arXiv subjects

Thomas Romary

Publications and source records attributed to Thomas Romary.

7 recordsLinked to original sources

arXiv2026

Generalizing Score-based generative models for Heavy-tailed Distributions

Score-based generative models (SGMs) have achieved remarkable empirical success, motivating their application to a broad range of data distributions. However, extending them to heavy-tailed targets remains a largely open problem. Although dedicated models for heavy-tailed distributions have been proposed, their generative fidelity remains unclear and they lack solid theoretical foundations, leaving important questions open in this regime. In this paper, we address this gap through two theoretical contributions. First, we show that combining early stopping with a suitable initialization is sufficient to extend the diffusion framework to any target distribution; in particular, we establish the well-posedness of the backward process and prove convergence of the approximated diffusion in KL divergence. Second, we derive novel theoretical guarantees for generation with normalizing flows, obtaining convergence results that hold under mild conditions on the flow family and without any assumption on the tail behavior of the target distribution. Building on these results, we propose a unified generative framework for heavy-tailed distributions: a normalizing flow is first trained to capture the tail behavior and is then used as an initialization prior for an SGM, which refines the samples by recovering fine-grained structural details. This design leverages the complementary strengths of the two model classes within a theoretically principled pipeline, overcoming the limitations of existing approaches.

stat.ML↗

arXiv2025

Efficient bayesian spatially varying coefficients modeling for censored data using the vecchia approximation

Spatially varying coefficients (SVC) models allow for marginal effects to be non-stationary over space and thus offer a higher degree of flexibility with respect to standard geostatistical models with external drift. At the same time, SVC models have the advantage that they are easily interpretable. They offer a flexible framework for understanding how the relationships between dependent and independent variables vary across space. The most common methods for modelling such data are the Geographically Weighted Regression (GWR) and Bayesian Gaussian Process (Bayes-GP). The Bayesian SVC model, which assumes that the coefficients follow Gaussian processes, provides a rigorous approach to account for spatial non-stationarity. However, the computational cost of Bayes-GP models can be prohibitively high when dealing with large datasets or/and when using a large number of covariates, due to the repeated inversion of dense covariance matrices required at each Markov chain Monte Carlo (MCMC) iteration. In this study, we propose an efficient Bayes-GP modeling framework leveraging the Vecchia approximation to reduce computational complexity while maintaining accuracy. The proposed method is applied to a challenging soil pollution data set in Toulouse, France, characterized by a high degree of censorship (two-thirds censored observations) and spatial clustering. Our results demonstrate the ability of the Vecchia-based Bayes-GP model to capture spatially varying effects and provide meaningful insights into spatial heterogeneity, even under the constraints of censored data.

stat.ME↗

arXiv2025

Spline Interpolation on Compact Riemannian Manifolds

Spline interpolation is a widely used class of methods for solving interpolation problems by constructing smooth interpolants that minimize a regularized energy functional involving the Laplacian operator. While many existing approaches focus on Euclidean domains or the sphere, relying on the spectral properties of the Laplacian, this work introduces a method for spline interpolation on general manifolds by exploiting its equivalence with kriging. Specifically, the proposed approach uses finite element approximations of random fields defined over the manifold, based on Gaussian Markov Random Fields and a discretization of the Laplace-Beltrami operator on a triangulated mesh. This framework enables the modeling of spatial fields with smooth variations and local anisotropies via domain deformation. The method is first validated on the sphere using both analytical test cases and a pollution-related study, and is compared to the classical spherical harmonics-based method. Additional experiments on the surface of a cylinder further illustrate the generality of the approach.

