Infinite-dimensional Log-Determinant divergences between positive definite trace class operators

This work presents a novel parametrized family of Log-Determinant (Log-Det) divergences between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Log-Det divergences between symmetric, positive definite matrices to the infinite-dimensional setting....

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Bibliographic Details
Published in:Linear algebra and its applications 2017-09, Vol.528, p.331-383
Main Author: Minh, Hà Quang
Format: Article
Language:English
Subjects:
Banach spaces
Covariance
Extended Fredholm determinant
Extended trace
Gaussian measures on Hilbert spaces
Hilbert space
Infinite-dimensional Log-Determinant divergences
Linear algebra
Mathematical analysis
Matrix
Matrix methods
Operators
Positive definite operators
Reproducing kernel Hilbert spaces
Trace class operators
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Summary:This work presents a novel parametrized family of Log-Determinant (Log-Det) divergences between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Log-Det divergences between symmetric, positive definite matrices to the infinite-dimensional setting. For the Log-Det divergences between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), we obtain closed form solutions via the corresponding Gram matrices. By employing the Log-Det divergences, we then generalize the Bhattacharyya and Hellinger distances and the Kullback–Leibler and Rényi divergences between multivariate normal distributions to Gaussian measures on an infinite-dimensional Hilbert space.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2016.09.018