Structure and Exact Certification of Nonnegative Tensor Decompositions

Abstract

We study the structure and exact certification of nonnegative tensor decompositions supplied through rational factor matrices. Every failure of minimality or essential uniqueness contains an irreducible positive exchange between $p$ supplied terms and $q$ competing terms. A dimension count based on the Lovitz--Petrov splitting theorem bounds the effective order of such an exchange, the sum over the modes of the joint factor-span dimension minus one, by $p+q-2$, and this bound is attained at every supplied length $R\ge2$. After eliminating ambient-row variables by the Farkas lemma, the resulting semialgebraic systems have fewer than $2R^2$ real variables. For input bit length $L$, this gives a deterministic $G(R)L^{O(R^2)}$-time algorithm that computes the nonnegative rank and an exact minimum-length decomposition, decides minimality and essential uniqueness, and returns exact witnesses when either property fails. The exponent is independent of the tensor order and the mode dimensions. For Lebesgue moment tensors, positive quadrature determines the nonnegative rank and the uniqueness behavior, and some rational length-$R$ inputs force a common field of degree at least $\lfloor R/2\rfloor!$ for any exact minimum-length output. As an illustration, an implementation resolves all 114 instances left inconclusive by earlier positive-scattering criteria, of which 95 are minimal and essentially unique and 19 are minimal but nonunique.