Year: 2014
Type: article
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.
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Awards: G1523888516, INSPIRE: Nonlinear Data Reduction applied to Dense Granular Media, CDI-Type II: Collaborative Research: Computational Homology, Jamming, and Force Chains in Dense Granular Flows, G1988336051, G3617724979, Collaborative proposal: Computing Dynamics of Multiparameter Systems, CDI-TYPE II--COLLABORATIVE RESEARCH: Using Algebraic Topology to Connect Models with Measurements in Complex Nonequilibrium Systems, G6071252945, G7147979661, G7830320759
OpenAlex ID: W2142495072