34 research outputs found
Thresholding in Learning Theory
In this paper we investigate the problem of learning an unknown bounded
function. We be emphasize special cases where it is possible to provide very
simple (in terms of computation) estimates enjoying in addition the property of
being universal : their construction does not depend on a priori knowledge on
regularity conditions on the unknown object and still they have almost optimal
properties for a whole bunch of functions spaces. These estimates are
constructed using a thresholding schema, which has proven in the last decade in
statistics to have very good properties for recovering signals with
inhomogeneous smoothness but has not been extensively developed in Learning
Theory. We will basically consider two particular situations. In the first
case, we consider the RKHS situation. In this case, we produce a new algorithm
and investigate its performances in . The exponential rates
of convergences are proved to be almost optimal, and the regularity assumptions
are expressed in simple terms. The second case considers a more specified
situation where the 's are one dimensional and the estimator is a wavelet
thresholding estimate. The results are comparable in this setting to those
obtained in the RKHS situation as concern the critical value and the
exponential rates. The advantage here is that we are able to state the results
in the norm and the regularity conditions are expressed in
terms of standard H\"older spaces
Heat kernel generated frames in the setting of Dirichlet spaces
Wavelet bases and frames consisting of band limited functions of nearly
exponential localization on Rd are a powerful tool in harmonic analysis by
making various spaces of functions and distributions more accessible for study
and utilization, and providing sparse representation of natural function spaces
(e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in
more general homogeneous spaces, on the interval and ball. The purpose of this
article is to develop band limited well-localized frames in the general setting
of Dirichlet spaces with doubling measure and a local scale-invariant
Poincar\'e inequality which lead to heat kernels with small time Gaussian
bounds and H\"older continuity. As an application of this construction, band
limited frames are developed in the context of Lie groups or homogeneous spaces
with polynomial volume growth, complete Riemannian manifolds with Ricci
curvature bounded from below and satisfying the volume doubling property, and
other settings. The new frames are used for decomposition of Besov spaces in
this general setting
Radon needlet thresholding
We provide a new algorithm for the treatment of the noisy inversion of the
Radon transform using an appropriate thresholding technique adapted to a
well-chosen new localized basis. We establish minimax results and prove their
optimality. In particular, we prove that the procedures provided here are able
to attain minimax bounds for any loss. It s important to notice
that most of the minimax bounds obtained here are new to our knowledge. It is
also important to emphasize the adaptation properties of our procedures with
respect to the regularity (sparsity) of the object to recover and to
inhomogeneous smoothness. We perform a numerical study that is of importance
since we especially have to discuss the cubature problems and propose an
averaging procedure that is mostly in the spirit of the cycle spinning
performed for periodic signals
Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators
We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of
a doubling metric measure space in the presence of a non-negative self-adjoint
operator whose heat kernel has Gaussian localization and the Markov property.
The class of almost diagonal operators on the associated sequence spaces is
developed and it is shown that this class is an algebra. The boundedness of
almost diagonal operators is utilized for establishing smooth molecular and
atomic decompositions for the above homogeneous Besov and Triebel-Lizorkin
spaces. Spectral multipliers for these spaces are established as well
Spin Needlets for Cosmic Microwave Background Polarization Data Analysis
Scalar wavelets have been used extensively in the analysis of Cosmic
Microwave Background (CMB) temperature maps. Spin needlets are a new form of
(spin) wavelets which were introduced in the mathematical literature by Geller
and Marinucci (2008) as a tool for the analysis of spin random fields. Here we
adopt the spin needlet approach for the analysis of CMB polarization
measurements. The outcome of experiments measuring the polarization of the CMB
are maps of the Stokes Q and U parameters which are spin 2 quantities. Here we
discuss how to transform these spin 2 maps into spin 2 needlet coefficients and
outline briefly how these coefficients can be used in the analysis of CMB
polarization data. We review the most important properties of spin needlets,
such as localization in pixel and harmonic space and asymptotic uncorrelation.
We discuss several statistical applications, including the relation of angular
power spectra to the needlet coefficients, testing for non-Gaussianity on
polarization data, and reconstruction of the E and B scalar maps.Comment: Accepted for publication in Phys. Rev.
Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds
Let be a random sample from some unknown probability density
defined on a compact homogeneous manifold of dimension . Consider a 'needlet frame' describing a localised
projection onto the space of eigenfunctions of the Laplace operator on with corresponding eigenvalues less than , as constructed in
\cite{GP10}. We prove non-asymptotic concentration inequalities for the uniform
deviations of the linear needlet density estimator obtained from an
empirical estimate of the needlet projection of . We apply these results to construct risk-adaptive
estimators and nonasymptotic confidence bands for the unknown density . The
confidence bands are adaptive over classes of differentiable and
H\"{older}-continuous functions on that attain their H\"{o}lder
exponents.Comment: Probability Theory and Related Fields, to appea
Gaussian bounds for the heat kernelson the ball and the simplex: classical approach
International audienceTwo-sided Gaussian bounds are established for the weighted heat kernels on the unit ball and the simplex in Rd generated by classical differential operators whose eigenfunctions are algebraic polynomials