On the Properties of Kullback-Leibler Divergence Between Multivariate Gaussian Distributions

Kullback-Leibler (KL) divergence is one of the most important divergence measures between probability distributions. In this paper, we prove several properties of KL divergence between multivariate Gaussian distributions. First, for any two n-dimensional Gaussian distributions N1 and N2, we give the supremum of KL(N1||N2) when KL(N2||N1)≤ varepsilon (varepsilon>0). For small varepsilon, we show that the supremum is varepsilon + 2varepsilon1.5 + O(varepsilon2). This quantifies the approximate symmetry of small KL divergence between Gaussians. We also find the infimum of KL(N1||N2) when KL(N2||N1)≥ M (M>0). We give the conditions when the supremum and infimum can be attained. Second, for any three n-dimensional Gaussians N1, N2, and N3, we find an upper bound of KL(N1||N3) if KL(N1||N2)≤ varepsilon1 and KL(N2||N3)≤ varepsilon2 for varepsilon1,varepsilon2≥ 0. For small varepsilon1 and varepsilon2, we show the upper bound is 3varepsilon1+3varepsilon2+2√varepsilon1varepsilon2+o(varepsilon1)+o(varepsilon2). This reveals that 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. Finally, We discuss the applications of our theorems in explaining counterintuitive phenomenon of flow-based model, deriving deep anomaly detection algorithm, and extending one-step robustness guarantee to multiple steps in safe reinforcement learning.