stat.CO↗

arXiv2023

A stable deep adversarial learning approach for geological facies generation

The simulation of geological facies in an unobservable volume is essential in various geoscience applications. Given the complexity of the problem, deep generative learning is a promising approach to overcome the limitations of traditional geostatistical simulation models, in particular their lack of physical realism. This research aims to investigate the application of generative adversarial networks and deep variational inference for conditionally simulating meandering channels in underground volumes. In this paper, we review the generative deep learning approaches, in particular the adversarial ones and the stabilization techniques that aim to facilitate their training. The proposed approach is tested on 2D and 3D simulations generated by the stochastic process-based model Flumy. Morphological metrics are utilized to compare our proposed method with earlier iterations of generative adversarial networks. The results indicate that by utilizing recent stabilization techniques, generative adversarial networks can efficiently sample from target data distributions. Moreover, we demonstrate the ability to simulate conditioned simulations through the latent variable model property of the proposed approach.

physics.geo-ph↗

arXiv2022

The SPDE approach for spatio-temporal datasets with advection and diffusion

In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest. Large space-time datasets call for new numerical methods to efficiently process them. The Stochastic Partial Differential Equation (SPDE) approach has proven to be effective for the estimation and the prediction in a spatial context. We present here the advection-diffusion SPDE with first order derivative in time which defines a large class of nonseparable spatio-temporal models. A Gaussian Markov random field approximation of the solution to the SPDE is built by discretizing the temporal derivative with a finite difference method (implicit Euler) and by solving the spatial SPDE with a finite element method (continuous Galerkin) at each time step. The ''Streamline Diffusion'' stabilization technique is introduced when the advection term dominates the diffusion. Computationally efficient methods are proposed to estimate the parameters of the SPDE and to predict the spatio-temporal field by kriging, as well as to perform conditional simulations. The approach is applied to a solar radiation dataset. Its advantages and limitations are discussed.

stat.ME↗

arXiv2018

Combining covariance tapering and lasso driven low rank decomposition for the kriging of large spatial datasets

Large spatial datasets are becoming ubiquitous in environmental sciences with the explosion in the amount of data produced by sensors that monitor and measure the Earth system. Consequently, the geostatistical analysis of these data requires adequate methods. Richer datasets lead to more complex modeling but may also prevent from using classical techniques. Indeed, the kriging predictor is not straightforwarldly available as it requires the inversion of the covariance matrix of the data. The challenge of handling such datasets is therefore to extract the maximum of information they contain while ensuring the numerical tractability of the associated inference and prediction algorithms. The different approaches that have been developed in the literature to address this problem can be classified into two families, both aiming at making the inversion of the covariance matrix computationally feasible. The covariance tapering approach circumvents the problem by enforcing the sparsity of the covariance matrix, making it invertible in a reasonable computation time. The second available approach assumes a low rank representation of the covariance function. While both approaches have their drawbacks, we propose a way to combine them and benefit from their advantages. The covariance model is assumed to have the form low rank plus sparse. The choice of the basis functions sustaining the low rank component is data driven and is achieved through a selection procedure, thus alleviating the computational burden of the low rank part. This model expresses as a spatial random effects model and the estimation of the parameters is conducted through a step by step approach treating each scale separately. The resulting model can account for second order non stationarity and handle large volumes of data.

math.ST↗

arXiv2014

Estimation of Space Deformation Model for Non-stationary Random Functions

Stationary Random Functions have been successfully applied in geostatistical applications for decades. In some instances, the assumption of a homogeneous spatial dependence structure across the entire domain of interest is unrealistic. A practical approach for modelling and estimating non-stationary spatial dependence structure is considered. This consists in transforming a non-stationary Random Function into a stationary and isotropic one via a bijective continuous deformation of the index space. So far, this approach has been successfully applied in the context of data from several independent realizations of a Random Function. In this work, we propose an approach for non-stationary geostatistical modelling using space deformation in the context of a single realization with possibly irregularly spaced data. The estimation method is based on a non-stationary variogram kernel estimator which serves as a dissimilarity measure between two locations in the geographical space. The proposed procedure combines aspects of kernel smoothing, weighted non-metric multi-dimensional scaling and thin-plate spline radial basis functions. On a simulated data, the method is able to retrieve the true deformation. Performances are assessed on both synthetic and real datasets. It is shown in particular that our approach outperforms the stationary approach. Beyond the prediction, the proposed method can also serve as a tool for exploratory analysis of the non-stationarity.

stat.ME↗