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

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 de...

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Bibliographic Details
Published in:Positivity : an international journal devoted to the theory and applications of positivity in analysis 2020-07, Vol.24 (3), p.631-662
Main Author: Minh, Hà Quang
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Divergence
Econometrics
Fourier Analysis
Hilbert space
Mathematics
Mathematics and Statistics
Operator Theory
Operators
Potential Theory
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Summary: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 affine-invariant Riemannian distance and the infinite-dimensional generalization of the symmetric Stein divergence.
ISSN:1385-1292
1572-9281
DOI:10.1007/s11117-019-00701-4