The Kaczmarz method is an iterative algorithm for solving overdetermined linear systems by consecutive projections onto the hyperplanes defined by the system equations. The method has a wide range of applications in signal processing, notably for biomedical imaging in X-ray tomography. It has been shown that selecting the hyperplane randomly at each iteration guarantees exponential convergence to the solution. We propose here a new implementation of the Kaczmarz method for clustered equations. When the hyperplanes are grouped into directional clusters, we draw the projection promoting sparse high-variance clusters. This leads to an improvement in performance, as we show in several numerical experiments. Some applications to image reconstruction are presented.
Lanteri, A., Maggioni, M., Vigogna, S. (2019). A biased kaczmarz algorithm for clustered equations. In Springer Proceedings in Mathematics and Statistics (pp.447-456). Springer [10.1007/978-3-030-21158-5_33].
A biased kaczmarz algorithm for clustered equations
Vigogna S.
2019-01-01
Abstract
The Kaczmarz method is an iterative algorithm for solving overdetermined linear systems by consecutive projections onto the hyperplanes defined by the system equations. The method has a wide range of applications in signal processing, notably for biomedical imaging in X-ray tomography. It has been shown that selecting the hyperplane randomly at each iteration guarantees exponential convergence to the solution. We propose here a new implementation of the Kaczmarz method for clustered equations. When the hyperplanes are grouped into directional clusters, we draw the projection promoting sparse high-variance clusters. This leads to an improvement in performance, as we show in several numerical experiments. Some applications to image reconstruction are presented.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
