This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted c... more This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature.
... to compute the inverse diffeomorphisms of Φt v, it is enough to revert the time in equation (... more ... to compute the inverse diffeomorphisms of Φt v, it is enough to revert the time in equation (2). One may refer to Younes, (2004) for ... fields generated by the tensor product of 2 one-dimensional B-splines (hence K = 4). An example of deformation of the classical Lena image is ...
Statistical Applications in Genetics and Molecular Biology, 2000
We investigate an important issue of a meta-algorithm for selecting variables in the framework of... more We investigate an important issue of a meta-algorithm for selecting variables in the framework of microarray data. This wrapper method starts from any classification algorithm and weights each variable (i.e. gene) relative to its efficiency for classification. An optimization procedure is then inferred which exhibits important genes for the studied biological process.
In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothi... more In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one dimensional setting incorporating diffeomorphic constraints is equivalent to impose monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to monotonize any unconstrained estimator of a regression function, and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances, and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching.
... k = 1,...,K for vector fields generated by the tensor prod-uct of two one-dimensional B-splin... more ... k = 1,...,K for vector fields generated by the tensor prod-uct of two one-dimensional B-splines (hence K = 4). An ex-ample of deformation of the classical Lena image is shown Page 4. ... Fig. 3 Random deformation of the Lena image with A = 0.1 and A = 0.5 in Fig. ...
... of the 22nd International Technical Meeting of The Satellite Division of the Institute of Nav... more ... of the 22nd International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009 ... of the error distributions into the tails, CNES and TAS have established a research action with French Universities of Lyon I and ...
This paper considers the problem of estimating a mean pattern in the setting of Grenander's patte... more This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a data set of curves or images is modeled by the random action of elements in a compact Lie group on an infinite dimensional space. In the case of observations contaminated by an additive Gaussian white noise, it is shown that estimating a reference template in the setting of Grenander's pattern theory falls into the category of deconvolution problems over Lie groups. To obtain this result, we build an estimator of a mean pattern by using Fourier deconvolution and harmonic analysis on compact Lie groups. In an asymptotic setting where the number of observed curves or images tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Sobolev balls. This rate depends on the smoothness of the density of the random Lie group elements representing shape variability in the data, which makes a connection between estimating a mean pattern and standard deconvolution problems in nonparametric statistics.
This paper considers the problem of adaptive estimation of a non-homogeneous intensity function f... more This paper considers the problem of adaptive estimation of a non-homogeneous intensity function from the observation of n independent Poisson processes having a common intensity that is randomly shifted for each observed trajectory. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes.
This paper considers the problem of adaptive estimation of a template in a randomly shifted curve... more This paper considers the problem of adaptive estimation of a template in a randomly shifted curve model. Using the Fourier transform of the data, we show that this problem can be transformed into a stochastic linear inverse problem. Our aim is to approach the estimator that has the smallest risk on the true template over a finite set of linear estimators defined in the Fourier domain. Based on the principle of unbiased empirical risk minimization, we derive a nonasymptotic oracle inequality in the case where the law of the random shifts is known. This inequality can then be used to obtain adaptive results on Sobolev spaces as the number of observed curves tend to infinity. Some numerical experiments are given to illustrate the performances of our approach.
... nonparametric regression Jérémie Bigot & Sébastien Gadat Laboratoire de Statistique et Pr... more ... nonparametric regression Jérémie Bigot & Sébastien Gadat Laboratoire de Statistique et Probabilités ... methods) can be found in Ramsay [22], Kelly and Rice [16], Mammen [17], Mam-men and Thomas-Agnan [19], Hall and Huang [13], Mammen, Marron, Turlach and ...
Le travail présenté ici concerne l'analyse, la compréhension et la représentation de grands graph... more Le travail présenté ici concerne l'analyse, la compréhension et la représentation de grands graphes. En effet, ce type de données se retrouve de manière naturelle dans un nombre croissant de problèmes concrets : web, recherche d'information, réseaux sociaux, réseaux d'interaction biologiques... La progression des moyens de recueil et de stockage des données rend la taille de ces graphes croissante : le développement de méthodes permettant leur analyse et leur représentation est donc un domaine de recherche dynamique et important à l'interface de nombreuses disciplines (mathématiques, statistique, informatique, ...). Dans cet article, nous développons une méthode de représentation de graphes basée sur une classification préalable des sommets. En effet, dans [18], les auteurs font remarquer que « reducing [the] level of complexity [of a network] to one that can be interpreted readily by the human eye, will be invaluable in helping us to understand the largescale structure of these new network data » : nous nous appuyons sur ce constat pour utiliser une classification des sommets du graphe comme étape préliminaire et simplificatrice à la représentation du graphe dans son ensemble. La phase de classification consiste en l'optimisation d'une mesure de qualité spécialement adaptée à la recherche de groupes denses dans les graphes : la modularité. Cette mesure quantifie la « distance » à un modèle nul dans lequel les arêtes sont placées indépendamment de la classification. Elle a montré sa pertinence dans la résolution de problèmes de recherche de groupes denses dans un graphe. L'optimisation de la modularité est effectuée par le biais d'un algorithme stochastique de recuit simulé. Enfin, la représentation, en tant que telle, est basée sur un algorithme de « forces » contraint décrit dans . Après une courte introduction au problème (partie 1), nous détaillons, dans la partie 2, la procédure de classification des sommets et, en partie 3, l'algorithme de représentation utilisé. Un exemple issu de l'analyse de réseaux sociaux est ensuite présenté en partie 4.
This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted c... more This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature.
