Continuation methods for nonnegative rank-1 approximation of nonnegative tensors

In this article, the rank-1 approximation of a nonnegative tensor.. Rn1x. xnm.0 is considered. Mathematically, the approximation problem can be formulated as an optimization problem. The Karush-Kuhn-Tucker (KKT) point of the optimization problem can be obtained by computing the nonnegative Z-eigenvector y of enlarged tensor.. Therefore, we propose an iterative method with prediction and correction steps for computing nonnegative Z-eigenvector y of enlarged tensor., called the continuation method. In the theoretical part, we show that the computation requires only O(Pi(m)(i=1) n(i)) flops for each iteration and the computed Z-eigenvector y has nonzero component block, and hence, the KKT point can be obtained. In addition, we show that the KKT point is a local optimizer of the optimization problem. Numerical experiments are provided to support the theoretical results.