Efficient Multilevel Eigensolvers with Applications to Data Analysis Tasks

Abstract

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.

Spectral Graph Reduction for Efficient Image and Streaming Video Segmentation

“…Eigensolvers based on multigrid methods temporarily reduce the number of nodes and speed up computation while keeping the solution of the original problem [15], yet the coarser grids do not fit well to image structures. In [6] an approximate algebraic multigrid has been proposed, which focuses on the use of different features at different scales rather than a fast exact solution of the single-grid problem.…”

Fast Hybrid Level Set Model for Non-homogenous Image Segmentation Solving by Algebraic Multigrid

“…In order to overcome this problem effectively, we adopt a multi-resolution thought-based partial differential equation solver called AMG [15] to reduce the number of iterations by using a larger time step.…”

“…In the field of solving linear equations, a highly efficient multi-resolution scheme, named algebraic multigrid (AMG) [15] strategy, has emerged in the past research of mathematics. The algebraic multigrid is independent of the geometrical properties of the problem to be solved, and it uses only the information of the system itself to solve the linear equations, allowing the solution on unstructured grids, making it easier to extend to areas such as image processing.…”

Progressive Multigrid Eigensolvers for Multiscale Spectral Segmentation

“…Unlike generic multigrid methods [4,2,8], we explicitly address constrained problems [17,5] with the intuition that the constraints themselves can guide the schedule of computations within the solver. Figure 1 provides a comparison.…”

“…connections between neighboring pixels, for image segmentation, as shown in green). Generic multigrid eigensolvers [4,2,8], applied to the corresponding Normalized Cuts eigenproblem [14], coarsen the problem by subsampling nodes and interpolating weights. Solution eigenvectors from iteratively coarsened problems, Ψ(W ) and Ψ(Ψ(W )), initialize the solver on the next finer problems, W and Ψ(W ), respectively (blue arrows).…”