Sébastien Gadat

A deconvolution approach to estimation of a common shape in a shifted curves model

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.

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

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.

Selection of biologically relevant genes with a wrapper stochastic algorithm

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.Theory and application with the SVM classifier were presented in Gadat and Younes, 2007 and we extend this method with CART. The classification error rates are computed on three famous public databases (Leukemia, Colon and Prostate) and compared with those from other wrapper methods (RFE, lo norm SVM, Random Forests). This allows the assessment of the statistical relevance of the proposed algorithm. Furthermore, a biological interpretation with the Ingenuity Pathway Analysis software outputs clearly shows that the gene selections from the different wrapper methods raise very relevant biological information, compared to a classical filter gene selection with T-test.

Individual Human Cytotoxic T Lymphocytes Exhibit Intraclonal Heterogeneity during Sustained Killing

The killing of antigen-bearing cells by clonal populations of cytotoxic T lymphocytes (CTLs) is thought to be a rapid phenomenon executed uniformly by individual CTLs. We combined bulk and single-CTL killing assays over a prolonged time period to provide the killing statistics of clonal human CTLs against an excess of target cells. Our data reveal efficiency in sustained killing at the population level, which relied on a highly heterogeneous multiple killing performance at the individual level. Although intraclonal functional heterogeneity was a stable trait in clonal populations, it was reset in the progeny of individual CTLs. In-depth mathematical analysis of individual CTL killing data revealed a substantial proportion of high-rate killer CTLs with burst killing activity. Importantly, such activity was delayed and required activation with strong antigenic stimulation. Our study implies that functional heterogeneity allows CTL populations to calibrate prolonged cytotoxic activity to the size of target cell populations.

Intensity estimation of non-homogeneous Poisson processes from shifted trajectories

In this paper, we consider the problem of estimating nonparametrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N1,…,Nn on the interval [0,1]. This problem arises when data (counts) are collected independently from n individuals according to similar Poisson processes. 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.

Long time behaviour and stationary regime of memory gradient diffusions

In this paper, we are interested in a diffusion process based on a gradient descent. The process is non Markov and has a memory term which is built as a weighted average of the drift term all along the past of the trajectory. For this type of diffusion, we study the long time behaviour of the process in terms of the memory. We exhibit some conditions for the long-time stability of the dynamical system and then provide, when stable, some convergence properties of the occupation measures and of the marginal distribution, to the associated steady regimes. When the memory is too long, we show that in general, the dynamical system has a tendency to explode, and in the particular Gaussian case, we explicitly obtain the rate of divergence.

Random Action of Compact Lie Groups and Minimax Estimation of a Mean Pattern

This paper considers the problem of estimating a mean pattern in the setting of Grenanders pattern theory. Shape variability in a dataset 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 Grenanders 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.

Smoothing under Diffeomorphic Constraints with Homeomorphic Splines

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 imposing monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to make monotone 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.

Stochastic Heavy Ball

This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f. The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f. We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms.

Bayesian methods for the Shape Invariant Model

In this paper, we consider the so-called Shape Invariant Model that is used to model a function f submitted to a random translation of law g in a white noise. This model is of interest when the law of the deformations is unknown. Our objective is to recover the law of the process Pf0,g0 as well as f and g. To do this, we adopt a Bayesian point of view and find priors on f and g so that the posterior distribution concentrates at a polynomial rate around Pf0,g0 when n goes to +∞. We then derive results on the identifiability of the SIM, as well as results on the functional objects themselves. We intensively use Bayesian non-parametric tools coupled with mixture models, which may be of independent interest in model selection from a frequentist point of view. AMS 2000 subject classifications: Primary 62G05, 62F15; secondary 62G20.