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The Kaczmarz method is an iterative algorithm for solving systems of linearequalities and inequalities, that iteratively projects onto these constraints.Recently, Strohmer and Vershynin [J. Fourier Anal. Appl., 15(2):262-278, 2009]gave a non-asymptotic convergence rate analysis for this algorithm, spurringnumerous extensions and generalizations of the Kaczmarz method. Rather than therandomized selection rule analyzed in that work, in this paper we insteaddiscuss greedy and approximate greedy selection rules. We show that in someapplications the computational costs of greedy and random selection arecomparable, and that in many cases greedy selection rules give fasterconvergence rates than random selection rules. Further, we give the firstmulti-step analysis of Kaczmarz methods for a particular greedy rule, andpropose a provably-faster randomized selection rule for matrices with manypairwise-orthogonal rows.

Convergence Rates for Greedy Kaczmarz Algorithms, and Faster Randomized Kaczmarz Rules Using the Orthogonality Graph