Plane Stochastic Tensors

We study combinatorial properties of nonnegative tensors. We make the following contributions: (1) we obtain equivalent conditions for sign nonsingular tensors and relationships between the combinatorial determinant and the permanent of nonnegative tensors, in Theorems 5.2.1 and 5.2.2; (2) the sets of plane stochastic tensors and totally plane stochastic tensors are closed, bounded and convex sets, and an nonnegative tensor has a plane stochastic pattern if and only if its positive entries are contained in a positive diagonal, in Lemma 5.3.1 and Theorem 5.3.2; (3) from a nonnegative tensor, we propose a normalization algorithm which converges to a plane stochastic tensor, in Theorem 5.3.8; (4) we discuss the boundlessness of the diagonal products of any nonnegative tensor and obtain a probabilistic algorithm after Theorem 5.4.4 for locating a positive diagonal in a (0, 1)-tensor; (5) we explore the axial N-index assignment problem via the set of plane stochastic tensors in Sect. 5.5.