IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Wavelets: a link between statistics, geometry, and probability

  • April 19, 2016
  • 4:30 p.m.
  • LeConte 412

Abstract

Wavelet theory was developed by Meyer, Daubechies, Lemarie, Mallat, Cohen and scores of other mathematicians more than thirty years ago, after the work of Frazier, Jawerth, and Weiss. Lots of applications have been made thereafter. For example, the Littlewood-Paley analysis and wavelet theory have proved to be a very useful tool in nonparametric statistic analysis. This is essentially due to the fact that most of the regularity (Sobolev and Besov) spaces can be characterized by wavelet sparse coefficients. In turn, in the nineties the wavelet theory allowed to develop ([3]) an adaptive estimator of the density of a probability law with no apriori knowledge of the regularity. Then it appeared that the Euclidian analysis is not always appropriate because many statistical problems have their own geometry. For instance, this is the situation in Tomography, where one uses Harmonic analysis of the ball, and in the study of the Cosmological Microwave Background, which requires Harmonic analysis on the sphere ([2]).

At the same time the wavelet theory was extended in various geometric and nonclassical frameworks. Extensions of this kind have already been implemented in the cases of the interval ([9]), the ball ([10]), the sphere ([7, 8]), and have been extensively used in statistical applications (see for instance ([4])).

In recent years the Littlewood-Paley analysis and wavelet theory were developed in the general framework of Riemannian manifolds and furthermore in the general setting of a positive operator associated to a suitable Dirichlet space with a good behavior of the associated heat kernel ([1, 5]). Among other things this theory allows to revisit the old problem of almost everywhere regularity of Gaussian fields ([6]).

In this talk we will review the topics mentioned above and present some new results.

References:
[1] T. Coulhon, G. Kerkyacharian, P. Petrushev, Heat kernel generated frames in the setting of Dirichlet spaces, J. Fourier Anal. Appl. 18(5) (2012), 995-1066.
[2] P. Baldi, G. Kerkyacharian, D. Marinucci and D. Picard, Asymptotic for spherical needlets, Annals of Statistics, Vol 37, N3, 2009.
[3] D. Donoho, I. Johnstone, G. Kerkyacharian, D. Picard, Wavelet shrinkage : Asymptotia, Journal of the Royal Statistical Society as "Special Read Paper". 57, No. 2, (1995), 301-369.
[4] G. Kerkyacharian, G. Kyriazis, E. Le Pennec, P. Petrushev, and D. Picard, Inversion of noisy radon transform by svd based needlet, Appl. Comput. Harmon. Anal. 28 (2010), 24-45.
[5] G. Kerkyacharian, P. Petrushev, Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces, Trans. Amer. Math. Soc. 367 (2015), 121-189.
[6] G. Kerkyacharian, P. Petrushev, D. Picard, S. Ogawa, Regularity of Gaussian processes on Dirichlet spaces, arXiv:1508.00822 (4 August 2015).
[7] F. Narcowich, P. Petrushev, and J. Ward, Local tight frames on spheres, SIAM J. Math. Anal. 38 (2006), 574-594.
[8] F. J. Narcowich, P. Petrushev, and J. Ward, Decomposition of Besov and Triebel-Lizorkin spaces on the sphere, J. Funct. Anal. 238 (2006), 530-564.
[9] P. Petrushev and Yuan Xu, Localized polynomial frames on the interval with Jacobi weights, J. Fourier Anal. Appl. 11(5) (2005), 557-575.
[10] P. Petrushev and Yuan Xu, Localized polynomials frames on the ball, Constr. Approx. 27 (2008), 121-148.

Brief Bio:
G. Kerkyacharian, has been professor at the University of Amiens and Paris 10. Currently he is a member of LPMA (Laboratoire de Probabilite et Modele Aleatoire) at Paris VI and Paris VII, and CREST (Centre de recherche en Statistique et Economie) in Paris. He has done some important work in several different areas: Statistics, Wavelets, Harmonic Analysis, Probability.

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