On Higher-Order Probabilistic Subrecursion

We study the expressive power of subrecursive probabilistic higher-order calculi. More specifically, we show that endowing a very expressive deterministic calculus like Godel's T with various forms of probabilistic choice operators may result in calculi which are not equivalent as for the class of distributions they give rise to, although they all guarantee almost-sure termination. Along the way, we introduce a probabilistic variation of the classic reducibility technique, and we prove that the simplest form of probabilistic choice leaves the expressive power of T essentially unaltered. The paper ends with some observations about the functional expressive power: expectedly, all the considered calculi capture the functions which T itself represents, at least when standard notions of observations are considered

Transport optimal semi-discret symétrique pour l'interpolation de maillages

Cette thèse a pour but de développer des méthodes géométriques pour approximer l'interpolation de déplacement, issue du transport optimal. Le transport optimal est une théorie mathématiques modélisant des déplacements de matière sous une contrainte de minimisation de coût, avec de nombreuses applications en physique, en informatique graphique et en géométrie. Le coût minimal du déplacement entre deux distributions définit une distance, qui elle-même est à l'origine de l'interpolation de déplacement. Ces interpolations peuvent sous certaines conditions présenter des discontinuités, que les approximations discrétisées du transport optimal n'arrivent pas toujours à bien capturer. Le travail de cette thèse vise à développer une approximation qui capture bien ces discontinuités. Notre méthode s'appuie sur le transport optimal semi-discret, où seul l'une des distributions est discrétisée, capturant ainsi avec précision les discontinuités de la distribution restée continue. Les plans de transport ainsi obtenus partitionnent la distribution continue en cellules associées aux échantillons de la discrétisation. Cette variante du transport optimal a cependant l'inconvénient de briser la symétrie entre les deux distributions. Nous commençons par présenter une approche réintroduisant cette symétrie, en calculant deux plans de transport couplés par le biais des barycentres de leurs cellules. Nous présentons ensuite un premier algorithme pour le calcul de ces plans de transport couplés. Il repose sur un schéma classique d'algorithme alterné, calculant successivement des plans de transport et les barycentres de leur cellules jusqu'à convergence. Les résultats obtenus à partir de cet algorithme permettent d'interpoler entre les distributions initiales en conservant une précision satisfaisante, en particulier au niveau des discontinuités, et y compris lorsque la discrétisation des distributions est faites avec relativement peu de points. Nous présentons ensuite notre exploration de méthodes d'optimisation pour résoudre le même problème. Ces méthodes expriment les contraintes de notre problème comme un optimum ou un point critique d'une fonctionnelle, et cherchent à atteindre ces points à l'aide d'algorithmes tels que la descente de gradient ou l'algorithme de Newton. Cette approche n'a cependant pas donné de résultats concluants, les fonctions considérées étant trop bruitées pour se prêter à des algorithmes d'optimisation.This thesis aims to develop geometric methods to approximate displacement interpolation, derived from optimal transport. Optimal transport is a mathematical theory modeling movements of matter under a cost minimization constraint, with many applications in physics, computer graphics and geometry. The minimum displacement cost between two distributions defines a distance, which itself is at the origin of displacement interpolation. This interpolation may under certain conditions present discontinuities, that the discretized approximations of the optimal transport do not always successfully capture. The work of this thesis aims to develop an approximation that captures these discontinuities well. Our method relies on semi-discrete optimal transport, where only one of the distributions is discretized, thus accurately capturing the discontinuities of the distribution that remains continuous. The transport plans thus obtained partition the continuous distribution into cells associated with the samples of the discretization. This variant of optimal transport however has the disadvantage of breaking the symmetry between the two distributions. We start by presenting an approach reintroducing this symmetry, by calculating two transport plans coupled through the barycenters of their cells. We then present a first algorithm for calculating these coupled transport plans. This first algorithm is based on a classical alternating algorithm scheme, successively computing the transport plans and the barycenters of their cells until convergence. The results obtained from this algorithm allow to interpolate between the initial distributions while maintaining a satisfactory precision, in particular when it comes to discontinuities, including when the discretization of the distributions is done with relatively few points. We then present our exploration of optimization methods for solving the same problem. These methods express the constraints of our problem as an optimum or a critical point of a functional, and aim to reach these points using algorithms such as gradient descent or Newton's method. However, this approach did not yield conclusive results, as the functions involved were too noisy to lend themselves well to optimization algorithms

