99 research outputs found
Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions
A pair of dual frames with almost exponentially localized elements (needlets)
are constructed on \RR_+^d based on Laguerre functions. It is shown that the
Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be
characterized in terms of respective sequence spaces that involve the needlet
coefficients.Comment: 42 page
Asymptotics for spherical needlets
We investigate invariant random fields on the sphere using a new type of
spherical wavelets, called needlets. These are compactly supported in frequency
and enjoy excellent localization properties in real space, with
quasi-exponentially decaying tails. We show that, for random fields on the
sphere, the needlet coefficients are asymptotically uncorrelated for any fixed
angular distance. This property is used to derive CLT and functional CLT
convergence results for polynomial functionals of the needlet coefficients:
here the asymptotic theory is considered in the high-frequency sense. Our
proposals emerge from strong empirical motivations, especially in connection
with the analysis of cosmological data sets.Comment: Published in at http://dx.doi.org/10.1214/08-AOS601 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Subsampling needlet coefficients on the sphere
In a recent paper, we analyzed the properties of a new kind of spherical
wavelets (called needlets) for statistical inference procedures on spherical
random fields; the investigation was mainly motivated by applications to
cosmological data. In the present work, we exploit the asymptotic uncorrelation
of random needlet coefficients at fixed angular distances to construct
subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate
how such statistics can be used for isotropy tests and for bootstrap estimation
of nuisance parameters, even when a single realization of the spherical random
field is observed. The asymptotic theory is developed in detail in the high
resolution sense.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ164 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus
We prove a multivariate CLT for skewness and kurtosis of the wavelets coefficients of a stationary field on the torus. The results are in the framework of the fixed-domain asymptotics, i.e. we refer to observations of a single field which is sampled at higher and higher frequencies. We consider also studentized statistics for the case of an unknown correlation structure. The results are motivated by the analysis of high-frequency financial data or cosmological data sets, with a particular interest towards testing for Gaussianity and isotropy. (c) 2007 Elsevier Inc. All rights reserved
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
Spherical Needlets for CMB Data Analysis
We discuss Spherical Needlets and their properties. Needlets are a form of
spherical wavelets which do not rely on any kind of tangent plane approximation
and enjoy good localization properties in both pixel and harmonic space;
moreover needlets coefficients are asymptotically uncorrelated at any fixed
angular distance, which makes their use in statistical procedures very
promising. In view of these properties, we believe needlets may turn out to be
especially useful in the analysis of Cosmic Microwave Background (CMB) data on
the incomplete sky, as well as of other cosmological observations. As a final
advantage, we stress that the implementation of needlets is computationally
very convenient and may rely completely on standard data analysis packages such
as HEALPix.Comment: 7 pages, 7 figure
Efficiently Learning Structured Distributions from Untrusted Batches
We study the problem, introduced by Qiao and Valiant, of learning from
untrusted batches. Here, we assume users, all of whom have samples from
some underlying distribution over . Each user sends a batch
of i.i.d. samples from this distribution; however an -fraction of
users are untrustworthy and can send adversarially chosen responses. The goal
is then to learn in total variation distance. When this is the
standard robust univariate density estimation setting and it is well-understood
that error is unavoidable. Suprisingly, Qiao and Valiant
gave an estimator which improves upon this rate when is large.
Unfortunately, their algorithms run in time exponential in either or .
We first give a sequence of polynomial time algorithms whose estimation error
approaches the information-theoretically optimal bound for this problem. Our
approach is based on recent algorithms derived from the sum-of-squares
hierarchy, in the context of high-dimensional robust estimation. We show that
algorithms for learning from untrusted batches can also be cast in this
framework, but by working with a more complicated set of test functions.
It turns out this abstraction is quite powerful and can be generalized to
incorporate additional problem specific constraints. Our second and main result
is to show that this technology can be leveraged to build in prior knowledge
about the shape of the distribution. Crucially, this allows us to reduce the
sample complexity of learning from untrusted batches to polylogarithmic in
for most natural classes of distributions, which is important in many
applications. To do so, we demonstrate that these sum-of-squares algorithms for
robust mean estimation can be made to handle complex combinatorial constraints
(e.g. those arising from VC theory), which may be of independent technical
interest.Comment: 46 page
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