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

Abstract

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 N_1 and N_2, we give the supremum of KL(N_1||N_2) when KL(N_2||N_1) \ (>0). For small , we show that the supremum is + 2^1.5 + O(^2). This quantifies the approximate symmetry of small KL divergence between Gaussians. We also find the infimum of KL(N_1||N_2) when KL(N_2||N_1) M\ (M>0). We give the conditions when the supremum and infimum can be attained. Second, for any three n-dimensional Gaussians N_1, N_2, and N_3, we find an upper bound of KL(N_1||N_3) if KL(N_1||N_2) _1 and KL(N_2||N_3) _2 for _1,_2 0. For small _1 and _2, we show the upper bound is 3_1+3_2+2_1_2+o(_1)+o(_2). 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.

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