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Statistical inverse problems and geometric "wavelet" construction

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Post-edited
Authors : Kerkyacharian, Gérard (Author of the conference)
CIRM (Publisher )

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inverse problem : toy model wavelet frame on R construction of needlet regularity of gaussian process in geometrical framework
Abstract : In the fist part of the talk, we will look to some statistical inverse problems for which the natural framework is no more an Euclidian one.
In the second part we will try to give the initial construction of (not orthogonal) wavelets -of the 80 - by Frazier, Jawerth,Weiss, before the Yves Meyer ORTHOGONAL wavelets theory.
In the third part we will propose a construction of a geometric wavelet theory. In the Euclidian case, Fourier transform plays a fundamental role. In the geometric situation this role is given to some "Laplacian operator" with some properties.
In the last part we will show that the previous theory could help to revisit the topic of regularity of Gaussian processes, and to give a criterium only based on the regularity of the covariance operator.

MSC Codes :
42C15 - General harmonic expansions, frames
 43A80 - Analysis on other specific Lie groups, See also {22Exx}
 43A85 - Analysis on homogeneous spaces
 46E35 - Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
 58J35 - Heat and other parabolic equation methods
 62G05 - Nonparametric estimation
 62G10 - Nonparametric hypothesis testing
 62G20 - Nonparametric asymptotic efficiency
Information on the Event

Event Title : Thematic month on statistics - Week 2 : Mathematical statistics and inverse problems / Mois thématique sur les statistiques - Semaine 2 : statistiques mathématiques et problèmes inverses
Event Organizers : Autin, Florent ; Golubev, Yuri ; Pouet, Christophe
Dates : 08/02/2016 - 12/02/2016
Event Year : 2016
Event URL : http://conferences.cirm-math.fr/1616.html

Citation Data

DOI : 10.24350/CIRM.V.18926703
Cite this video as: Kerkyacharian, Gérard (2016). Statistical inverse problems and geometric "wavelet" construction. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18926703
URI : http://dx.doi.org/10.24350/CIRM.V.18926703

See Also

Bibliography

  • P. Baldi, G.Kerkyacharian , D. Marinucci, D. Picard, Adaptive density estimation for directional data using needlets. Ann. Statist. 37 (2009), no. 6A. 3362-3395. - dx.doi.org/10.1214/09-AOS682

  • T. Coulhon, G. Kerkyacharian, P. Petrushev, Heat kernel generated frames in the setting of Dirichlet spaces, J. Fourier Anal. Appl. 18 (2012), 995-1066. - dx.doi.org/10.1007/s00041-012-9232-7

  • M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS No 79 (1991), AMS. - http://bookstore.ams.org/cbms-79

  • D. Geller, I. Z. Pesenson, Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds, J. Geom. Anal. 21 (2011), 334-371. - dx.doi.org/10.1007/s12220-010-9150-3

  • G.Kerkyacharian, T.M. Pham Ngoc, D. Picard, Localized spherical deconvolution. Ann. Statist. 39 (2011), no. 2, 1042-1068. - dx.doi.org/10.1214/10-AOS858

  • G.Kerkyacharian, R.Nickl, D. Picard, Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probab. Theory Related Fields 153 (2012), no. 1-2, 363{404. - dx.doi.org/10.1007/s00440-011-0348-5

  • G.Kerkyacharian, G.Kyriazys, E. Le Pennec, P. Petrushev, D. Picard, Inversion of noisy Radon transform by SVD based needlets. Appl. Comput. Harmon. Anal. 28 (2010), no. 1, 24-45. - dx.doi.org/10.1016/j.acha.2009.06.001

  • G.Kerkyacharian,E. Le Pennec, D. Picard, Radon needlet thresholding, Bernoulli 18 (2012), no. 2, 391- 433. - dx.doi.org/10.3150/10-BEJ340

  • G. Kerkyacharian, P. Petrushev, Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Trans. Amer. Math. Soc. 367 (2015), no. 1, 121-189. - dx.doi.org/10.1090/S0002-9947-2014-05993-X

  • G.Kerkyacharian, S.Ogawa, P. Petrushev, D. Picard, Regularity of Gaussian Processes on Dirichlet spaces, arXiv:1508.00822v1 [math.PR] 4 Aug 2015 - http://arxiv.org/abs/1508.00822

  • G. Kyriazis, P. Petrushev, and Y. Xu, Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces, Studia Math. 186 (2008), 161- 202. - dx.doi.org/10.4064/sm186-2-3

  • G. Kyriazis, P. Petrushev, and Y. Xu, Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball, Proc. London Math. Soc. 97 (2008), 477-513. - dx.doi.org/10.1112/plms/pdn010

  • F. Narcowich, P. Petrushev, J. Ward, Localized tight frames on spheres, SIAM J. Math. Anal. 38 (2006), 574 - 594. - dx.doi.org/10.1137/040614359

  • F. Narcowich, P. Petrushev, and J. Ward, Decomposition of Besov and Triebel-Lizorkin spaces on the sphere, J. Funct. Anal. 238 (2006), 530-564. - dx.doi.org/10.1016/j.jfa.2006.02.011

  • P. Petrushev, Y. Xu, Localized polynomial frames on the interval with Jacobi weights, J. Four. Anal. Appl. 11 (2005), 557-575. - dx.doi.org/10.1007/s00041-005-4072-3

  • P. Petrushev, Y. Xu, Localized polynomial frames on the ball, Constr. Approx. 27 (2008), 121-148. - dx.doi.org/10.1007/s00365-007-0678-9


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