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Smooth function extension based on high dimensional unstructured data

Authors: Charles K. Chui and H. N. Mhaskar
Journal: Math. Comp. 83 (2014), 2865-2891
MSC (2010): Primary 41A25, 42C15, 68Q32
Published electronically: June 18, 2014
MathSciNet review: 3246813
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Abstract: 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-set as a point-cloud $ {\mathcal P}$ in some high dimensional Euclidean space $ {\mathbb{R}}^{s}$. Then the problem is to extend a desirable objective function $ f$ from a certain relatively smaller training subset $ \mathcal {C}\subset {\mathcal P}$ to some continuous manifold $ {\mathbb{X}}\subset {\mathbb{R}}^{s}$ that contains $ {\mathcal P}$, at least approximately. More precisely, when the point cloud $ {\mathcal P}$ of the given data-set is modeled in the abstract by some unknown compact manifold embedded in the ambient Euclidean space $ {\mathbb{R}}^{s}$, the extension problem can be considered as the interpolation problem of seeking the objective function on the manifold $ {\mathbb{X}}$ that agrees with $ f$ on $ \mathcal {C}$ under certain desirable specifications. For instance, by considering groups of cardinality $ s$ of data values as points in a point-cloud in $ {\mathbb{R}}^{s}$, such groups that are far apart in the original spatial data domain in $ {\mathbb{R}}^{1}$ or $ {\mathbb{R}}^{2}$, 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.

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Additional Information

Charles K. Chui
Affiliation: Department of Statistics, Stanford University, Stanford, California 94305

H. N. Mhaskar
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125 — and — Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California 91711

Received by editor(s): February 17, 2012
Received by editor(s) in revised form: February 23, 2013
Published electronically: June 18, 2014
Additional Notes: The first author’s research was supported by ARO Grants # W911NF-07-1-0525 and # W911NF-11-1-0426.
The second author’s research was supported, in part, by grant DMS-0908037 from the National Science Foundation and grant W911NF-09-1-0465 from the U.S. Army Research Office.
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