Journal of Computational Dynamics (JCD)

Reconstructing functions from random samples

Pages: 233 - 248, Volume 1, Issue 2, December 2014      doi:10.3934/jcd.2014.1.233

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Steve Ferry - Department of Mathematics, Rutgers University, Piscataway, NJ 08854, United States (email)
Konstantin Mischaikow - Rutgers University, 110 Frelinghusen Road, Piscataway, NJ 08854, United States (email)
Vidit Nanda - Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, United States (email)

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.

Keywords:  Homology, homotopy, nonlinear maps, topological inference.
Mathematics Subject Classification:  Primary: 55U05, 55U10, 55U15; Secondary: 62-07.

Received: May 2012;      Revised: May 2013;      Available Online: December 2014.