ALGORITHM 1.

Continuation of gradient extremal curves on manifolds defined by point clouds.

Input: initial point $p=p_0$ near minimum, samples per iteration $N$, continuation parameters, threshold $\rho >0$ used in the convergence

criterion.

Output: equilibrium $p_{\star }$ of $X$.

for $n=1,2,3,\dots$ do

Sample $N$ points from a neighborhood $U\subset M$ of $p_{n\text{-}1}$.

Use manifold learning to obtain local coordinates $\phi :U\to {\mathbb{R}}^d$

Obtain a parameterization $\psi :V\to U$, where $V=\phi (U)\subseteq {\mathbb{R}}^d$.

Approximate $Z={\psi }^{\star }E$ via Gaussian process regression.

$q\leftarrow \phi (p_{n-1})$

if $n=1$ then

$v\leftarrow$ Random direction in ${\mathbb{R}}^d$

else

$v\leftarrow {\phi }_{\star }(q)w_{n-1}$

end if

$q_n,v_n\leftarrow \mathrm{resolve}\mathrm{\_}\mathrm{extremal}\mathrm{\_}\mathrm{curve}(q,v)$ $▹$ See Algorithm 2.

$p_n,w_n\leftarrow \psi (q_n),{\psi }_{\star }(q_n)v_n$

if $\Vert \mathrm{grad}Z(q_n)\Vert <\rho$ and $n>1$ then

return $p_{\star }\leftarrow p_n$

end if

end for