The Projected Newton Iteration Approach for Computing the Nonnegative Z-Eigenpairs of Nonnegative Tensors

Abstract

In this paper, we propose a new projected Newton iteration for computing the nonnegative Z-eigenpairs of nonnegative tensors. We show that the required iteration has a local quadratic convergence. More specially, the formulation aims to solve the tensor equation arising from the multilinear PageRank problem. Numerical experiments are provided to illustrate the effectiveness and superiority of the proposed approach.

A new modified Newton iteration for computing nonnegative Z-eigenpairs of nonnegative tensors

“…is nonsingular. To put our new modified Newton iteration in a proper setting, we give below a brief description of MNI in [5] and PNI in [1]. We also mention that a Newton-based method has been studied in [6] for computing Z-eigenpairs of symmetric tensors.…”

“…The PNI in [1] can be described through a comparison with MNI. In PNI, line 6 of MNI is not performed.…”

“…So the w k in line 7 is still the w k in line 5. It is assumed in [1] that e T w k = 0. Then e T x k+1 = m − 1 > 0 and x k+1 has some positive components.…”

“…But it has also created a new problem: in line 9 of MNI the definition in (12) must be modified. For a pair of nonnegative vectors w and v = 0, it is defined in [1] that…”

“…It is observed in [5] that MNI can also compute nonnegative eigenpairs with some zero components in the eigenvectors, but the convergence is much slower. Recently, another modified Newton iteration (called PNI) is proposed in [1], where a projection is used at a later stage than in [5]. PNI can significantly speed up the convergence in computing nonnegative eigenpairs with some zero components in the eigenvectors.…”