Nonnegative Tensor Factorization, Completely Positive Tensors, and a Hierarchical Elimination Algorithm

Nonnegative tensor factorization has applications in statistics, computer vision, exploratory multiway data analysis and blind source separation. A symmetric nonnegative tensor, which has an exact symmetric nonnegative factorization, is called a completely positive tensor. This concept extends the concept of completely positive matrices. A classical result in the theory of completely positive matrices is that a symmetric, diagonally dominated nonnegative matrix is a completely positive matrix. In this paper, we introduce strongly symmetric tensors and show that a symmetric tensor has a symmetric binary decomposition if and only if it is strongly symmetric. Then we show that a strongly symmetric, hierarchically dominated nonnegative tensor is a completely positive tensor, and present a hierarchical elimination algorithm for checking this. Numerical examples are given to illustrate this. Some other properties of completely positive tensors are discussed. In particular, we show that the completely positive tensor cone and the copositive tensor cone of the same order are dual to each other.