On the Uniqueness Result of Theorem 6 in “Relative Entropy and the Multivariable Multidimensional Moment Problem”

Matrix-valued covariance extension and multivariate spectral estimation are formulated as generalized moment problems in the “THREE” approach and its extensions. Under this context, we discuss Theorem 6 in the title paper concerning the bijectivity of a moment map defined over a parametric family of spectral densities. In particular, we provide a counterexample in which the moment map under consideration is shown to have a critical point, namely a point at which the Jacobian loses rank. Then, with standard techniques in bifurcation theory, we conclude further that the computed critical point is a bifurcation point, which means that the moment map is not injective.