140 research outputs found
Large Deviations for Random Spectral Measures and Sum Rules
We prove a Large Deviation Principle for the random spec- tral measure
associated to the pair where is sampled in the GUE(N) and e is
a fixed unit vector (and more generally in the - extension of this
model). The rate function consists of two parts. The contribution of the
absolutely continuous part of the measure is the reversed Kullback information
with respect to the semicircle distribution and the contribution of the
singular part is connected to the rate function of the extreme eigenvalue in
the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but
in thoses cases the expression of the rate function is not so explicit
Exact Reconstruction using Beurling Minimal Extrapolation
We show that measures with finite support on the real line are the unique
solution to an algorithm, named generalized minimal extrapolation, involving
only a finite number of generalized moments (which encompass the standard
moments, the Laplace transform, the Stieltjes transformation, etc). Generalized
minimal extrapolation shares related geometric properties with basis pursuit of
Chen, Donoho and Saunders [CDS98]. Indeed we also extend some standard results
of compressed sensing (the dual polynomial, the nullspace property) to the
signed measure framework. We express exact reconstruction in terms of a simple
interpolation problem. We prove that every nonnegative measure, supported by a
set containing s points,can be exactly recovered from only 2s + 1 generalized
moments. This result leads to a new construction of deterministic sensing
matrices for compressed sensing.Comment: 27 pages, 3 figures version 2 : minor changes and new titl
Sum rules and large deviations for spectral matrix measures
A sum rule relative to a reference measure on R is a relationship between the
reversed Kullback-Leibler divergence of a positive measure on R and some
non-linear functional built on spectral elements related to this measure (see
for example Killip and Simon 2003). In this paper, using only probabilistic
tools of large deviations, we extend the sum rules obtained in Gamboa, Nagel
and Rouault (2015) to the case of Hermitian matrix-valued measures. We recover
the earlier result of Damanik, Killip and Simon (2010) when the reference
measure is the (matrix-valued) semicircle law and obtain a new sum rule when
the reference measure is the (matrix-valued) Marchenko-Pastur law
Estimation of Translation, Rotation, and Scaling between Noisy Images Using the FourierâMellin Transform
In this paper we focus on extended Euclidean registration of a set of noisy images. We provide an appropriate statistical model for this kind of registration problems, and a new criterion based on Fourier-type transforms is proposed to estimate the translation, rotation and scaling parameters to align a set of images. This criterion is a two step procedure which does not require the use of a reference template onto which aligning all the images. Our approach is based on M-estimation and we prove the consistency of the resulting estimators. A small scale simulation study and real examples are used to illustrate the numerical performances of our procedure
Large Deviations for Random Power Moment Problem
We consider the set M_n of all n-truncated power moment sequences of
probability measures on [0,1]. We endow this set with the uniform probability.
Picking randomly a point in M_n, we show that the upper canonical measure
associated with this point satisfies a large deviation principle. Moderate
deviation are also studied completing earlier results on asymptotic normality
given by \citeauthorChKS93 [Ann. Probab. 21 (1993) 1295-1309]. Surprisingly,
our large deviations results allow us to compute explicitly the (n+1)th moment
range size of the set of all probability measures having the same n first
moments. The main tool to obtain these results is the representation of M_n on
canonical moments [see the book of \citeauthorDS97].Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000055
Semi-parametric estimation of shifts
We observe a large number of functions differing from each other only by a
translation parameter. While the main pattern is unknown, we propose to
estimate the shift parameters using -estimators. Fourier transform enables
to transform this statistical problem into a semi-parametric framework. We
study the convergence of the estimator and provide its asymptotic behavior.
Moreover, we use the method in the applied case of velocity curve forecasting.Comment: Published in at http://dx.doi.org/10.1214/07-EJS026 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Estimation error for blind Gaussian time series prediction
We tackle the issue of the blind prediction of a Gaussian time series. For
this, we construct a projection operator build by plugging an empirical
covariance estimation into a Schur complement decomposition of the projector.
This operator is then used to compute the predictor. Rates of convergence of
the estimates are given
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