Log-Determinant Divergences Between Positive Definite Hilbert-Schmidt Operators

Abstract

The current work generalizes the author’s previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert-Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive definite unitized Hilbert-Schmidt operators as special cases, including the infinite-dimensional generalizations of the affine-invariant Riemannian distance and symmetric Stein divergence.