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The current work generalizes the author's previous work on theinfinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-BetaLog-Det divergences, defined on the set of positive definite unitized traceclass operators on a Hilbert space, to the entire Hilbert manifold of positivedefinite unitized Hilbert-Schmidt operators. This generalization is carried outvia the introduction of the extended Hilbert-Carleman determinant for unitizedHilbert-Schmidt operators, in addition to the previously introduced extendedFredholm determinant for unitized trace class operators. The resultingparametrized family of Alpha-Beta Log-Det divergences is general and containsmany divergences between positive definite unitized Hilbert-Schmidt operatorsas special cases, including the infinite-dimensional affine-invariantRiemannian distance and the infinite-dimensional generalization of thesymmetric Stein divergence.

Infinite-dimensional Log-Determinant divergences between positive definite Hilbert-Schmidt operators