Affiliation:
1. College of Science & Engineering , Ritsumeikan University , Japan
2. Graduate School of Engineering , Nagoya University , Japan
Abstract
Abstract
Recently, the Lanczos bidiagonalization method over tensor space has been proposed for computing the maximum and minimum singular values of a tensor sum T. The method over tensor space is practical in memory and has a simple implementation due to recent developments in tensor computations; however, there is still room for improvement in the convergence to the minimum singular value. This study reconstructed an invert Lanczos bidiagonalization method from vector space to tensor space. The resulting algorithm requires solving linear systems at each iteration step. Using standard direct methods, such as the LU decomposition for solving the linear systems requires a huge memory of O(n
6). Therefore, this paper proposes a tensor-structure-preserving direct methods of T whose memory requirements are of O(n
3), which is equivalent to the order of iterative methods. Numerical examples indicate that the number of iterations tends to be much smaller than that of the conventional method.
Subject
Geometry and Topology,Algebra and Number Theory
Reference13 articles.
1. [1] A. Ohashi, T. Sogabe, On computing maximum/minimum singular values of a generalized tensor sum, Electron. T. Numer. Ana., 43(1), 244–254 (2015)
2. [2] J. Ballani, L. Grasedyck, A projection method to solve linear systems in tensor format, Numer. Linear Algebr., 20(1), 27–43 (2013)
3. [3] D. Kressner, C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. A., 31(4), 1688–1714 (2010)
4. [4] T.G. Kolda, B.W. Bader, Tensor decompositions and applications, SIAM Rev., 51(3), 455–500 (2009)
5. [5] G. Golub, W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., 2(2), 205–224 (1965)
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献