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.
The classical variants of the Kaczmarz method rely on deterministic mechanisms for selecting a row at each iteration. Indeed, the simplest version just cycles through the rows in order. It has long been known that the cyclic control scheme performs badly when the rows are arranged in an unhappy order @cite_5 . The literature contains empirical evidence @cite_32 that control mechanisms may be more effective, but until recently there was no compelling theoretical analysis to support this observation.
@cite_5 @cite_32
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Spring balance apparatus including a weighing lever supported on a knife edge balance point with one side of the lever acted on by a mass to be weighed against a weighing spring coupled with the other side of the weighing lever, and including an arrangement for the compensation of a measuring error due to temperature change by modifying the ratio of transmission of forces applied to the weighing lever by means of a bimet allic element in which the inventive improvement comprises a knife edge member transferring the forces of the weighing spring to the other side of the weighing lever which knife-edge member has connected rigidly to it an actuating lever coupled to the bimet allic element to alter responsive to temperature changes the orientation of the knife edge member relative to the other side of the weighing lever and thereby modify the ratio of transmission of forces to compensate for measuring error that would otherwise occur due to temperature change. Algebraic reconstruction techniques (ART) are iterative procedures for recovering objects from their projections. It is claimed that by a careful adjustment of the order in which the collected data are accessed during the reconstruction procedure and of the so-called relaxation parameters that are to be chosen in an algebraic reconstruction technique, ART can produce high-quality reconstructions with excellent computational efficiency. This is demonstrated by an example based on a particular (but realistic) medical imaging task, showing that ART can match the performance of the standard expectation-maximization approach for maximizing likelihood (from the point of view of that particular medical task), but at an order of magnitude less computational cost. >