Capturing Ridge Functions in High Dimensions from Point Queries

Handle URI:
http://hdl.handle.net/10754/597725
Title:
Capturing Ridge Functions in High Dimensions from Point Queries
Authors:
Cohen, Albert; Daubechies, Ingrid; DeVore, Ronald; Kerkyacharian, Gerard; Picard, Dominique
Abstract:
Constructing a good approximation to a function of many variables suffers from the "curse of dimensionality". Namely, functions on ℝ N with smoothness of order s can in general be captured with accuracy at most O(n -s/N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x)=g(a · x) when both a∈ℝ N and g ∈ C[0,1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g ∈ C s[0,1]. We also study the role of sparsity or compressibility of a in such query problems. © 2011 Springer Science+Business Media, LLC.
Citation:
Cohen A, Daubechies I, DeVore R, Kerkyacharian G, Picard D (2011) Capturing Ridge Functions in High Dimensions from Point Queries. Constructive Approximation 35: 225–243. Available: http://dx.doi.org/10.1007/s00365-011-9147-6.
Publisher:
Springer Nature
Journal:
Constructive Approximation
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
21-Dec-2011
DOI:
10.1007/s00365-011-9147-6
Type:
Article
ISSN:
0176-4276; 1432-0940
Sponsors:
This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the ARO/DoD Contract W911NF-07-1-0185; the NSF Grant DMS 0915231; the French-German PROCOPE contract 11418YB; the Agence Nationale de la Recherche (ANR) project ECHANGE (ANR-08-EMER-006); the excellence chair of the Fondation "Sciences Mathematiques de Paris" held by Ronald DeVore. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
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Full metadata record

DC FieldValue Language
dc.contributor.authorCohen, Alberten
dc.contributor.authorDaubechies, Ingriden
dc.contributor.authorDeVore, Ronalden
dc.contributor.authorKerkyacharian, Gerarden
dc.contributor.authorPicard, Dominiqueen
dc.date.accessioned2016-02-25T12:55:36Zen
dc.date.available2016-02-25T12:55:36Zen
dc.date.issued2011-12-21en
dc.identifier.citationCohen A, Daubechies I, DeVore R, Kerkyacharian G, Picard D (2011) Capturing Ridge Functions in High Dimensions from Point Queries. Constructive Approximation 35: 225–243. Available: http://dx.doi.org/10.1007/s00365-011-9147-6.en
dc.identifier.issn0176-4276en
dc.identifier.issn1432-0940en
dc.identifier.doi10.1007/s00365-011-9147-6en
dc.identifier.urihttp://hdl.handle.net/10754/597725en
dc.description.abstractConstructing a good approximation to a function of many variables suffers from the "curse of dimensionality". Namely, functions on ℝ N with smoothness of order s can in general be captured with accuracy at most O(n -s/N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x)=g(a · x) when both a∈ℝ N and g ∈ C[0,1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g ∈ C s[0,1]. We also study the role of sparsity or compressibility of a in such query problems. © 2011 Springer Science+Business Media, LLC.en
dc.description.sponsorshipThis research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107; the AFOSR Contract FA95500910500; the ARO/DoD Contract W911NF-07-1-0185; the NSF Grant DMS 0915231; the French-German PROCOPE contract 11418YB; the Agence Nationale de la Recherche (ANR) project ECHANGE (ANR-08-EMER-006); the excellence chair of the Fondation "Sciences Mathematiques de Paris" held by Ronald DeVore. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSpringer Natureen
dc.subjectActive learningen
dc.subjectCompressed sensingen
dc.subjectHigh-dimensional interpolationen
dc.subjectRidge functionsen
dc.titleCapturing Ridge Functions in High Dimensions from Point Queriesen
dc.typeArticleen
dc.identifier.journalConstructive Approximationen
dc.contributor.institutionUniversite Pierre et Marie Curie, Paris, Franceen
dc.contributor.institutionPrinceton University, Princeton, United Statesen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionUniversite Paris 7- Denis Diderot, Paris, Franceen
kaust.grant.numberKUS-C1-016-04en
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