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Many applications, including the image search engine, image inpainting, hyperspectral image dimensionality reduction, pattern recognition, and time series prediction, can be facilitated by considering the given discrete data\u2013set as a point-cloudP<\/mml:mi><\/mml:mrow><\/mml:mrow>{\\mathcal P}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>in some high dimensional Euclidean spaceR<\/mml:mi><\/mml:mrow><\/mml:mrow>s<\/mml:mi><\/mml:mrow><\/mml:msup>{\\mathbb R}^{s}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>. Then the problem is to extend a desirable objective functionf<\/mml:mi>f<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>from a certain relatively smaller training subsetC<\/mml:mi><\/mml:mrow>\u2282<\/mml:mo>P<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:mrow>\\mathcal {C}\\subset {\\mathcal P}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>to some continuous manifoldX<\/mml:mi><\/mml:mrow><\/mml:mrow>\u2282<\/mml:mo>R<\/mml:mi><\/mml:mrow><\/mml:mrow>s<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow>{\\mathbb X}\\subset {\\mathbb R}^{s}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>that containsP<\/mml:mi><\/mml:mrow><\/mml:mrow>{\\mathcal P}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>, at least approximately. More precisely, when the point cloudP<\/mml:mi><\/mml:mrow><\/mml:mrow>{\\mathcal P}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>of the given data\u2013set is modeled in the abstract by some unknown compact manifold embedded in the ambient Euclidean spaceR<\/mml:mi><\/mml:mrow><\/mml:mrow>s<\/mml:mi><\/mml:mrow><\/mml:msup>{\\mathbb R}^{s}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>, the extension problem can be considered as the interpolation problem of seeking the objective function on the manifoldX<\/mml:mi><\/mml:mrow><\/mml:mrow>{\\mathbb X}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>that agrees withf<\/mml:mi>f<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>onC<\/mml:mi><\/mml:mrow>\\mathcal {C}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>under certain desirable specifications. For instance, by considering groups of cardinalitys<\/mml:mi>s<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>of data values as points in a point-cloud inR<\/mml:mi><\/mml:mrow><\/mml:mrow>s<\/mml:mi><\/mml:mrow><\/mml:msup>{\\mathbb R}^{s}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>, such groups that are far apart in the original spatial data domain inR<\/mml:mi><\/mml:mrow><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msup>{\\mathbb R}^{1}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>orR<\/mml:mi><\/mml:mrow><\/mml:mrow>2<\/mml:mn><\/mml:mrow><\/mml:msup>{\\mathbb R}^{2}<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>, but have similar geometric properties, can be arranged to be close neighbors on the manifold. The objective of this paper is to incorporate the consideration of data geometry and spatial approximation, with immediate implications to the various directions of application areas. Our main result is a point-cloud interpolation formula that provides a near-optimal degree of approximation to the target objective function on the unknown manifold.<\/p>","DOI":"10.1090\/s0025-5718-2014-02819-6","type":"journal-article","created":{"date-parts":[[2014,6,18]],"date-time":"2014-06-18T13:45:19Z","timestamp":1403099119000},"page":"2865-2891","source":"Crossref","is-referenced-by-count":9,"title":["Smooth function extension based on high dimensional unstructured data"],"prefix":"10.1090","volume":"83","author":[{"given":"Charles","family":"Chui","sequence":"first","affiliation":[]},{"given":"H.","family":"Mhaskar","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,6,18]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"M. Belkin and P. 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Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"1","key":"31","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.acha.2009.08.006","article-title":"Eignets for function approximation on manifolds","volume":"29","author":"Mhaskar, H. N.","year":"2010","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"key":"32","doi-asserted-by":"crossref","unstructured":"H.N. Mhaskar, A generalized diffusion frame for parsimonious representation of functions on data defined manifolds, Neural Networks 24 (2011), 345\u2013359.","DOI":"10.1016\/j.neunet.2010.12.007"},{"issue":"1","key":"33","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.acha.2007.09.005","article-title":"Data analysis and representation on a general domain using eigenfunctions of Laplacian","volume":"25","author":"Saito, Naoki","year":"2008","journal-title":"Appl. Comput. Harmon. 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