基于总最小二乘和近端分裂的扰动压缩感知快速重建算法
Fast Reconstruction Algorithm for Perturbed Compressive Sensing Based on Total Least-Squares and Proximal Splitting
AI总结：
本文提出一种基于总最小二乘和近端分裂的快速算法，用于解决受扰动的压缩感知问题，通过最小化l1正则化Rayleigh商成本函数，提升收敛速度和稳定性，实验验证了其有效性。
摘要：
我们考虑在系统两边已知参数受扰动的情况下，求解欠定线性方程组稀疏解的问题。这在完全受扰动的压缩感知设置中尤为相关，其中未知稀疏向量的投影测量和相关投影矩阵的知识均受噪声、误差、不匹配等影响。我们提出了一种新的迭代算法来解决此问题。所提算法利用近端梯度法，通过最小化l1正则化Rayleigh商成本函数来寻找稀疏总最小二乘解。我们通过自适应规则配合回溯线搜索确定算法在每次迭代中的步长，以提高算法的收敛速度并保持其稳定性。所提算法在计算效率上显著优于一种流行的先前提出算法，后者采用交替方向法和坐标下降迭代。该算法需要显著更少的计算量以达到相同精度。我们通过模拟结果展示了所提算法的有效性。
英文摘要：
We consider the problem of finding a sparse solution for an underdetermined linear system of equations when the known parameters on both sides of the system are subject to perturbation. This problem is particularly relevant to reconstruction in fully-perturbed compressive-sensing setups where both the projected measurements of an unknown sparse vector and the knowledge of the associated projection matrix are perturbed due to noise, error, mismatch, etc. We propose a new iterative algorithm for tackling this problem. The proposed algorithm utilizes the proximal-gradient method to find a sparse total least-squares solution by minimizing an l1-regularized Rayleigh-quotient cost function. We determine the step-size of the algorithm at each iteration using an adaptive rule accompanied by backtracking line search to improve the algorithm's convergence speed and preserve its stability. The proposed algorithm is considerably faster than a popular previously-proposed algorithm, which employs the alternating-direction method and coordinate-descent iterations, as it requires significantly fewer computations to deliver the same accuracy. We demonstrate the effectiveness of the proposed algorithm via simulation results.
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