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 tight 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 $n$.