Alpha-beta log-determinant divergences between positive definite trace class operators. (English) Zbl 1519.47040
Summary: 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.
MSC:
| 47B65 | Positive linear operators and order-bounded operators |
| 47L07 | Convex sets and cones of operators |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
Keywords:
positive definite operators; trace class operators; infinite-dimensional log-determinant divergences; alpha-beta divergences; affine-invariant Riemannian distance; Stein divergence; extended trace; extended Fredholm determinant; reproducing kernel Hilbert spaces; covariance operatorsSoftware:
Matrix Means ToolboxReferences:
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