Find the second derivative of the loss function

Gram-matrix-parameterization.md

@@ -95,7 +95,16 @@ This matrix is stored as `neg_grad` in the Rust and Julia implementations of the
 
 #### The second derivative of the loss function
 
-*Writeup in progress. Implemented in `app-proto/src/engine.rs` and `engine-proto/gram-test/Engine.jl`.*
+Recalling that
+\[ -d\Delta = \mathcal{P}(dA^\top Q A + A^\top Q\,dA), \]
+we can express the derivative of $\operatorname{grad}(f)$ as
+\[
+\begin{align*}
+d\operatorname{grad}(f) & = -4 Q\,dA\,\mathcal{P}(\Delta) - 4 Q A\,\mathcal{P}(d\Delta) \\
+& = -4 Q\big[dA\,\mathcal{P}(\Delta) + A\,\mathcal{P}(-d\Delta)\big].
+\end{align*}
+\]
+In the Rust and Julia implementations of the realization routine, we express $d\operatorname{grad}(f)$ as a matrix in the standard basis for $\operatorname{End}(\mathbb{R}^n)$. We apply the cotangent vector $d\operatorname{grad}(f)$ to each standard basis matrix $E_{ij}$ by setting the value of the matrix-valued 1-form $dA$ to $E_{ij}$.
 
 #### Finding minima