论文标题
扩展的随机Kaczmarz方法，用于稀疏最小二乘和冲动噪声问题
Extended Randomized Kaczmarz Method for Sparse Least Squares and Impulsive Noise Problems
论文作者
论文摘要
扩展的随机kaczmarz方法是一种众所周知的迭代方案，可以找到可能不一致的线性系统的摩尔 - 柔性逆解，并且与标准的随机kaczmarz方法相比，每次迭代中只需要一个额外的系统矩阵列。同样，稀疏的随机kaczmarz方法已被证明可以线性收敛到一致的线性系统的稀疏解。在这里，我们结合了思想，并提出了一种扩展的稀疏随机kaczmarz方法。我们显示了线性的预期收敛到稀疏最小二乘解决方案，从某种意义上说，正规化基础追踪问题的扩展变体得到了解决。此外，我们将该方法的附加步骤推广到了一个更抽象的优化问题。我们从数值上证明，如果噪声集中在系统矩阵范围的补充中，并且我们的概括可以处理冲动的噪声，则我们的方法可以找到真实和复杂系统的稀疏最小二乘解决方案。
The Extended Randomized Kaczmarz method is a well known iterative scheme which can find the Moore-Penrose inverse solution of a possibly inconsistent linear system and requires only one additional column of the system matrix in each iteration in comparison with the standard randomized Kaczmarz method. Also, the Sparse Randomized Kaczmarz method has been shown to converge linearly to a sparse solution of a consistent linear system. Here, we combine both ideas and propose an Extended Sparse Randomized Kaczmarz method. We show linear expected convergence to a sparse least squares solution in the sense that an extended variant of the regularized basis pursuit problem is solved. Moreover, we generalize the additional step in the method and prove convergence to a more abstract optimization problem. We demonstrate numerically that our method can find sparse least squares solutions of real and complex systems if the noise is concentrated in the complement of the range of the system matrix and that our generalization can handle impulsive noise.
