Showing papers by "Jean-Michel Loubes published in 2009"

Statistical M-Estimation and Consistency in Large Deformable Models for Image Warping

Abstract: The problem of defining appropriate distances between shapes or images and modeling the variability of natural images by group transformations is at the heart of modern image analysis. A current trend is the study of probabilistic and statistical aspects of deformation models, and the development of consistent statistical procedure for the estimation of template images. In this paper, we consider a set of images randomly warped from a mean template which has to be recovered. For this, we define an appropriate statistical parametric model to generate random diffeomorphic deformations in two-dimensions. Then, we focus on the problem of estimating the mean pattern when the images are observed with noise. This problem is challenging both from a theoretical and a practical point of view. M-estimation theory enables us to build an estimator defined as a minimizer of a well-tailored empirical criterion. We prove the convergence of this estimator and propose a gradient descent algorithm to compute this M-estimator in practice. Simulations of template extraction and an application to image clustering and classification are also provided.

Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

Abstract: The aim of this paper is to establish non-asymptotic minimax rates of testing for goodness-of-fit hypotheses in a heteroscedastic setting. More precisely, we deal with sequences $(Y_j)_{j\in J}$ of independent Gaussian random variables, having mean $(\theta_j)_{j\in J}$ and variance $(\sigma_j)_{j\in J}$. The set $J$ will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to $l_2$ and $l_{\infty}$ norms, without assumption on $(\sigma_j)_{j\in J}$ and on several functions spaces. Our point of view is completely non-asymptotic.

Estimation of the distribution of random shifts deformation

Abstract: Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution μ modeling the variability in the response of each individual. Our aim is to construct a nonparametric estimator of the density of these random translation deformations using semiparametric preliminary estimates of the shifts. Based on the results of Dalalyan et al. [7], semiparametric estimators are obtained in our discrete framework and their performance studied. From these estimates we construct a nonparametric estimator of the target density. Both rates of convergence and an algorithm to construct the estimator are provided.

Nonparametric estimation of covariance functions by model selection

Abstract: We propose a model selection approach for covariance estimation of a multi-dimensional stochastic process. Under very general assumptions, observing i.i.d replications of the process at fixed observation points, we construct an estimator of the covariance function by expanding the process onto a collection of basis functions. We study the non asymptotic property of this estimate and give a tractable way of selecting the best estimator among a possible set of candidates. The optimality of the procedure is proved via an oracle inequality which warrants that the best model is selected.

Review of rates of convergence and regularity conditions for inverse problems

Abstract: The aim of this article is to review the different rates of convergence encountered in inverse problems, with both deterministic and stochastic noise. Indeed, in the litterature, several regularity conditions are often assumed leading to apparently different rates. We point out the different points of view and provide global assumptions that handle most of the cases encountered. Moreover we discuss optimality of some different usual estimators in the minimax but also the maxiset framework.

Oracle Inequality for Instrumental Variable Regression

Abstract: We tackle the problem of estimating a regression function observed in an instrumental regression framework. This model is an inverse problem with unknown operator. We provide a spectral cut-off estimation procedure which enables to derive oracle inequalities which warrants that our estimate, built without any prior knowledge, behaves as well as, up to log term, if the best model were known.

A Statistical Framework for Road Traffic Prediction

Abstract: The subject of this article is the prediction of traffic mean speeds on a road network in order to enhance live traffic information available to users. The approach in this work relies on the hypothesis that traffic dynamics can be summarized by two components: a deterministic and periodical trend and a short term component linked to the physics of traffic. The article will use statistical methods built upon historical databases in order to propose automatable, modular and efficient methods which also meet the industrial constraints currently faced.

Statistical M-Estimationand Consistencyinlargedeformable modelsfor Image Warping

Abstract: The problem of defining appropriate distances between shape s or images and modeling the variability of natural images by group transformations is a t the heart of modern image analysis. A current trend is the study of probabilistic and statistica l aspects of deformation models, and the development of consistent statistical procedure for the es timation of template images. In this paper, we consider a set of images randomly warped from a mean template which has to be recovered. For this, we define an appropriate statistical parametric mo del to generate random diffeomorphic deformations in two-dimensions. Then, we focus on the problem of estimating the mean pattern when the images are observed with noise. This problem is challenging both from a theoretical and a practical point of view. M-estimation theory enables us to build an estimator defined as a minimizer of a well-tailored empirical criterion. We prove the convergence of this estimator and propose a gradient descent algorithm to compute this M-estimator in practice. Simulations of template extraction and an application to image clustering and classification are also provided.

Adaptive estimation of covariance functions via wavelet thresholding and information projection

Abstract: In this paper, we study the problem of nonparametric adaptive estimation of the covariance function of a stationary Gaussian process. For this purpose, we consider a wavelet-based method which combines the ideas of wavelet approximation and estimation by information projection in order to warrants the positive semidefiniteness property of the solution. The spectral density of the process is estimated by projecting the wavelet thresholding expansion of the periodogram onto a family of exponential functions. This ensures that the spectral density estimator is a strictly positive function. Then, by Bochner theorem, we obtain a semidefinite positive estimator of the covariance function. The theoretical behavior of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. We also show the excellent practical performance of the estimator in some numerical experiments.

Maximum Entropy Estimation for Survey sampling

Abstract: Calibration methods have been widely studied in survey sampling over the last decades. Viewing calibration as an inverse problem, we extend the calibration technique by using a maximum entropy method. Finding the optimal weights is achieved by considering random weights and looking for a discrete distribution which maximizes an entropy under the calibration constraint. This method points a new frame for the computation of such estimates and the investigation of its statistical properties.

Regularization with Approximated $L^2$ Maximum Entropy Method

Abstract: We tackle the inverse problem of reconstructing an unknown finite measure $\mu$ from a noisy observation of a generalized moment of $\mu$ defined as the integral of a continuous and bounded operator $\Phi$ with respect to $\mu$. When only a quadratic approximation $\Phi_m$ of the operator is known, we introduce the $L^2$ approximate maximum entropy solution as a minimizer of a convex functional subject to a sequence of convex constraints. Under several assumptions on the convex functional, the convergence of the approximate solution is established and rates of convergence are provided.