推导多元高斯分布间KL散度的雅可比与海森矩阵，便于优化计算
The Jacobian and Hessian of the Kullback-Leibler Divergence between Multivariate Gaussian Distributions (Technical Report)
- 基于微分理论推导KL散度对均值和协方差的梯度与二阶导
- 给出解析表达式，适用于变分推断等需要二阶信息的场景
- 适合研究变分贝叶斯、生成模型优化的读者参考
本文推导了两个多元高斯分布之间KL散度的雅可比矩阵和海森矩阵，基于一阶和二阶微分理论。推导过程参考了Magnus & Neudecker（1999）的矩阵微分理论，并借鉴了Minka的部分推导技巧。文档分为结果总结与详细推导两部分，逐项说明每个元素的推导方法，标注所用技巧及基础概念，旨在清晰展示推导逻辑，便于理解与应用。
原文摘要 · Abstract (English)
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 \cite{magnus99}. I've also got great inspiration from some of the derivations in \cite{minka}. 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.
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