Transport optimal semi-discret symétrique pour l'interpolation de maillages

This thesis aims to develop geometric methods to approximate displacement interpolation, derived from optimal transport. Optimal transport is a mathematical theory modeling movements of matter under a cost minimization constraint, with many applications in physics, computer graphics and geometry. The minimum displacement cost between two distributions defines a distance, which itself is at the origin of displacement interpolation. This interpolation may under certain conditions present discontinuities, that the discretized approximations of the optimal transport do not always successfully capture. The work of this thesis aims to develop an approximation that captures these discontinuities well. Our method relies on semi-discrete optimal transport, where only one of the distributions is discretized, thus accurately capturing the discontinuities of the distribution that remains continuous. The transport plans thus obtained partition the continuous distribution into cells associated with the samples of the discretization. This variant of optimal transport however has the disadvantage of breaking the symmetry between the two distributions. We start by presenting an approach reintroducing this symmetry, by calculating two transport plans coupled through the barycenters of their cells. We then present a first algorithm for calculating these coupled transport plans. This first algorithm is based on a classical alternating algorithm scheme, successively computing the transport plans and the barycenters of their cells until convergence. The results obtained from this algorithm allow to interpolate between the initial distributions while maintaining a satisfactory precision, in particular when it comes to discontinuities, including when the discretization of the distributions is done with relatively few points. We then present our exploration of optimization methods for solving the same problem. These methods express the constraints of our problem as an optimum or a critical point of a functional, and aim to reach these points using algorithms such as gradient descent or Newton's method. However, this approach did not yield conclusive results, as the functions involved were too noisy to lend themselves well to optimization algorithms.Cette thèse a pour but de développer des méthodes géométriques pour approximer l'interpolation de déplacement, issue du transport optimal. Le transport optimal est une théorie mathématiques modélisant des déplacements de matière sous une contrainte de minimisation de coût, avec de nombreuses applications en physique, en informatique graphique et en géométrie. Le coût minimal du déplacement entre deux distributions définit une distance, qui elle-même est à l'origine de l'interpolation de déplacement. Ces interpolations peuvent sous certaines conditions présenter des discontinuités, que les approximations discrétisées du transport optimal n'arrivent pas toujours à bien capturer. Le travail de cette thèse vise à développer une approximation qui capture bien ces discontinuités. Notre méthode s'appuie sur le transport optimal semi-discret, où seul l'une des distributions est discrétisée, capturant ainsi avec précision les discontinuités de la distribution restée continue. Les plans de transport ainsi obtenus partitionnent la distribution continue en cellules associées aux échantillons de la discrétisation. Cette variante du transport optimal a cependant l'inconvénient de briser la symétrie entre les deux distributions. Nous commençons par présenter une approche réintroduisant cette symétrie, en calculant deux plans de transport couplés par le biais des barycentres de leurs cellules. Nous présentons ensuite un premier algorithme pour le calcul de ces plans de transport couplés. Il repose sur un schéma classique d'algorithme alterné, calculant successivement des plans de transport et les barycentres de leur cellules jusqu'à convergence. Les résultats obtenus à partir de cet algorithme permettent d'interpoler entre les distributions initiales en conservant une précision satisfaisante, en particulier au niveau des discontinuités, et y compris lorsque la discrétisation des distributions est faites avec relativement peu de points. Nous présentons ensuite notre exploration de méthodes d'optimisation pour résoudre le même problème. Ces méthodes expriment les contraintes de notre problème comme un optimum ou un point critique d'une fonctionnelle, et cherchent à atteindre ces points à l'aide d'algorithmes tels que la descente de gradient ou l'algorithme de Newton. Cette approche n'a cependant pas donné de résultats concluants, les fonctions considérées étant trop bruitées pour se prêter à des algorithmes d'optimisation

