Infinite-dimensional Log-Determinant divergences II: Alpha-Beta divergences
Abstract
This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infinite-dimensional setting. The family of Alpha-Beta Log-Det divergences is highly general and contains many divergences as special cases, including the recently formulated infinite dimensional affine-invariant Riemannian distance and the infinite-dimensional Alpha Log-Det divergences between positive definite unitized trace class operators. In particular, it includes a parametrized family of metrics between positive definite trace class operators, with the affine-invariant Riemannian distance and the square root of the symmetric Stein divergence being special cases. For the Alpha-Beta Log-Det divergences between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), we obtain closed form formulas via the corresponding Gram matrices.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2016
- DOI:
- 10.48550/arXiv.1610.08087
- arXiv:
- arXiv:1610.08087
- Bibcode:
- 2016arXiv161008087Q
- Keywords:
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- Mathematics - Functional Analysis;
- Computer Science - Artificial Intelligence;
- Computer Science - Information Theory;
- Statistics - Machine Learning
- E-Print:
- 71 pages