Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917304839700480 |
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| author | Xiao, Shiji Zhang, Yufeng Liu, Chubo Ding, Yan Li, Keqin Li, Kenli |
| author_facet | Xiao, Shiji Zhang, Yufeng Liu, Chubo Ding, Yan Li, Keqin Li, Kenli |
| contents | 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 $ε_1+ε_2+2\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_02577 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions Xiao, Shiji Zhang, Yufeng Liu, Chubo Ding, Yan Li, Keqin Li, Kenli Machine Learning Information Theory 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 $ε_1+ε_2+2\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning. |
| title | Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions |
| topic | Machine Learning Information Theory |
| url | https://arxiv.org/abs/2602.02577 |