Reconstruction of a Riemannian manifold from noisy intrinsic distances

We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian manifold $(M,g)$ is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let $\{X_1,X_2,\dots ,X_N\}$ bea set of $N$ sample points sampled randomly from an unknown Riemannian $M$ manifold. We assume that we are given the numbers $D_{jk}=d_M(X_j,X_k)+{\eta }_{jk}$, where $j,k\in {1,2,\dots,N}$\mathrm{.}Here,$d_M(X_j,X_k)$aregeodesicdistances,$\eta_{jk}$ are independent, identically distributed random variables such that $\mathbb E e^{|\eta_{jk}|}$isfinite\mathrm{.}Weshowthatwhen$N$ is large enough, it is possible to construct an approximation of the Riemannian manifold $(M,g)$ with a large probability. This problem is a generalization of the geometric Whitney problem with random measurement errors. We consider also the case when the information on noisy distance $D_{jk}$ of points $X_j$ and $X_k$ is missing with some probability. In particular, we consider the case when we have no information on points that are far away.