... to compute the inverse diffeomorphisms of Φt v, it is enough to revert the time in equation (... more ... to compute the inverse diffeomorphisms of Φt v, it is enough to revert the time in equation (2). One may refer to Younes, (2004) for ... fields generated by the tensor product of 2 one-dimensional B-splines (hence K = 4). An example of deformation of the classical Lena image is ...
Statistical Applications in Genetics and Molecular Biology, 2000
We investigate an important issue of a meta-algorithm for selecting variables in the framework of... more We investigate an important issue of a meta-algorithm for selecting variables in the framework of microarray data. This wrapper method starts from any classification algorithm and weights each variable (i.e. gene) relative to its efficiency for classification. An optimization procedure is then inferred which exhibits important genes for the studied biological process.
In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothi... more In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one dimensional setting incorporating diffeomorphic constraints is equivalent to impose monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to monotonize any unconstrained estimator of a regression function, and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances, and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching.
... k = 1,...,K for vector fields generated by the tensor prod-uct of two one-dimensional B-splin... more ... k = 1,...,K for vector fields generated by the tensor prod-uct of two one-dimensional B-splines (hence K = 4). An ex-ample of deformation of the classical Lena image is shown Page 4. ... Fig. 3 Random deformation of the Lena image with A = 0.1 and A = 0.5 in Fig. ...
... of the 22nd International Technical Meeting of The Satellite Division of the Institute of Nav... more ... of the 22nd International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009 ... of the error distributions into the tails, CNES and TAS have established a research action with French Universities of Lyon I and ...
This paper considers the problem of estimating a mean pattern in the setting of Grenander's patte... more This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a data set of curves or images is modeled by the random action of elements in a compact Lie group on an infinite dimensional space. In the case of observations contaminated by an additive Gaussian white noise, it is shown that estimating a reference template in the setting of Grenander's pattern theory falls into the category of deconvolution problems over Lie groups. To obtain this result, we build an estimator of a mean pattern by using Fourier deconvolution and harmonic analysis on compact Lie groups. In an asymptotic setting where the number of observed curves or images tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Sobolev balls. This rate depends on the smoothness of the density of the random Lie group elements representing shape variability in the data, which makes a connection between estimating a mean pattern and standard deconvolution problems in nonparametric statistics.
This paper considers the problem of adaptive estimation of a non-homogeneous intensity function f... more This paper considers the problem of adaptive estimation of a non-homogeneous intensity function from the observation of n independent Poisson processes having a common intensity that is randomly shifted for each observed trajectory. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes.
This paper considers the problem of adaptive estimation of a template in a randomly shifted curve... more This paper considers the problem of adaptive estimation of a template in a randomly shifted curve model. Using the Fourier transform of the data, we show that this problem can be transformed into a stochastic linear inverse problem. Our aim is to approach the estimator that has the smallest risk on the true template over a finite set of linear estimators defined in the Fourier domain. Based on the principle of unbiased empirical risk minimization, we derive a nonasymptotic oracle inequality in the case where the law of the random shifts is known. This inequality can then be used to obtain adaptive results on Sobolev spaces as the number of observed curves tend to infinity. Some numerical experiments are given to illustrate the performances of our approach.
... nonparametric regression Jérémie Bigot & Sébastien Gadat Laboratoire de Statistique et Pr... more ... nonparametric regression Jérémie Bigot & Sébastien Gadat Laboratoire de Statistique et Probabilités ... methods) can be found in Ramsay [22], Kelly and Rice [16], Mammen [17], Mam-men and Thomas-Agnan [19], Hall and Huang [13], Mammen, Marron, Turlach and ...
Le travail présenté ici concerne l'analyse, la compréhension et la représentation de grands graph... more Le travail présenté ici concerne l'analyse, la compréhension et la représentation de grands graphes. En effet, ce type de données se retrouve de manière naturelle dans un nombre croissant de problèmes concrets : web, recherche d'information, réseaux sociaux, réseaux d'interaction biologiques... La progression des moyens de recueil et de stockage des données rend la taille de ces graphes croissante : le développement de méthodes permettant leur analyse et leur représentation est donc un domaine de recherche dynamique et important à l'interface de nombreuses disciplines (mathématiques, statistique, informatique, ...). Dans cet article, nous développons une méthode de représentation de graphes basée sur une classification préalable des sommets. En effet, dans [18], les auteurs font remarquer que « reducing [the] level of complexity [of a network] to one that can be interpreted readily by the human eye, will be invaluable in helping us to understand the largescale structure of these new network data » : nous nous appuyons sur ce constat pour utiliser une classification des sommets du graphe comme étape préliminaire et simplificatrice à la représentation du graphe dans son ensemble. La phase de classification consiste en l'optimisation d'une mesure de qualité spécialement adaptée à la recherche de groupes denses dans les graphes : la modularité. Cette mesure quantifie la « distance » à un modèle nul dans lequel les arêtes sont placées indépendamment de la classification. Elle a montré sa pertinence dans la résolution de problèmes de recherche de groupes denses dans un graphe. L'optimisation de la modularité est effectuée par le biais d'un algorithme stochastique de recuit simulé. Enfin, la représentation, en tant que telle, est basée sur un algorithme de « forces » contraint décrit dans . Après une courte introduction au problème (partie 1), nous détaillons, dans la partie 2, la procédure de classification des sommets et, en partie 3, l'algorithme de représentation utilisé. Un exemple issu de l'analyse de réseaux sociaux est ensuite présenté en partie 4.
Uploads
Papers by S. Gadat