Reconstructing functions from random samples

Author: 

Ferry, S
Mischaikow, K
Nanda, V

Publication Date: 

1 December 2014

Journal: 

Journal of Computational Dynamics

Last Updated: 

2020-05-13T11:41:56.39+01:00

Issue: 

2

Volume: 

1

DOI: 

10.3934/jcd.2014.1.233

page: 

233-248

abstract: 

From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding result for a Lipschitz-continuous function between two such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy and homology groups with high confidence using only finite sampled data from the domain and range, as well as knowledge of the image of every point sampled from the domain. We provide explicit bounds on the size of the point samples required for such reconstruction in terms of intrinsic properties of the domain, the co-domain and the function. This reconstruction is robust to certain types of bounded sampling and evaluation noise.

Symplectic id: 

673279

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article