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 ${X_1,X_2,\dots,X_N}$ be a set of $N$ sample points sampled randomly from an unknown Riemannian manifold $M$. 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.

This is joint work with Charles Fefferman, Sergei Ivanov and Matti Lassas.

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Last updated: 16 Oct 2019