Reconstructing functions from random samples
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 mani...
| Main Authors: | , , |
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| Format: | Journal article |
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American Institute of Mathematical Sciences
2014
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| _version_ | 1826292148465565696 |
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| author | Ferry, S Mischaikow, K Nanda, V |
| author_facet | Ferry, S Mischaikow, K Nanda, V |
| author_sort | Ferry, S |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T03:10:13Z |
| format | Journal article |
| id | oxford-uuid:b3f08ed6-42bf-48cb-bdc1-4453176db347 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:10:13Z |
| publishDate | 2014 |
| publisher | American Institute of Mathematical Sciences |
| record_format | dspace |
| spelling | oxford-uuid:b3f08ed6-42bf-48cb-bdc1-4453176db3472022-03-27T04:22:39ZReconstructing functions from random samplesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b3f08ed6-42bf-48cb-bdc1-4453176db347Symplectic Elements at OxfordAmerican Institute of Mathematical Sciences2014Ferry, SMischaikow, KNanda, VFrom 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. |
| spellingShingle | Ferry, S Mischaikow, K Nanda, V Reconstructing functions from random samples |
| title | Reconstructing functions from random samples |
| title_full | Reconstructing functions from random samples |
| title_fullStr | Reconstructing functions from random samples |
| title_full_unstemmed | Reconstructing functions from random samples |
| title_short | Reconstructing functions from random samples |
| title_sort | reconstructing functions from random samples |
| work_keys_str_mv | AT ferrys reconstructingfunctionsfromrandomsamples AT mischaikowk reconstructingfunctionsfromrandomsamples AT nandav reconstructingfunctionsfromrandomsamples |