  We recover the Riemannian gradient of a given function defined on interior
points of a Riemannian submanifold in the Euclidean space based on a sample of
function evaluations at points in the submanifold. This approach is based on
the estimates of the Laplace-Beltrami operator proposed in the diffusion-maps
theory. The Riemannian gradient estimates do not involve differential terms.
Analytical convergence results of the Riemannian gradient expansion are proved.
We apply the Riemannian gradient estimate in a gradient-based algorithm
providing a derivative-free optimization method. We test and validate several
applications, including tomographic reconstruction from an unknown random angle
distribution, and the sphere packing problem in dimensions 2 and 3.
