On the consistency of Fr\'echet means in deformable models for curve and image analysis

A new class of statistical deformable models is introduced to study high-dimensional curves or images. In addition to the standard measurement error term, these deformable models include an extra error term modeling the individual variations in intensity around a mean pattern. It is shown that an appropriate tool for statistical inference in such models is the notion of sample Fr\'echet means, which leads to estimators of the deformation parameters and the mean pattern. The main contribution of this paper is to study how the behavior of these estimators depends on the number n of design points and the number J of observed curves (or images). Numerical experiments are given to illustrate the finite sample performances of the procedure

Necessary and sufficient condition for the existence of a Fréchet mean on the circle

First submission : Advances in Applied Probability (AAP) on May 17th 2011 (ref. AP/13983)Let $(\S^1,d_{\S^1})$ be the unit circle in $\R^2$ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure $\mu$ to admit a well defined Fréchet mean on $(\S^1,d_{\S^1})$. %This criterion allows to recover already known sufficient conditions of existence. We derive a new sufficient condition of existence $P(\alpha,\varphi)$ with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it

Consistent estimation of a mean planar curve modulo similarities

We consider the problem of estimating a mean planar curve from a set of $J$ random planar curves observed on a $k$-points deterministic design. We study the consistency of a smoothed Procrustean mean curve when the observations obey a deformable model including some nuisance parameters such as random translations, rotations and scaling. The main contribution of the paper is to analyze the influence of the dimension $k$ of the data and of the number $J$ of observed configurations on the convergence of the smoothed Procrustean estimator to the mean curve of the model. Some numerical experiments illustrate these results

Etude des propriétés statistiques des moyennes de Fréchet dans des modèles de déformations pour l'analyse de courbes et d'images en grande dimension

Cette thèse porte sur l'analyse statistique de données sur lesquelles agissent des déformations. Dans un premier temps, nous présentons une nouvelle classe de modèles statistiques semiparamétriques dits de déformations. Ces modèles peuvent s'appliquer à l'étude de courbes temporelles ou d'images de grande dimension. Les données sont supposées être générées par une courbe/image moyenne qui est bruitée et sur laquelle agit un opérateur de déformation. Nous étudions l'estimation des paramètres d'intérêt de ces modèles dans le cas général, puis dans le cas particulier des courbes du plan sur lesquelles agissent les rotations, translations et homothéties. Dans un second temps, nous considérons les structures non-euclidiennes induites par les actions de groupes de déformations. Un des enjeux des statistiques dans de tels espaces est de généraliser la notion de moyenne euclidienne. C'est ainsi que nous étudions les propriétés qui garantissent l'existence de la moyenne de Fréchet dans le cas particulier du cercle unité muni de la distance de la longueur d'arc.We are concerned with the statistical analysis of data observed with extra nuisance deformations. To this end, we first introduce a new class of semi-parametric deformable models. These models can be used to study the variability of time dependent curves or high dimensional images. We suppose that the curves or images at hand are generated by a noisy ideal mean pattern on which act some deformations operators. We then study the estimation of the parameters of interest of such models in the general case and in the particular case of planar curves observed with some rotation, translation and scaling. In a second part, we study the notion of mean in non-Euclidean spaces. More precisely, we study the conditions of existence of the Fréchet mean in the unit circle of the plane endowed with the arclength distance

Improve learning combining crowdsourced labels by weighting Areas Under the Margin

In supervised learning -- for instance in image classification -- modern massive datasets are commonly labeled by a crowd of workers. The obtained labels in this crowdsourcing setting are then aggregated for training. The aggregation step generally leverages a per worker trust score. Yet, such worker-centric approaches discard each task ambiguity. Some intrinsically ambiguous tasks might even fool expert workers, which could eventually be harmful for the learning step. In a standard supervised learning setting -- with one label per task and balanced classes -- the Area Under the Margin (AUM) statistic is tailored to identify mislabeled data. We adapt the AUM to identify ambiguous tasks in crowdsourced learning scenarios, introducing the Weighted AUM (WAUM). The WAUM is an average of AUMs weighted by worker and task dependent scores. We show that the WAUM can help discarding ambiguous tasks from the training set, leading to better generalization or calibration performance. We report improvements with respect to feature-blind aggregation strategies both for simulated settings and for the CIFAR-10H crowdsourced dataset

Hydro-mechanically Coupled Interface Finite Element for the Modeling of Soil–Structure Interactions: Application to Offshore Constructions

peer reviewe

Learning Riemannian geometry for mixed-effect models using deep generative networks.

We take up on recent work on the Riemannian geometry of generative networks to propose a new approach for learning both a manifold structure and a Riemannian metric from data. It allows the derivation of statistical analysis on manifolds without the need for the user to design new Riemannian structure for each specific problem. In high-dimensional data, it can learn non diagonal metrics, whereas manual design is often limited to the diagonal case. We illustrate how the method allows the construction of a meaningful low-dimensional representation of data and exhibit the geometry of the space of brain images during Alzheimer's progression