| | |
| | |
Stat |
Members: 3643 Articles: 2'487'895 Articles rated: 2609
28 March 2024 |
|
| | | |
|
Article overview
| |
|
Reconstruction of a Riemannian manifold from noisy intrinsic distances | Charles Fefferman
; Sergei Ivanov
; Matti Lassas
; Hariharan Narayanan
; | Date: |
17 May 2019 | Abstract: | 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 $j,kin
{1,2,dots,N}$. 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 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
| News, job offers and information for researchers and scientists:
| |