# 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 𝒩_1 and 𝒩_2, we find the supremum of KL(𝒩_1||𝒩_2) when KL(𝒩_2||𝒩_1)≤ϵ for ϵ>0. This reveals 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 for M>0. Secondly, for any three n-dimensional Gaussians 𝒩_1, 𝒩_2 and 𝒩_3, we find a bound of KL(𝒩_1||𝒩_3) if KL(𝒩_1||𝒩_2) and KL(𝒩_2||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.

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