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l₀-范数属于NP问题,最小化l₁-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l₀-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。

Objective Compressed sensing signal reconstruction involves finding the sparse solution of an underdetermined system of linear equations. This problem has three common solutions:minimizing the l₀-norm, l₁-norm, and l_p-norm. Minimizing the l₀-norm can solve this problem, but the method belongs to NP complete. Meanwhile, the l₁-norm has no solution in theory, and the l_p-norm is not a convex function. Under this background, this paper presents a recovery method of compressed sensing signals based on regularized smooth convex optimization. Method To obtain the global optimal solution, a convex function in the entire space is first designed as the objective function of optimization to fit the l₀-norm of a signal. Second, the optimization problem with n variables is transformed into n optimization problems with one variable. Finally, a fast iterative shrinkage-thresholding algorithm is proposed to find the solution, with the convergence speed being that of quadratic convergence. Result Experimental results show that the method is robust and fast compared with three other types of algorithms. Specifically, the reconstruction time of the proposed method is approximately 50% of that of the other algorithms when the signal has more than 150 dimensions. In the real data experiments, the F-norm of the reconstructed signal and the original signal difference with the proposed method is approximately 70% of the other algorithms. Conclusion The proposed algorithm with a high compression ratio and a good restoration effect is well-suited for processing large data. It meets the requirements for application in information retrieval, dictionary learning, image compression, and so on.