732 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
Anisotropic Denoising in Functional Deconvolution Model with Dimension-free Convergence Rates
In the present paper we consider the problem of estimating a periodic
-dimensional function based on observations from its noisy
convolution. We construct a wavelet estimator of , derive minimax lower
bounds for the -risk when belongs to a Besov ball of mixed smoothness
and demonstrate that the wavelet estimator is adaptive and asymptotically
near-optimal within a logarithmic factor, in a wide range of Besov balls. We
prove in particular that choosing this type of mixed smoothness leads to rates
of convergence which are free of the "curse of dimensionality" and, hence, are
higher than usual convergence rates when is large. The problem studied in
the paper is motivated by seismic inversion which can be reduced to solution of
noisy two-dimensional convolution equations that allow to draw inference on
underground layer structures along the chosen profiles. The common practice in
seismology is to recover layer structures separately for each profile and then
to combine the derived estimates into a two-dimensional function. By studying
the two-dimensional version of the model, we demonstrate that this strategy
usually leads to estimators which are less accurate than the ones obtained as
two-dimensional functional deconvolutions. Indeed, we show that unless the
function is very smooth in the direction of the profiles, very spatially
inhomogeneous along the other direction and the number of profiles is very
limited, the functional deconvolution solution has a much better precision
compared to a combination of solutions of separate convolution equations. A
limited simulation study in the case of confirms theoretical claims of
the paper.Comment: 29 pages, 1 figure, 1 tabl
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
Localized spherical deconvolution
We provide a new algorithm for the treatment of the deconvolution problem on
the sphere which combines the traditional SVD inversion with an appropriate
thresholding technique in a well chosen new basis. We establish upper bounds
for the behavior of our procedure for any loss. It is important
to emphasize the adaptation properties of our procedures with respect to the
regularity (sparsity) of the object to recover as well as to inhomogeneous
smoothness. We also perform a numerical study which proves that the procedure
shows very promising properties in practice as well.Comment: Published in at http://dx.doi.org/10.1214/10-AOS858 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Testing the isotropy of high energy cosmic rays using spherical needlets
For many decades, ultrahigh energy charged particles of unknown origin that
can be observed from the ground have been a puzzle for particle physicists and
astrophysicists. As an attempt to discriminate among several possible
production scenarios, astrophysicists try to test the statistical isotropy of
the directions of arrival of these cosmic rays. At the highest energies, they
are supposed to point toward their sources with good accuracy. However, the
observations are so rare that testing the distribution of such samples of
directional data on the sphere is nontrivial. In this paper, we choose a
nonparametric framework that makes weak hypotheses on the alternative
distributions and allows in turn to detect various and possibly unexpected
forms of anisotropy. We explore two particular procedures. Both are derived
from fitting the empirical distribution with wavelet expansions of densities.
We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574-594
(electronic)], the so-called needlets. The expansions are truncated at scale
indices no larger than some , and the distances between
those estimates and the null density are computed. One family of tests (called
Multiple) is based on the idea of testing the distance from the null for each
choice of , whereas the so-called PlugIn approach is
based on the single full expansion, but with thresholded wavelet
coefficients. We describe the practical implementation of these two procedures
and compare them to other methods in the literature. As alternatives to
isotropy, we consider both very simple toy models and more realistic
nonisotropic models based on Physics-inspired simulations. The Monte Carlo
study shows good performance of the Multiple test, even at moderate sample
size, for a wide sample of alternative hypotheses and for different choices of
the parameter . On the 69 most energetic events published by the
Pierre Auger Collaboration, the needlet-based procedures suggest statistical
evidence for anisotropy. Using several values for the parameters of the
methods, our procedures yield -values below 1%, but with uncontrolled
multiplicity issues. The flexibility of this method and the possibility to
modify it to take into account a large variety of extensions of the problem
make it an interesting option for future investigation of the origin of
ultrahigh energy cosmic rays.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS619 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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