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

## TL;DR

This paper extends the class of infinite-dimensional Log-Det divergences to positive definite Hilbert-Schmidt operators using new determinants, unifying various divergences including Riemannian and Stein types.

## Contribution

It introduces the extended Hilbert-Carleman determinant and generalizes the Alpha-Beta Log-Det divergences to the entire Hilbert manifold of positive definite Hilbert-Schmidt operators.

## Key findings

- Unified framework for infinite-dimensional divergences
- Includes Riemannian and Stein divergences as special cases
- Provides new tools for analysis on Hilbert manifolds

## Abstract

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.