Issue No. 08 - August (2010 vol. 32)
Dan Kushnir , Yale University, New Haven
Meirav Galun , Weizmann Institute of Science, Rehovot
Achi Brandt , Weizmann Institute of Science, Rehovot
Multigrid solvers proved very efficient for solving massive systems of equations in various fields. These solvers are based on iterative relaxation schemes together with the approximation of the “smooth” error function on a coarser level (grid). We present two efficient multilevel eigensolvers for solving massive eigenvalue problems that emerge in data analysis tasks. The first solver, a version of classical algebraic multigrid (AMG), is applied to eigenproblems arising in clustering, image segmentation, and dimensionality reduction, demonstrating an order of magnitude speedup compared to the popular Lanczos algorithm. The second solver is based on a new, much more accurate interpolation scheme. It enables calculating a large number of eigenvectors very inexpensively.
Eigenvalues and eigenvectors, multigrid and multilevel methods, graph algorithms, segmentation, clustering.
A. Brandt, D. Kushnir and M. Galun, "Efficient Multilevel Eigensolvers with Applications to Data Analysis Tasks," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 32, no. , pp. 1377-1391, 2009.