上,都取得了优于其他3种类型算法的效果。在仿真实验中,当信号维数大于150维时,该方法重构时间为其他算法的50%左右,具有快速性;在真实数据实验中,该方法重构出的信号与原始信号差的F-范数为其他算法的70%,具有良好的鲁棒性。结论本文算法为二阶收敛的凸优化算法,可确保快速收敛到全局最优解,适合处理大型数据,在信息检索、字典学习和图像压缩等领域具有较大的潜在应用价值。
The purpose of compressed sensing signal reconstruction process is to solve the linear system of indefinite sparse solution process. Three solutions to the problem of unsteady linear systems are not robust enough: minimizing the l_0-norm belongs to the NP problem, minimizing the non-solution of the l_1-norm and the non-convex problem of minimizing the l_p-norm. The solution is based on the smooth regular convex optimization method. Methods In order to obtain the global optimal solution and ensure the robustness of the algorithm, firstly, the full-space signal l_0-norm convex fitting function is designed as the objective function of optimization. Secondly, the n-ary function optimization problem is transformed into n unary functions In the end, the solution is solved by the fast shrinking algorithm, which converges the convergence speed to second-order convergence. Results The algorithm achieved better results than the other three types of algorithms both in the simulation data set and in the real data set. In the simulation experiment, when the signal dimension is more than 150, the reconstruction time of this method is about 50% of other algorithms, which is fast. In the real data experiment, the difference between the reconstructed signal and the original signal F - The norm is 70% of other algorithms, with good robustness. Conclusion This algorithm is a convex optimization algorithm with second-order convergence, which ensures fast convergence to the global optimal solution and is suitable for large data processing. It has great potential applications in information retrieval, dictionary learning and image compression.