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28 March 2024
 
  » arxiv » 1905.7182

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Reconstruction of a Riemannian manifold from noisy intrinsic distances
Charles Fefferman ; Sergei Ivanov ; Matti Lassas ; Hariharan Narayanan ;
Date 17 May 2019
AbstractWe 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 $j,kin {1,2,dots,N}$. 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.
Source arXiv, 1905.7182
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