arXiv subjects

Anne Dutfoy

Publications and source records attributed to Anne Dutfoy.

13 recordsLinked to original sources

arXiv2026

Csisz\'ar indices and interpolating copulas

We study various properties of $f$-divergences and Csisz\'ar indices between two probability distributions in very general setups for the convex function $f$ and for the probability distributions. We establish general structural properties of $f$-divergences and show how they are inherited by the associated Csisz\'ar indices, including monotonicity and invariance under suitable transformations. We also study the relationship between Csisz\'ar indices and copula representations of random vectors. When the marginal distributions have atoms, the copula representation is not unique and the Csisz\'ar index of the transformed vectors may increase. We build a large family of interpolating copulas which minimize the Csisz\'ar index and thus preserve the dependence structure of the initial vector.

math.ST↗

arXiv2025

A tree-based Polynomial Chaos expansion for surrogate modeling and sensitivity analysis of complex numerical models

This paper introduces Tree-based Polynomial Chaos Expansion (Tree-PCE), a novel surrogate modeling technique designed to efficiently approximate complex numerical models exhibiting nonlinearities and discontinuities. Tree-PCE combines the expressive power of Polynomial Chaos Expansion (PCE) with an adaptive partitioning strategy inspired by regression trees. By recursively dividing the input space into hyperrectangular subdomains and fitting localized PCEs, Tree-PCE constructs a piecewise polynomial surrogate that improves both accuracy and computational efficiency. The method is particularly well-suited for global sensitivity analysis, enabling direct computation of Sobol' indices from local expansion coefficients and introducing a new class of sensitivity indices derived from the tree structure itself. Numerical experiments on synthetic and real-world models, including a 2D morphodynamic case, demonstrate that Tree-PCE offers a favorable balance between accuracy and complexity, especially in the presence of discontinuities. While its performance depends on the compromise between the number of subdomains and the degree of local polynomials, this trade-off can be explored using automated hyperparameter optimization frameworks. This opens promising perspectives for systematically identifying optimal configurations and enhancing the robustness of surrogate modeling in complex systems.

stat.ME↗

arXiv2022

Off-the-grid prediction and testing for linear combination of translated features

We consider a model where a signal (discrete or continuous) is observed with an additive Gaussian noise process. The signal is issued from a linear combination of a finite but increasing number of translated features. The features are continuously parameterized by their location and depend on some scale parameter. First, we extend previous prediction results for off-the-grid estimators by taking into account here that the scale parameter may vary. The prediction bounds are analogous, but we improve the minimal distance between two consecutive features locations in order to achieve these bounds. Next, we propose a goodness-of-fit test for the model and give non-asymptotic upper bounds of the testing risk and of the minimax separation rate between two distinguishable signals. In particular, our test encompasses the signal detection framework. We deduce upper bounds on the minimal energy,expressed as the $\ell_2$-norm of the linear coefficients, to successfully detect a signal in presence of noise. The general model considered in this paper is a non-linear extension of the classical high-dimensional regression model. It turns out that,in this framework, our upper bound on the minimax separation rate matches (up to a logarithmic factor) the lower bound on the minimax separation rate for signal detection in the high-dimensional linear model associated to a fixed dictionary of features. We also propose a procedure to test whether the features of the observed signal belong to a given finite collection under the assumption that the linear coefficients may vary, but have prescribed signs under the null hypothesis. A non-asymptotic upper bound on the testing risk is given.We illustrate our results on the spikes deconvolution model with Gaussian features on the real line and with the Dirichlet kernel, frequently used in the compressed sensing literature, on the torus.

math.ST↗

arXiv2022

Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary

In this paper we observe a set, possibly a continuum, of signals corrupted by noise. Each signal is a finite mixture of an unknown number of features belonging to a continuous dictionary. The continuous dictionary is parametrized by a real non-linear parameter. We shall assume that the signals share an underlying structure by assuming that each signal has its active features included in a finite and sparse set. We formulate regularized optimization problem to estimate simultaneously the linear coefficients in the mixtures and the non-linear parameters of the features. The optimization problem is composed of a data fidelity term and a $(\ell_1,L^p)$-penalty. We call its solution the Group-Nonlinear-Lasso and provide high probability bounds on the prediction error using certificate functions. Following recent works on the geometry of off-the-grid methods, we show that such functions can be constructed provided the parameters of the active features are pairwise separated by a constant with respect to a Riemannian metric.When the number of signals is finite and the noise is assumed Gaussian, we give refinements of our results for $p=1$ and $p=2$ using tail bounds on suprema of Gaussian and $\chi^2$ random processes. When $p=2$, our prediction error reaches the rates obtained by the Group-Lasso estimator in the multi-task linear regression model. Furthermore, for $p=2$ these prediction rates are faster than for $p=1$ when all signals share most of the non-linear parameters.

stat.ML↗

arXiv2022

Reparameterization of extreme value framework for improved Bayesian workflow

Using Bayesian methods for extreme value analysis offers an alternative to frequentist ones, with several advantages such as easily dealing with parametric uncertainty or studying irregular models. However, computations can be challenging and the efficiency of algorithms can be altered by poor parametrization choices. The focus is on the Poisson process characterization of univariate extremes and outline two key benefits of an orthogonal parameterization. First, Markov chain Monte Carlo convergence is improved when applied on orthogonal parameters. This analysis relies on convergence diagnostics computed on several simulations. Second, orthogonalization also helps deriving Jeffreys and penalized complexity priors, and establishing posterior propriety thereof. The proposed framework is applied to return level estimation of Garonne flow data (France).

