Infinite-dimensional Log-Determinant divergences II: Alpha-Beta divergences
This work addresses a theoretical problem for researchers in functional analysis and machine learning, offering a generalized framework for divergences in infinite-dimensional spaces, but it appears incremental as it builds on prior matrix-based divergences.
The authors tackled the problem of generalizing divergences between positive definite matrices to infinite-dimensional operators, resulting in a parametrized family called Alpha-Beta Log-Determinant divergences that includes existing metrics as special cases and provides closed-form formulas for covariance operators in RKHS.
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.