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 $KL(\mathcal{N}_1, \mathcal{N}_2)\leq ε_1$ and $KL(\mathcal{N}_2, \mathcal{N}_3)\leq ε_2$, then $KL(\mathcal{N}_1, \mathcal{N}_3)< 3ε_1+3ε_2+2\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. 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 $ε_1$ and $ε_2$ are small, the supremum is $ε_1+ε_2+\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.