stat.ME↗

arXiv2022

Adaptive importance sampling based on fault tree analysis for piecewise deterministic Markov process

Piecewise deterministic Markov processes (PDMPs) can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling (AIS) method based on a cross-entropy (CE) procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In this paper original families are proposed. They are well adapted to high-dimensional industrial systems. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry.

stat.CO↗

arXiv2022

Off-the-grid learning of mixtures from a continuous dictionary

We consider a general non-linear model where the signal is a finite mixture of an unknown, possibly increasing, number of features issued from a continuous dictionary parameterized by a real non-linear parameter. The signal is observed with Gaussian (possibly correlated) noise in either a continuous or a discrete setup. We propose an off-the-grid optimization method, that is, a method which does not use any discretization scheme on the parameter space, to estimate both the non-linear parameters of the features and the linear parameters of the mixture. We use recent results on the geometry of off-the-grid methods to give minimal separation on the true underlying non-linear parameters such that interpolating certificate functions can be constructed. Using also tail bounds for suprema of Gaussian processes we bound the prediction error with high probability. Assuming that the certificate functions can be constructed, our prediction error bound is up to $\log$-factors similar to the rates attained by the Lasso predictor in the linear regression model. We also establish convergence rates that quantify with high probability the quality of estimation for both the linear and the non-linear parameters. We develop in full details our main results for two applications: the Gaussian spike deconvolution and the scaled exponential model.

stat.ML↗

arXiv2022

On the use of a local $\hat{R}$ to improve MCMC convergence diagnostic

Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the $\hat{R}$ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator $\hat{R}_\infty$, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other $\hat{R}$ versions.

math.ST↗

arXiv2016

Fast adaptive estimation of log-additive exponential models in Kullback-Leibler divergence

We study the problem of nonparametric estimation of density functions with a product form on the domain $\triangle=\{( x_1, \ldots, x_d)\in \mathbb{R}^d, 0\leq x_1\leq \dots \leq x_d \leq 1\}$. Such densities appear in the random truncation model as the joint density function of observations. They are also obtained as maximum entropy distributions of order statistics with given marginals. We propose an estimation method based on the approximation of the logarithm of the density by a carefully chosen family of basis functions. We show that the method achieves a fast convergence rate in probability with respect to the Kullback-Leibler divergence for densities whose logarithm belongs to a Sobolev function class with known regularity. In the case when the regularity is unknown, we propose an estimation procedure using convex aggregation of the log-densities to obtain adaptability. The performance of this method is illustrated in a simulation study.

math.ST↗

arXiv2016

Optimal exponential bounds for aggregation of estimators for the Kullback-Leibler loss

We study the problem of model selection type aggregation with respect to the Kullback-Leibler divergence for various probabilistic models. Rather than considering a convex combination of the initial estimators $f_1, \ldots, f_N$, our aggregation procedures rely on the convex combination of the logarithms of these functions. The first method is designed for probability density estimation as it gives an aggregate estimator that is also a proper density function, whereas the second method concerns spectral density estimation and has no such mass-conserving feature. We select the aggregation weights based on a penalized maximum likelihood criterion. We give sharp oracle inequalities that hold with high probability, with a remainder term that is decomposed into a bias and a variance part. We also show the optimality of the remainder terms by providing the corresponding lower bound results.

math.ST↗

arXiv2015

Maximum entropy distribution of order statistics with given marginals

We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.

math.ST↗

arXiv2015

Open TURNS: An industrial software for uncertainty quantification in simulation

The needs to assess robust performances for complex systems and to answer tighter regulatory processes (security, safety, environmental control, and health impacts, etc.) have led to the emergence of a new industrial simulation challenge: to take uncertainties into account when dealing with complex numerical simulation frameworks. Therefore, a generic methodology has emerged from the joint effort of several industrial companies and academic institutions. EDF R&D, Airbus Group and Phimeca Engineering started a collaboration at the beginning of 2005, joined by IMACS in 2014, for the development of an Open Source software platform dedicated to uncertainty propagation by probabilistic methods, named OpenTURNS for Open source Treatment of Uncertainty, Risk 'N Statistics. OpenTURNS addresses the specific industrial challenges attached to uncertainties, which are transparency, genericity, modularity and multi-accessibility. This paper focuses on OpenTURNS and presents its main features: openTURNS is an open source software under the LGPL license, that presents itself as a C++ library and a Python TUI, and which works under Linux and Windows environment. All the methodological tools are described in the different sections of this paper: uncertainty quantification, uncertainty propagation, sensitivity analysis and metamodeling. A section also explains the generic wrappers way to link openTURNS to any external code. The paper illustrates as much as possible the methodological tools on an educational example that simulates the height of a river and compares it to the height of a dyke that protects industrial facilities. At last, it gives an overview of the main developments planned for the next few years.

stat.CO↗

arXiv2013

Maximum entropy copula with given diagonal section

We consider copulas with a given diagonal section and compute the explicit density of the unique optimal copula which maximizes the entropy. In this sense, this copula is the least informative among the copulas with a given diagonal section. We give an explicit criterion on the diagonal section for the existence of the optimal copula and give a closed formula for its entropy. We also provide examples for some diagonal sections of usual bivariate copulas and illustrate the differences between them and the maximum entropy copula with the same diagonal section.

math.ST↗