Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum

Asuka Ohashi,Tomohiro Sogabe

doi:10.3390/axioms10030211

Abstract

We consider computing an arbitrary singular value of a tensor sum: T:=In⊗Im⊗A+In⊗B⊗Iℓ+C⊗Im⊗Iℓ∈Rℓmn×ℓmn, where A∈Rℓ×ℓ, B∈Rm×m, C∈Rn×n. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of (TTT−σ˜2Iℓmn)−1, where σ˜ is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix TTT−σ˜2Iℓmn. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of TTT and the PCG method for TTT−σ˜2Iℓmn using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments.

Highlights

We consider computing an arbitrary singular value of a tensor sum:T := In ⊗ Im ⊗ A + In ⊗ B ⊗ I + C ⊗ Im ⊗ I ∈ R mn× mn, (1)where A ∈ R ×, B ∈ Rm×m, C ∈ Rn×n, In is the n × n identity matrix, and the symbol “⊗” denotes the Kronecker product
The convergence rate of the shift and invert Lanczos method depends on the ratio of gaps between the maximum, the second maximum, and the minimum singular values σ1, σ2, σm of (TTT − σ 2 I mn)−1 as follows: (σ12 − σ22)/(σ12 − σm2 )
The number of iterations of the shift-and-invert Lanczos method (“the Lanczos method" hereafter) and the average of the number of iterations of the CG or preconditioned conjugate gradient (PCG) method based on the eigendecomposition or the Schur decomposition are summarized

Summary

Introduction

We consider computing an arbitrary singular value of a tensor sum:. Where A ∈ R × , B ∈ Rm×m, C ∈ Rn×n, In is the n × n identity matrix, and the symbol “⊗” denotes the Kronecker product. Previous studies [1,2] provided methods to compute the maximum and minimum singular values of T. One can compute only the maximum and minimum singular values of T without shift. The previous studies are based on the Lanczos bidiagonalization method (see, e.g., [3]), which computes the maximum and minimum singular values of a matrix. For insights on Lanczos bidiagonalization method, see, e.g., [4,5,6]. The Lanczos bidiagonalization method for T was implemented using tensors and their operations to reduce the memory requirement

Methods

Results

Conclusion