We measure the out-of-domain uncertainty in the prediction of Neural Networks using a statistical notion called "Lens Depth'' (LD) combined with Fermat Distance, which is able to capture precisely the "depth'' of a point with respect to a distribution in feature space, without any distributional assumption. Our method also has no trainable parameter. The method is applied directly in the feature space at test time and does not intervene in training process. As such, it does not impact the performance of the original model. The proposed method gives excellent qualitative results on toy datasets and can give competitive or better uncertainty estimation on standard deep learning datasets compared to strong baseline methods.

In the fair classification setup, we recast the links between fairness and predictability in terms of probability metrics. We analyze repair methods based on mapping conditional distributions to the Wasserstein barycenter. We propose a Random Repair which yields a tradeoff between minimal information loss and a certain amount of fairness.

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to these kernels can be efficiently forecast, opening new perspectives in Gaussian process modeling.

We investigate the multi-dimensional super resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the l1-minimization in the space of Radon measures in the multi-dimensional frame on semi-algebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal l1-minimization exact recovery problem of super resolution that unleashes standard techniques (such as moment-sum-of-squares hierarchies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the super resolution problem in dimension greater than 2 and for a large family of domains described by semi-algebraic sets.

To overcome the cost of numerical simulations of transient phenomena, the goal is to construct a robust spatio-temporal reduced model capable of long-term in time predictions. The construction proposed in this article has to deal with several constraints: the reference model is a black box with high dimensional inputs and outputs, long-term in time prediction, few learning samples available, non-linear behaviour and the construction time must remain reasonable while the prediction time must be negligible. Recurrent neural networks are predictive models adapted to this dynamic framework. The improvements of the construction methodology detailed in this paper are the weights optimization through a multilevel optimization approach, a robust construction based on cross-validation and an application of sensitivity analysis in order to reduce the input dimension of the network. Finally, this construction is validated on an industrial test case predicting the temperature of an electronic equipment located in the avionic bay and subjected to fluctuations of its boundary conditions.

We present a group lasso procedure for generalized linear models (GLMs) and we study the properties of this estimator applied to sparse high-dimensional GLMs. Under general conditions on the covariates and on the joint distribution of the pair covariates, we provide oracle inequalities promoting group sparsity of the covariables. We get convergence rates for the prediction and estimation error and we show the ability of this estimator to recover good sparse approximation of the true model. Then, we extend this procedure to the case of an elastic net penalty. At last, we apply these results to the so-called Poisson regression model (the output is modeled as a Poisson process whose intensity relies on a linear combination of the covariables). The group lasso method enables to select few groups of meaningful variables among the set of inputs.

In this paper we focus on extended Euclidean registration of a set of noisy images. We provide an appropriate statistical model for this kind of registration problem, and a new criterion based on Fourier-type transforms is proposed to estimate the translation, rotation, and scaling parameters to align a set of images. This criterion is a two-step procedure which does not require the use of a reference template onto which all the images are aligned. Our approach is based on M-estimation, and we prove the consistency of the resulting estimators. A small-scale simulation study and real examples are used to illustrate the numerical performances of our procedure.

In generalized moment problems (signed) measures are searched to fit given observations, or continuous functions are searched to fit given constraints. Known convex methods for solving such problems, and their stochastic interpretations via maximum entropy on the mean (MEM) and in a Bayesian sense are reviewed, with some improvements on previous results. Then the MEM and Bayesian approaches are extended to default models with a dependence structure, yielding new families of solutions. One family involves a transfer kernel, and allows using prior information such as modality, convexity, or Sobolev norms. Another family of solutions with possibly nonconvex criteria, is arrived at using default models with exchangeable random variables. The main technical tools are convex analysis and large deviations theory.

Reconstruction of a spectral density function from a finite set of covariances can be performed by maximizing an entropy functional. The method of the maximum entropy on the mean is used For computing a discrete version of this spectral density and allows one to give a new interpretation of these reconstruction methods. In fact, the authors show that the choice of the entropy is directly related to a prior distribution. In particular, they consider processes on Z/sup 2/. Steepest descent procedures permit the numerical computation of discrete realizations for a wide class of entropies. To ensure the nonnegativity of the solution related to the Burg entropy, they present a new algorithm based on a fixed-point method and the Yule-Walker equations to compute this solution. Then, the solution of the dual problem is obtained as the limit of the trajectory of an ordinary differential equation.>