We consider the reconstruction of a manifold (or, invariant manifold learning), where a smooth Riemannian manifold M is determined from the 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 \{ X1, X2, . . . , XN \} be a set of N sample points sampled randomly from an unknown Riemannian M manifold. We assume that we are given the numbers D jk = dM (Xj, X k ) + \eta jk , where j, k \in \{ 1, 2, . . . , N \} . Here, dM (Xj, X k ) are geodesic distances, and \eta jk are independent, identically distributed random variables such that the exponential moment \BbbE e | \eta jk | is finite. We show that when N \sim C0\delta - 3n (log(1/\delta )) 5 log(1/\theta ), with the probability 1 -\theta , it is possible to construct a manifold that approximates the Riemannian manifold (M, g) with the error \delta . Here, C0 depends on the intrinsic dimension n of M and the bounds for the diameter, sectional curvature, the injectivity radius of (M, g), and the the exponential moment of the noise. This problem is a generalization of the geometric Whitney problem with random measurement errors. We also consider the case when the information on the noisy distance D jk of points Xj and X k is missing with a certain probability. In particular, we consider the case when we have no information on points that are far away.