# The Jacobian and Hessian of the Kullback-Leibler Divergence between Multivariate Gaussian Distributions (Technical Report)

arXiv: https://arxiv.org/abs/2506.23996 · HTML: https://mlsanity.com/paper/2506.23996/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 3 references that are in the map.
- First posted on arXiv 30 Jun 2025.

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

- [An Analysis of Posterior Collapse, Parameterization and Initialization in Variational Deep Gaussian Processes](https://mlsanity.com/paper/2606.25882/) (2026)

## Abstract (arXiv)

This document shows how to obtain the Jacobian and Hessian matrices of the Kullback-Leibler divergence between two multivariate Gaussian distributions, using the first and second-order differentials. The presented derivations are based on the theory presented by citemagnus99. I've also got great inspiration from some of the derivations in citeminka. Since I pretend to be at most didactic, the document is split into a summary of results and detailed derivations on each of the elements involved, with specific references to the tricks used in the derivations, and to many of the underlying concepts.