# Reverse-mode Hessian matrix of the Cholesky factor

Let $f(L)$ be a scalar function of the lower triangular Cholesky decomposition of the covariance matrix $\Sigma$ such as $\Sigma = LL'$. Let's assume that we know the first and second derivative of $f(L)$ wrt to $L{_i}{_j}$: $$dL=\frac{\text{d } f(L)}{\text{d vech}(L)}$$ $$hL=\frac{\text{d}^2 f(L)}{\text{d vech}(L)^2}$$ where $dL$ is a vector of the same size of $vech(L)$ and $hL$ is the corresponding hessian matrix.

To pass from the Cholesky factor to the (lower triangular) covariance matrix, Iain Murray has found an interesting formula for the first derivative: $$\text{tril}(d\Sigma) = \Phi(L^{-T}(P+P^T)L^{-1}) \text{ where } P=\Phi(L^TdL)$$ and $\Phi()$ takes the lower-triangular part of a matrix and halves its diagonal.

What I would like to find is the formula for $h\Sigma$ as a function of $dL$ and $hL$.

A similar question has been asked here for the first derivative, and Iain Murray has written an interesting paper on the question of first derivatives.

Thanks!