Phase retrieval of complex-valued objects via a randomized Kaczmarz method

Abstract

This paper investigates the convergence of the randomized Kaczmarz algorithm for the problem of phase retrieval of complex-valued objects. Although this algorithm has been studied for the real-valued case in [ 28], its generalization to the complex-valued case is nontrivial and has been left as a conjecture. This paper applies a different approach by establishing the connection between the convergence of the algorithm and the convexity of an objective function. Based on the connection, it demonstrates that when the sensing vectors are sampled uniformly from a unit sphere in ${\mathcal{C}}^n$ and the number of sensing vectors $m$ satisfies $m>O(n\log n)$ as $n, m\rightarrow \infty $, then this algorithm with a good initialization achieves linear convergence to the solution with high probability. The method can be applied to other statistical models of sensing vectors as well. A similar convergence result is established for the unitary model, where the sensing vectors are from the columns of random orthogonal matrices.2000 Math Subject Classification: 68W20, 68W27, 92D25.