IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

From statistical estimation to well localized frames, via heat kernel

  • April 2, 2012
  • 2:30 p.m.
  • LeConte 312

Abstract

(Joint work with T. Coulhoun, P. Petrushev)

Since during the last twenty years, wavelet theory has proved to be a very useful tool for theoretical purposes as well as for applications. One of the main reasons is that they provide a sparse representation of signals. In this talk, we will revisit some statistical results due to sparse representations and provide an extension of this theory in a general geometric framework.

This extension has already been performed for different cases: the interval [PX], the ball [PX2], the sphere [NPW1] and [NPW2].

This has been extensively used in statistical applications and we will recall some of them. ([KKLPP], [GKP], [BKMP], [KNP], [KPPW]).

Our general framework here, will be a metric space $(M, \rho)$ equipped with a positive Radon measure, such that $(M,\rho,\mu)$ is a homogeneous space in the sense of Harmonic Analysis (there exists $d>0$, which plays the role of an "upper dimension", such that for all $x\in M$, and $r>0, \mu(B(x,2r)) \leq 2^d \mu(B(x,r))$. Moreover, the geometry of the space is related to a positive self-adjoint operator $L$ and to the associated semi-group $e^{-tL}$. We suppose in addition that $e^{-tL}$ is markovian.

Here is the main hypothesis : $e^{-tL}$ is a kernel operator, and this kernel $P _ t(x,y)$ is Lipschitz regular and has the following Gaussian estimate : for all $x,y$ in $M$, $t>0$ ,

$$P _ t(x,y) \leq \frac{ C _ 2e^{ -c \frac{\rho^2(x,y)}t }}{\sqrt{\mu(B(x, \sqrt t) ) \mu(B(y, \sqrt t) )}} .$$

It is well known that this property is verified for the Laplacian of a Riemannian manifold with non-negative Ricci curvature, for Nilpotent Lie Groups, compact Lie Groups and their homogeneous spaces (see [G], [CS], [O], [S], ...) and many other examples.

One can define spaces of "low frequencies", Besov spaces $B^s _ {p,q},$ Tribel-Lizorkin $F^s _ {p,q}$ and Sobolev spaces $H^s _ p$. These spaces have several equivalent definitions: through Littlewood-Paley theory, through interpolation, or through semi-group theory.

Our task is to provide localized frames in duality : $\psi _ {j, \xi}, \tilde{\psi} _ {j,\xi} , j\in \mathbb{N}, \xi \in A _ j$ such that all these spaces are characterized by their coefficient in the representation: $$f(x)= \displaystyle\sum\limits _ j \displaystyle\sum\limits _ {\xi \in A _ j} \langle f, \psi _ {j, \xi} \rangle \tilde{\psi} _ {j,\xi}(x)$$

We can prove that Sobolev, Besov , and Triebel-Lizorkin spaces are characterized by the coefficients in this representation.

REFERENCES

[G] A. Grigor'yan. Heat kernel and analysis on manifolds, volume 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI, 2009.
[CS] T. Coulhon and A. Sikora. Gaussian heat kernel upper bounds via the Phragmén- Lindelöf theorem. Proc. Lond. Math. Soc. (3), 96(2):507-544, 2008.
[KKLPP] G. Kerkyacharian, G. Kyriazis, E. Le Pennec, P. Petrushev, and D. Picard. Inversion of noisy radon transform by svd based needlet. 2008.
[NPW1] F. Narcowich, P. Petrushev, and J. Ward. Local tight frames on spheres. SIAM J. Math. Anal., 2006.
[NPW2] F. J. Narcowich, P. Petrushev, and J. Ward. Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal., 2006.
[O] E. M. Ouhabaz. Analysis of heat equations on domains, volume 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2005.
[PX] P. Petrushev and Y. Xu. Localized polynomial frames on the interval with Jacobi weights. J. Fourier Anal. Appl., 11(5):557-575, 2005.
[PX2] P. Petrushev and Y. Xu. Localized polynomials frames on the ball. Constr. Approx., 27:121-148, 2008.
[S] L. Saloff-Coste. Aspects of Sobolev-type inequalities, volume 289 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2002.
[KPPW] Needlets algorithms for estimation in inverse problems G. KERKYACHARIAN, P. PETRUSHEV, D. PICARD et T.WILLER. Electronic Journal of Statistics, Vol. 1 (2007) 3076.
[KNP] Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. KERKYACHARIAN, G.; NICKL, R.; PICARD, D. Probability Theory and Related Field, to appear.
[BKMP] Adaptive density estimation for directional data. BALDI, P.; KERKYACHARIAN, G.; MARINUCCI, D.; PICARD, D. The Annals of Statistics 2009, Vol. 37, No. 6A, 33623395
[GKP] "Spin Needlets for Cosmic Microwave Background Polarization Data Analysis." GELLER, D.; HANSEN, F.; MARINUCCI, D.; KERKYACHARIAN, G.; PICARD,D. APS "Physical Review D" Vol 78, issue 12.
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