257 research outputs found
Quantitative Isoperimetric Inequalities on the Real Line
In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown
that, in the Gauss space, a set of given measure and almost minimal Gauss
boundary measure is necessarily close to be a half-space. Using only geometric
tools, we extend their result to all symmetric log-concave measures \mu on the
real line. We give sharp quantitative isoperimetric inequalities and prove that
among sets of given measure and given asymmetry (distance to half line, i.e.
distance to sets of minimal perimeter), the intervals or complements of
intervals have minimal perimeter.Comment: 14 pages, 3 figure
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
Non-uniform spline recovery from small degree polynomial approximation
We investigate the sparse spikes deconvolution problem onto spaces of
algebraic polynomials. Our framework encompasses the measure reconstruction
problem from a combination of noiseless and noisy moment measurements. We study
a TV-norm regularization procedure to localize the support and estimate the
weights of a target discrete measure in this frame. Furthermore, we derive
quantitative bounds on the support recovery and the amplitudes errors under a
Chebyshev-type minimal separation condition on its support. Incidentally, we
study the localization of the knots of non-uniform splines when a Gaussian
perturbation of their inner-products with a known polynomial basis is observed
(i.e. a small degree polynomial approximation is known) and the boundary
conditions are known. We prove that the knots can be recovered in a grid-free
manner using semidefinite programming
Randomized pick-freeze for sparse Sobol indices estimation in high dimension
This article investigates a new procedure to estimate the influence of each
variable of a given function defined on a high-dimensional space. More
precisely, we are concerned with describing a function of a large number of
parameters that depends only on a small number of them. Our proposed method
is an unconstrained -minimization based on the Sobol's method. We
prove that, with only evaluations of , one can find
which are the relevant parameters
Constructions déterministes pour la régression parcimonieuse
Dans cette thèse nous étudions certains designs déterministes pour la régression par-cimonieuse. Notre problématique est largement inspirée du " Compressed Sensing " où l'on cherche à acquérir et compresser simultanément un signal de grande taille à partir d'un petit nombre de mesures linéaires. Plus précisément, nous faisons le lien entre l'erreur d'estimation et l'erreur de prédiction des estimateurs classiques (lasso, sélecteur Dantzig et basis pursuit) et la distorsion (qui mesure l'" écart " entre la norme 1 et la norme Euclidienne) du noyau du design considéré. Notre étude montre que toute construction de sous-espaces de faibles distorsions (appelés sous-espaces " presque " Euclidiens) conduit à de " bons " designs. Dans un second temps, nous nous intéressons aux designs construits à partir de graphes expanseurs déséquilibrés. Nous en établissons de manière précise les performances en termes d'erreur d'estimation et d'erreur de prédiction. Enfin, nous traitons la reconstruction exacte de mesures signées sur la droite réelle. Nous démontrons que tout système de Vandermonde généralisé permet la reconstruction fidèle de n'importe quel vecteur parcimonieux à partir d'un très faible nombre d'observations. Dans une partie indépendante, nous étudions la stabilité de l'inégalité isopérimétrique sur la droite réelle pour des mesures log-concaves.In this thesis we investigate some deterministic designs for the sparse regression. Our issue is mainly inspired by Compressed Sensing which is concerned by simultane-ously acquire and compress a signal of large size from a small number of linear measurements. More precisely, we show that there exists a link between variable selection and prediction error with standard estimators (such as the lasso, the Dantzig selector, the basis pursuit) and the distortion, which measures how "far" is the Manhattan norm from the Euclidean norm, of the null-space of the design. Hence, we show that every construction of subspaces with low-distortion (called "almost" Euclidean subspaces) gives
"good" designs. In a second part, we are interested by designs constructed from un balanced expander graphs. We accurately established their performances in terms of
variable selection and prediction error. Finally, we are interested in the faithful reconstruction of signed measures on the real line. We show that every generalized Vander monde system gives design such one can exactly recover all the sparse vectors from a dramatically small number of observations. In an independent part, we investigate the stability of the isoperimetric inequalities for the log-concave measures on the real line
Power of the Spacing test for Least-Angle Regression
Recent advances in Post-Selection Inference have shown that conditional
testing is relevant and tractable in high-dimensions. In the Gaussian linear
model, further works have derived unconditional test statistics such as the
Kac-Rice Pivot for general penalized problems. In order to test the global
null, a prominent offspring of this breakthrough is the spacing test that
accounts the relative separation between the first two knots of the celebrated
least-angle regression (LARS) algorithm. However, no results have been shown
regarding the distribution of these test statistics under the alternative. For
the first time, this paper addresses this important issue for the spacing test
and shows that it is unconditionally unbiased. Furthermore, we provide the
first extension of the spacing test to the frame of unknown noise variance.
More precisely, we investigate the power of the spacing test for LARS and
prove that it is unbiased: its power is always greater or equal to the
significance level . In particular, we describe the power of this test
under various scenarii: we prove that its rejection region is optimal when the
predictors are orthogonal; as the level goes to zero, we show that the
probability of getting a true positive is much greater than ; and we
give a detailed description of its power in the case of two predictors.
Moreover, we numerically investigate a comparison between the spacing test for
LARS and the Pearson's chi-squared test (goodness of fit).Comment: 22 pages, 8 figure
Consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden Markov models
In this paper, we consider the filtering and smoothing recursions in
nonparametric finite state space hidden Markov models (HMMs) when the
parameters of the model are unknown and replaced by estimators. We provide an
explicit and time uniform control of the filtering and smoothing errors in
total variation norm as a function of the parameter estimation errors. We prove
that the risk for the filtering and smoothing errors may be uniformly upper
bounded by the risk of the estimators. It has been proved very recently that
statistical inference for finite state space nonparametric HMMs is possible. We
study how the recent spectral methods developed in the parametric setting may
be extended to the nonparametric framework and we give explicit upper bounds
for the L2-risk of the nonparametric spectral estimators. When the observation
space is compact, this provides explicit rates for the filtering and smoothing
errors in total variation norm. The performance of the spectral method is
assessed with simulated data for both the estimation of the (nonparametric)
conditional distribution of the observations and the estimation of the marginal
smoothing distributions.Comment: 27 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1501.0478
- …