In the following blog post: https://cswhjiang.github.io/2015/10/13/Roperator/
The author arrives at a Hessian-vector product $Hv = (\nabla f(w + rv) - \nabla f(w))/r + O(r)$
I am not sure why the Big O term is positive because when I try to replicate the result I get:
$\nabla f(w + \Delta w) = \nabla f(w) + H\Delta w + O(||\Delta w||^2)$
$\nabla f(w + rv) = \nabla f(w) + rHv + O(||rv||^2)$
$\nabla f(w + rv) = \nabla f(w) + rHv + O(r^2)$
$rHv = \nabla f(w + rv) - \nabla f(w) - O(r^2)$
$Hv = (\nabla f(w + rv) - \nabla f(w))/r - O(r)$
I feel like I might be misunderstanding something about the Big O term of the Taylor Expansion.
Start with the mean-value for a function of one variable: $$g(1)-g(0)=g^/(r), 0<r<1$$$$=g^/(0)+g^/(r)-g^/(0).$$ Now apply the mean-value theorem twice more $$g^/(r)-g^/(0)=rg^{//}(\rho),0<\rho<r$$ $$=r(g^{//}(0)+\rho g^{///}(\zeta)),0<\zeta<\rho$$ Now take two points in $\mathbb R^3$, $P_0=(a,b,c)$ and $P_1=(\alpha,\beta,\gamma)$ and let $$\mathbf v=[\alpha-a,\beta-b,\gamma-c]=[v_1,v_2,v_3]$$ Let $f:\mathbb R^3\rightarrow \mathbb R,g(r)=f(P_0+r \mathbf v)$. Then $$g^/(r)=\mathbf v \bullet \nabla f(P_0+r \mathbf v)$$ $$g^{//}(r)=\mathbf v \text H(P_0+r \mathbf v)\mathbf v^{\mathbf T}$$ $$g^{///}(r)=\sum_{i.j,k=1}^3v_1v_2v_3f_{ijk}(P_0+r \mathbf v)$$ With these substitutions you should be able to get started.