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

TL;DR

This paper examines the conditions under which a specific moment map related to spectral densities is injective, providing a counterexample and analyzing bifurcation points to challenge previous assumptions of uniqueness.

Contribution

It offers a counterexample demonstrating the non-injectivity of the moment map and applies bifurcation theory to analyze the critical points affecting uniqueness.

Findings

Counterexample shows the moment map has a critical point

Critical point identified as a bifurcation point

Theorem 6's uniqueness claim is challenged

Abstract

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 \cite{Georgiou-06} 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.