On computing the minimum singular value of a tensor sum

A Ohashi,T Sogabe

doi:10.1515/spma-2019-0009

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.

Highlights

Consider computing the minimum singular value of the following generalized tensor sum [1]: T := In ⊗ Im ⊗ A + In ⊗ B ⊗ I + C ⊗ Im ⊗ I ∈ R mn× mn, (1)
The method was obtained from reconstructing the Lanczos bidiagonalization method, which is widely known as the Golub-Kahan bidiagonalization method [5], over tensor space
This paper presented an invert Lanczos bidiagonalization method over tensor space with tensor-structurepreserving direct methods

Summary

Introduction

Ohashi and Sogabe [1] proposed the Lanczos bidiagonalization method over tensor space to compute the maximum/minimum singular values of T in (1). From the de nition of L, the mode products, (7), and (8), the following implementations of vec− (T− pi+ ) and vec− ((T− )Tqi) are obtained: Algorithm 3-1 Computation of vec− (T− pi+ ) using the eigendecomposition Vec− (T− pi+ ) and vec− ((T− )Tqi) are computed employing only smaller matrices R(A), R(B), R(C), Q(A), Q(B), Q(C) and tensors W and X, using Algorithms 4-1(A) and 4-2(A) with Algorithms 4-1(B) and 4-2(B), respectively.

Results

Conclusion