Matrix-free Hessian-vector product ($v\mapsto {\nabla }^{2}l(x)v$)

Applying a Hessian matrix to a vector without explicitly storing all its entries. For $H=A^{\mathsf{T}}\Delta A$ with diagonal $\Delta$, evaluate $Hv=A^{\mathsf{T}}(\Delta (Av))$. This uses two matrix-vector multiplications and a diagonal scaling: $O(nd)$ arithmetic for dense $A\in {\mathbb{R}}^{n\times d}$, rather than forming the $d\times d$ Hessian matrix. Iterative second-order methods can use this operation, with safeguards for indefinite curvature when the objective is nonconvex.