Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions

The Kullback-Leibler (KL) divergence is not a proper distance metric and does not satisfy the triangle inequality, posing theoretical challenges in certain practical applications. Existing work has demonstrated that KL divergence between multivariate Gaussian distributions follows a relaxed triangle inequality. Given any three multivariate Gaussian distributions $\mathcal{N}_1, \mathcal{N}_2$, and $\mathcal{N}_3$, if $KL(\mathcal{N}_1, \mathcal{N}_2)\leq \epsilon_1$ and $KL(\mathcal{N}_2, \mathcal{N}_3)\leq \epsilon_2$, then $KL(\mathcal{N}_1, \mathcal{N}_3)< 3\epsilon_1+3\epsilon_2+2\sqrt{\epsilon_1\epsilon_2}+o(\epsilon_1)+o(\epsilon_2)$. However, the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ is still unknown. In this paper, we investigate the relaxed triangle inequality for the KL divergence between multivariate Gaussian distributions and give the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ as well as the conditions when the supremum can be attained. When $\epsilon_1$ and $\epsilon_2$ are small, the supremum is $\epsilon_1+\epsilon_2+2\sqrt{\epsilon_1\epsilon_2}+o(\epsilon_1)+o(\epsilon_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.