  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 $j,k\in
\{1,2,\dots,N\}$. Here, $d_M(X_j,X_k)$ are geodesic distances, $\eta_{jk}$ are
independent, identically distributed random variables such that $\mathbb E
e^{|\eta_{jk}|}$ is finite. We show that when $N$ 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.
