A parametric transformation algorithm and linear convergence of the largest eigenvalue of quasi-symmetric positive tensors

In this paper, we introduce a novel class of quasi-symmetric positive tensors, which generalize nonnegative symmetric tensors. We propose a parametric transformation algorithm dedicated to calculating the largest eigenvalue of nonnegative tensors. Leveraging structural information encoded in the tensor’s associated directed graphs, we show that our algorithm has R-linear convergence for weakly irreducible quasi-symmetric positive tensors. Furthermore, we establish a general condition for the linear convergence of the algorithm, thus extending existing linear convergence theories, such as those underlying the Ng-Qi-Zhou (NQZ) algorithm for essentially positive tensors and the Liu-Zhou-Ibrahim (LZI) algorithm for weakly positive tensors. Meanwhile, we perform numerical experiments to compare the computational efficiency of our proposed algorithm with that of the NQZ and LZI algorithms.