Greedy Randomized and Maximal Weighted Residual Kaczmarz Methods with Oblique Projection

For solving large-scale consistent linear system, a greedy randomized Kaczmarz method with oblique projection and a maximal weighted residual Kaczmarz method with oblique projection are proposed. By using oblique projection, these two methods greatly reduce the number of iteration steps and running time to find the minimum norm solution, especially when the rows of matrix A are close to linear correlation. Theoretical proof and numerical results show that the greedy randomized Kaczmarz method with oblique projection and the maximal weighted residual Kaczmarz method with oblique projection are more effective than the greedy randomized Kaczmarz method and the maximal weighted residual Kaczmarz method respectively.

An Iterative Hard Thresholding Algorithm based on Sparse Randomized Kaczmarz Method for Compressed Sensing

The paper proposes a novel signal reconstruction algorithm through substituting the gradient descent method in the iterative hard thresholding algorithm with a faster sparse randomized Kaczmarz method. By designing a series of gradually attenuated weights for the matrix rows whose indexes lie outside of the support set of the original sparse signal, we can focus the iterations on the effective support rows of the measurement matrix. The experiment results show that the proposed algorithm presents a faster convergence rate and more accurate reconstruction accuracy than the state-of-the-art algorithms. Meanwhile, the successful reconstruction probability of the proposed algorithm is higher than that of other algorithms. Moreover, the characteristics of the proposed signal reconstruction algorithm are also analyzed in detail through numerical experiments.

Phase retrieval via randomized Kaczmarz: theoretical guarantees

We consider the problem of phase retrieval, i.e. that of solving systems of quadratic equations. A simple variant of the randomized Kaczmarz method was recently proposed for phase retrieval, and it was shown numerically to have a computational edge over state-of-the-art Wirtinger flow methods. In this paper, we provide the first theoretical guarantee for the convergence of the randomized Kaczmarz method for phase retrieval. We show that it is sufficient to have as many Gaussian measurements as the dimension, up to a constant factor. Along the way, we introduce a sufficient condition on measurement sets for which the randomized Kaczmarz method is guaranteed to work. We show that Gaussian sampling vectors satisfy this property with high probability; this is proved using a chaining argument coupled with bounds on Vapnik–Chervonenkis (VC) dimension and metric entropy.