Computation of extremal eigenvalues of high-dimensional lattice-theoretic tensors via tensor-train decompositions

TL;DR

This paper develops tensor-train algorithms to compute extremal eigenvalues of high-dimensional lattice-theoretic tensors, overcoming the curse of dimensionality and enabling tractable analysis of complex tensor eigenvalue problems.

Contribution

It introduces explicit low-rank tensor-train decompositions for meet and LCM tensors, making high-dimensional eigenvalue computations feasible and independent of tensor order.

Findings

Tensor-train decomposition effectively reduces complexity

Eigenvalue algorithms are adapted for tensor-train format

Numerical examples demonstrate method effectiveness

Abstract

This paper lies in the intersection of several fields: number theory, lattice theory, multilinear algebra, and scientific computing. We adapt existing solution algorithms for tensor eigenvalue problems to the tensor-train framework. As an application, we consider eigenvalue problems associated with a class of lattice-theoretic meet and join tensors, which may be regarded as multidimensional extensions of the classically studied meet and join matrices such as GCD and LCM matrices, respectively. In order to effectively apply the solution algorithms, we show that meet tensors have an explicit low-rank tensor-train decomposition with sparse tensor-train cores with respect to the dimension. Moreover, this representation is independent of tensor order, which eliminates the so-called curse of dimensionality from the numerical analysis of these objects and makes the solution of tensor eigenvalue problems tractable with increasing dimensionality and order. For LCM tensors it is shown that a tensor-train decomposition with an a priori known TT rank exists under certain assumptions. We present a series of easily reproducible numerical examples covering tensor eigenvalue and generalized eigenvalue problems that serve as future benchmarks. The numerical results are used to assess the sharpness of existing theoretical estimates.