  The Randomized Kaczmarz method (RK) is a stochastic iterative method for
solving linear systems that has recently grown in popularity due to its speed
and low memory requirement. Selectable Set Randomized Kaczmarz (SSRK) is an
variant of RK that leverages existing information about the Kaczmarz iterate to
identify an adaptive "selectable set" and thus yields an improved convergence
guarantee. In this paper, we propose a general perspective for selectable set
approaches and prove a convergence result for that framework. In addition, we
define two specific selectable set sampling strategies that have competitive
convergence guarantees to those of other variants of RK. One selectable set
sampling strategy leverages information about the previous iterate, while the
other leverages the orthogonality structure of the problem via the Gramian
matrix. We complement our theoretical results with numerical experiments that
compare our proposed rules with those existing in the literature.
