# Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions

arXiv: https://arxiv.org/abs/2602.02577 · HTML: https://mlsanity.com/paper/2602.02577/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 40 references that are in the map.
- First posted on arXiv 31 Jan 2026.

## Builds on (most cited first, 15 of 15)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Constrained Variational Policy Optimization for Safe Reinforcement Learning](https://mlsanity.com/paper/2201.11927/) (2022)
- [Detecting Rotated Objects as Gaussian Distributions and Its 3-D Generalization](https://mlsanity.com/paper/2209.10839/) (2022)
- [On the Properties of Kullback-Leibler Divergence Between Multivariate Gaussian Distributions](https://mlsanity.com/paper/2102.05485/) (2021)
- [A Less Biased Evaluation of Out-of-distribution Sample Detectors](https://mlsanity.com/paper/1809.04729/) (2018)
- [Are generative deep models for novelty detection truly better?](https://mlsanity.com/paper/1807.05027/) (2018)
- [Kullback-Leibler Divergence-Based Out-of-Distribution Detection with Flow-Based Generative Models](https://mlsanity.com/paper/2002.03328/) (2020)

## Built on by (most cited first, 1 of 1)

- [Optimal Stability of KL Divergence under Gaussian Perturbations](https://mlsanity.com/paper/2604.11026/) (2026)

## Abstract (arXiv)

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 N1, N2, and N3, if KL(N1, N2)≤ ε1 and KL(N2, N3)≤ ε2, then KL(N1, N3)< 3ε1+3ε2+2√ε1ε2+o(ε1)+o(ε2). However, the supremum of KL(N1, N3) 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(N1, N3) as well as the conditions when the supremum can be attained. When ε1 and ε2 are small, the supremum is ε1+ε2+2√ε1ε2+o(ε1)+o(ε2). Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.