A separated dimer mode gives a local inexact saddle error bound

Let $d\ge 2$, let $U\subseteq {\mathbb{R}}^d$ be open, and let $V\in C^4(U,\mathbb{R})$ using the convention of $C^k$ maps and multi-index derivative notation in Euclidean space. Fix $x\in U$. Put $H={\mathrm{\nabla }}^{2}V(x)$ and suppose a validated ordered eigenvalue enclosure certifies index one with gap $\gamma >0$ as in Validated ordered eigenvalue enclosures for an index-one Hessian, so ${\lambda }_1(H)\le -\gamma$ and ${\lambda }_j(H)\ge \gamma$ for $j\ge 2$. Let $ϵ>0$, let $v$ be a unit direction accepted by the rotation test of Finite-budget dimer rotation and saddle translation, and assume the whole segment $x+[-ϵ,ϵ]v$ lies in $U$. Let $D_{ϵ}$ be the paired-force estimate, with $\mathrm{\parallel }(I-v{v}^T)D_{ϵ}{\mathrm{\parallel }}_2\le {\tau }_{\mathrm{r}\mathrm{o}\mathrm{t}}$. Suppose each force evaluation has Euclidean error at most $\eta$ and the third derivatives of each gradient coordinate along this segment are bounded by $M_4$. Set

$\delta :=\frac{\sqrt{d}{M}_4}{6}{ϵ}^2+\frac{\eta }{ϵ}\mathrm{.}$

If $\delta \le \gamma \mathrm{/}4$ and the measured Rayleigh estimate $v^T{D}_{ϵ}\le -\gamma \mathrm{/}2$, then the sine of the angle from $v$ to the lowest-eigenvalue line of $H$ satisfies

$\sin \mathrm{\angle }(v,q_1):=\sqrt{1-\mathrm{\mid }{q}_1^{T}v{\mathrm{\mid }}^2}\le \frac{4({\tau }_{\mathrm{r}\mathrm{o}\mathrm{t}}+\delta )}{5\gamma },$

where $q_1$ is any unit eigenvector for ${\lambda }_1(H)$. This is a conditional mode bound; rotation success alone is not an index certificate.

Let ${ϵ}_g=\mathrm{\parallel }\mathrm{\nabla }V(x){\mathrm{\parallel }}_2$. If ${ϵ}_g=0$, $x$ itself is an index-one critical point. If ${ϵ}_g>0$, put $\rho =2{ϵ}_g\mathrm{/}\gamma$ and additionally assume $B_2(x,2\rho )\subset U$, $H(\cdot )={\mathrm{\nabla }}^{2}V(\cdot )$ is $L$-Lipschitz in the induced Euclidean operator norm on that ball, and $L\rho \mathrm{/}\gamma \le 1\mathrm{/}2$. Then there is a unique index-one critical point $x_{\ast }$ in ${\bar{B}}_2(x,\rho )$ and

$\mathrm{\parallel }{x}_{\ast }-x{\mathrm{\parallel }}_2\le \rho =\frac{2{ϵ}_g}{\gamma }\mathrm{.}$

The dimer direction bound does not replace the validated eigenvalue enclosure or the residual, inverse, and Hessian-Lipschitz conditions in this local saddle certificate.