The block Kaczmarz method is an iterative scheme for solving overdetermined
least-squares problems. At each step, the algorithm projects the current
iterate onto the solution space of a subset of the constraints. This paper
describes a block Kaczmarz algorithm that uses a randomized control scheme to
choose the subset at each step. This algorithm is the first block Kaczmarz
method with an (expected) linear rate of convergence that can be expressed in
terms of the geometric properties of the matrix and its submatrices. The
analysis reveals that the algorithm is most effective when it is given a good
row paving of the matrix, a partition of the rows into well-conditioned blocks.
The operator theory literature provides detailed information about the
existence and construction of good row pavings. Together, these results yield
an efficient block Kaczmarz scheme that applies to many overdetermined
least-squares problem
