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Learning functions on unknown manifolds.

The University of Chicago.

Learning functions on unknown manifolds.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Learning functions on unknown manifolds./
Author:	Zhou, Xueyuan.
Description:	126 p.
Notes:	Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: .
Contained By:	Dissertation Abstracts International73-04B.
Subject:	Statistics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3487652
ISBN:	9781267071637

Learning functions on unknown manifolds.
Zhou, Xueyuan.

Learning functions on unknown manifolds. - 126 p.

Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: .

Thesis (Ph.D.)--The University of Chicago, 2011.

As more and more complex data sources become available, the analysis of graph and manifold data has become an essential part of various sciences. In this thesis, learning functions through samples on a manifold is investigated. Toward this goal, several problems are studied. First, regularization in Sobolev spaces on manifolds are studied, which can be used to find a smooth function on the data manifold. The regularizer is implemented by the iterated Laplacian, which is a natural tool to study Sobolev spaces on manifolds. This way of regularization generalizes thin plate splines from regular grid on a known domain to random samples on an unknown manifold. Second, we study the asymptotic behavior of graph Laplacian eigenmaps method, which is a finite-dimensional approximation of the infinite-dimensional problem. The analysis shows that the method, as a nonparametric regression method, achieves the optimal integrated mean squares error rate in terms of the intrinsic dimensionality of the manifold. Third, the limit behavior of the graph Laplacian on a manifold boundary and its implications are discussed. Finally, we studied a ranking on data manifold algorithm in information retrieval from a functional analysis point of view, which goes beyond the physics-based intuition.

ISBN: 9781267071637Subjects--Topical Terms:

556824
Statistics.

Learning functions on unknown manifolds.
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020 $a 9781267071637
035 $a (UMI)AAI3487652
035 $a AAI3487652
040 $a UMI $c UMI
100 1 $a Zhou, Xueyuan. $3 845503
245 1 0 $a Learning functions on unknown manifolds.
300 $a 126 p.
500 $a Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: .
500 $a Advisers: Mikhail Belkin; Pedro Felzenszwalb.
502 $a Thesis (Ph.D.)--The University of Chicago, 2011.
520 $a As more and more complex data sources become available, the analysis of graph and manifold data has become an essential part of various sciences. In this thesis, learning functions through samples on a manifold is investigated. Toward this goal, several problems are studied. First, regularization in Sobolev spaces on manifolds are studied, which can be used to find a smooth function on the data manifold. The regularizer is implemented by the iterated Laplacian, which is a natural tool to study Sobolev spaces on manifolds. This way of regularization generalizes thin plate splines from regular grid on a known domain to random samples on an unknown manifold. Second, we study the asymptotic behavior of graph Laplacian eigenmaps method, which is a finite-dimensional approximation of the infinite-dimensional problem. The analysis shows that the method, as a nonparametric regression method, achieves the optimal integrated mean squares error rate in terms of the intrinsic dimensionality of the manifold. Third, the limit behavior of the graph Laplacian on a manifold boundary and its implications are discussed. Finally, we studied a ranking on data manifold algorithm in information retrieval from a functional analysis point of view, which goes beyond the physics-based intuition.
590 $a School code: 0330.
650 4 $a Statistics. $3 556824
650 4 $a Artificial Intelligence. $3 646849
690 $a 0463
690 $a 0800
710 2 $a The University of Chicago. $b Computer Science. $3 845504
773 0 $t Dissertation Abstracts International $g 73-04B.
790 1 0 $a Belkin, Mikhail, $e advisor
790 1 0 $a Felzenszwalb, Pedro, $e advisor
790 1 0 $a Belkin, Mikhail $e committee member
790 1 0 $a Felzenszwalb, Pedro $e committee member
790 1 0 $a Scott, L. Ridgway $e committee member
790 $a 0330
791 $a Ph.D.
792 $a 2011
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3487652

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