Coupled tensor models for probability mass function estimation: Part II, uniqueness of the model

In this paper, uniqueness properties of a coupled factorization of 3D marginal tensors (or PCTF3D) are studied. The PCTF3D method (detailed in the Part I article) performs estimation of probability mass functions (PMFs) by coupling 3D marginals, seen as order-3 tensors. The core novelty of PCTF3D’s approach relies on the partial coupling which consists in choosing a limited set of 3D marginals to be coupled. PCTF3D uniqueness is examined through the prism of polynomial mappings and their recoverability. A numerical algorithm is proposed for finding the maximal rank for which recoverability is guaranteed. This approach properly accounts for the coupling strategy and simplex constraints. Using the proposed algorithm, the different coupling strategies from Part I are examined with respect to their uniqueness properties. Finally, a new identifiability bound is given for a so-called Cartesian coupling which improves existing sufficient bounds available in the literature.