Two-subspace Projection Method for Coherent Overdetermined Systems

Abstract. We present a Projection onto Convex Sets (POCS) type algorithm for solving systems of linear equations. POCS methods have found many applications ranging from computer tomography to digital signal and image processing. The Kaczmarz method is one of the most popular solvers for overdetermined systems of linear equations due to its speed and simplicity. Here we introduce and analyze an extension of the Kaczmarz method which iteratively projects the estimate onto a solution space given from two randomly selected rows. We show that this projection algorithm provides exponential convergence to the solution in expectation. The convergence rate significantly improves upon that of the standard randomized Kaczmarz method when the system has coherent rows. We also show that the method is robust to noise, and converges exponentially in expectation to the noise floor. Experimental results are provided which confirm that in the coherent case our method significantly outperforms the randomized Kaczmarz method.