On Higher-Order Probabilistic Subrecursion

International audienceWe study the expressive power of subrecursive probabilistic higher-order calculi. More specifically, we show that endowing a very expressive deterministic calculus like Gödel's T with various forms of probabilistic choice operators may result in calculi which are not equivalent as for the class of distributions they give rise to, although they all guarantee almost-sure termination. Along the way, we introduce a probabilistic variation of the classic reducibility technique, and we prove that the simplest form of probabilistic choice leaves the expressive power of T essentially unaltered. The paper ends with some observations about functional expressivity: expectedly, all the considered calculi represent precisely the functions which T itself represents

Symmetrized semi-discrete optimal transport

Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set of samples, while the other one remains continuous. In this paper we present a method to introduce some symmetry into the solution using coupled power diagrams. This symmetry is key to capturing the discontinuities of the transport map reflected in the geometry of the power cells. We design our method as a fixed-point algorithm alternating between computations of semi-discrete transport maps and recentering of the sites. The resulting objects are coupled power diagrams with identical geometry, allowing us to approximate displacement interpolation through linear interpolation of the meshes vertices. Through these coupled power diagrams, we have a natural way of jointly sampling measures

Symmetrized semi-discrete optimal transport

Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set of samples, while the other one remains continuous. In this paper we present a method to introduce some symmetry into the solution using coupled power diagrams. This symmetry is key to capturing the discontinuities of the transport map reflected in the geometry of the power cells. We design our method as a fixed-point algorithm alternating between computations of semi-discrete transport maps and recentering of the sites. The resulting objects are coupled power diagrams with identical geometry, allowing us to approximate displacement interpolation through linear interpolation of the meshes vertices. Through these coupled power diagrams, we have a natural way of jointly sampling measures

Symmetrized semi-discrete optimal transport

Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set of samples, while the other one remains continuous. In this paper we present a method to introduce some symmetry into the solution using coupled power diagrams. This symmetry is key to capturing the discontinuities of the transport map reflected in the geometry of the power cells. We design our method as a fixed-point algorithm alternating between computations of semi-discrete transport maps and recentering of the sites. The resulting objects are coupled power diagrams with identical geometry, allowing us to approximate displacement interpolation through linear interpolation of the meshes vertices. Through these coupled power diagrams, we have a natural way of jointly sampling measures

Burosumab and Dental Abscesses in Children With X‐Linked Hypophosphatemia

International audienc

Prevalence of Enthesopathies in Adults With X-linked Hypophosphatemia: Analysis of Risk Factors

International audienceAbstract Context Enthesopathies are the determinant of a poor quality of life in adults with X-linked hypophosphatemia (XLH). Objective To describe the prevalence of patients with enthesopathies and to identify the risk factors of having enthesopathies. Methods Retrospective study in the French Reference Center for Rare Diseases of the Calcium and Phosphate Metabolism between June 2011 and December 2020. Adult XLH patients with full body X-rays performed using the EOS® low-dose radiation system and clinical data collected from medical records. The main outcome measures were demographics, PHEX mutation, conventional treatment, and dental disease with the presence of enthesopathies. Results Of the 114 patients included (68% women, mean age 42.2 ± 14.3 years), PHEX mutation was found in 105 patients (94.6%), 86 (77.5%) had been treated during childhood. Enthesopathies (spine and/or pelvis) were present in 67% of the patients (n = 76). Patients with enthesopathies were significantly older (P = .001) and more frequently reported dental disease collected from medical records (P = .03). There was no correlation between the PHEX mutations and the presence of enthesopathies. Sixty-two patients had a radiographic dental examination in a reference center. Severe dental disease (number of missing teeth, number of teeth endodontically treated, alveolar bone loss, and proportion of patients with 5 abscesses or more) was significantly higher in patients with enthesopathies. Conclusion Adult XLH patients have a high prevalence of enthesopathies in symptomatic adults patients with XLH seen in a reference center. Age and severe dental disease were significantly associated with the presence of enthesopathies