  The Kaczmarz method is an algorithm for finding the solution to an
overdetermined consistent system of linear equations Ax=b by iteratively
projecting onto the solution spaces. The randomized version put forth by
Strohmer and Vershynin yields provably exponential convergence in expectation,
which for highly overdetermined systems even outperforms the conjugate gradient
method. In this article we present a modified version of the randomized
Kaczmarz method which at each iteration selects the optimal projection from a
randomly chosen set, which in most cases significantly improves the convergence
rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep
the runtime on the same order as the original randomized version, adding only
extra preprocessing time. We present a series of empirical studies which
demonstrate the remarkable acceleration in convergence to the solution using
this modified approach.
