This work studies the Geometric Jensen-Shannon divergence, based on the notion of geometric mean of probability measures, in the setting of Gaussian measures on an infinite-dimensional Hilbert space. On the set of all Gaussian measures equivalent to a fixed one, we present a closed form expression for this divergence that directly generalizes the finite-dimensional version. Using the notion of Log-Determinant divergences between positive definite unitized trace class operators, we then define a Regularized Geometric Jensen-Shannon divergence that is valid for any pair of Gaussian measures and that recovers the exact Geometric Jensen-Shannon divergence between two equivalent Gaussian measures when the regularization parameter tends to zero.

翻译：本研究基于概率测度的几何平均概念，探讨了无限维希尔伯特空间上高斯测度间的几何Jensen-Shannon散度。在所有等价于某固定高斯测度的集合上，我们给出了该散度的闭式表达式，该表达式直接推广了有限维情形。通过利用正定单位化迹类算子间的对数行列式散度概念，我们进而定义了适用于任意一对高斯测度的正则化几何Jensen-Shannon散度，该散度在正则化参数趋于零时可还原为两个等价高斯测度间的精确几何Jensen-Shannon散度。