On the Properties of Kullback-Leibler Divergence Between Gaussians

Kullback-Leibler (KL) divergence is one of the most important divergence measures between probability distributions. In this paper, we investigate the properties of KL divergence between Gaussians. Firstly, for any two $n$-dimensional Gaussians $mathcal{N}_1$ and $mathcal{N}_2$, we find the supremum of $KL(mathcal{N}_1||mathcal{N}_2)$ when $KL(mathcal{N}_2||mathcal{N}_1)leq epsilon$ for $epsilon>0$. This reveals the approximate symmetry of small KL divergence between Gaussians. We also find the infimum of $KL(mathcal{N}_1||mathcal{N}_2)$ when $KL(mathcal{N}_2||mathcal{N}_1)geq M$ for $M>0$. Secondly, for any three $n$-dimensional Gaussians $mathcal{N}_1, mathcal{N}_2$ and $mathcal{N}_3$, we find a bound of $KL(mathcal{N}_1||mathcal{N}_3)$ if $KL(mathcal{N}_1||mathcal{N}_2)$ and $KL(mathcal{N}_2||mathcal{N}_3)$ are bounded. This reveals that the KL divergence between Gaussians follows a relaxed triangle inequality. Importantly, all the bounds in the theorems presented in this paper are independent of the dimension $n$.