On the rank of quaternion Hankel matrices

This paper discusses the left and right ranks of quaternion matrices with Hankel structure. While they are in general different for arbitrary quaternion matrices, we show that the left and right ranks of quaternion Hankel matrices are equal. Moreover, we establish the relation between Hankel matrices and the existence of linear recurrence relations with quaternion coefficients.

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

In this paper, uniqueness properties of a coupled tensor model are studied. This new coupled tensor model is used in a new method called Partial Coupled Tensor Factorization of 3D marginals or PCTF3D. This method performs estimation of probability mass functions by coupling 3D marginals, seen as order-3 tensors. The core novelty of PCTF3D's approach (detailed in the part I article) relies on the partial coupling which consists on the choice of 3D marginals to be coupled. Tensor methods are ubiquitous in many applications of statistical learning, with their biggest advantage of having strong uniqueness properties. In this paper, the uniqueness properties of PCTF3D's constrained coupled low-rank model is assessed. While probabilistic constraints of the coupled model are handled properly, it is shown that uniqueness highly depends on the coupling used in PCTF3D. After proposing a Jacobian algorithm providing maximum recoverable rank, different coupling strategies presented in the Part I article are examined with respect to their uniqueness properties. Finally, an identifiability bound is given for a so-called Cartesian coupling which permits enhancing sufficient bounds of the literature.