Power of the Spacing test for Least-Angle Regression

Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac-Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the spacing test to the frame of unknown noise variance. More precisely, we investigate the power of the spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level $\alpha$. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the spacing test for LARS and the Pearson's chi-squared test (goodness of fit).Comment: 22 pages, 8 figure

A Rice method proof of the Null-Space Property over the Grassmannian

The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of $\ell\_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.Comment: 18 Pages, some Figure

Gaussian geometry and tools for compressed sensing

Cette thèse s'inscrit dans le cadre de l'analyse statistique en grande dimension. Plus précisé- ment, l'objet de cette thèse est d'étudier la possible application d'outils issus de la théorie des processus Gaussiens afin de redémontrer certaines propriétés des matrices à entrées Gaussiennes et d'étendre certaines procédures de test du modèle linéaire Gaussien standard. Dans la première partie, nous nous concentrons sur les matrices Gaussiennes. Notre objectif est de démontrer, via des formules du type Kac-Rice, qu'une telle matrice satisfait, avec très grande probabilité, la Null Space Property (NSP) et la Propriété d'Isométrie Restreinte (RIP). De plus, nous déduisons des transitions de phases dépendant des paramètres classiques de la régression parcimonieuse, à savoir le nombre d'observations, le nombre de prédicteurs et le degré de sparsité. Dans la seconde partie, nous traitons le cas du test de nullité globale des paramètres pour le modèle linéaire Gaussien, afin de l'appliquer au cas de la sélection de modèle. Dans ces travaux, qui s'inscrivent dans la continuité de Taylor, Loftus et Tibshirani, nous proposons un test non conditionnel pour l'hypothèse de nullité globale dans le cadre du lasso et discutons autour de sa puissance. De plus, nous généralisons ces résultats aux processus Gaussiens, pour inclure, par exemple, la cas de la super-résolution. Dans une troisième partie, nous présentons quelques applications de la formule de Rice visantà calculer la fonction de répartition du maximum d'un processus Gaussien afin d'en déduire une version numériquement implémentable. Dans un deuxième temps, nous discutons de l'efficacité ou non de certaines approximations classiques pour la fonction de répartition du maximum. Enfin, nous étudions le comportement asymptotique du nombre de franchissements d'un niveau donné u sur un intervalle de temps [0,T] pour un processus Gaussien dérivable.This thesis fallin within the context of high-dimensional data analysis. More specificaly, the purpose is to study the possible application of some Gaussian tools to prove classical results on matrices with Gaussian entries and to extend existing test procedures for Gaussian linear models. In a first part, we focus on Gaussian matrices. Our aim is to prove, using a Kac-Rice formula on Gaussian processes, that such a matrice satisfies, with overwhelming probability, the Null Space Property (NSP) and the Restricted Isometry Property (RIP). Moreover, we derive phase transition graphs depending on the classical parameters of sparse regression, namely the number of observations, the number of predictors and the level of sparsity. In a second part, we deal with global null testing for Gaussian linear models, with application to Compressed Sensing. Following recent works of Taylor, Loftus and Tibshirani, we purpose a test for global null hypothesis in the lasso case and discuss about its power. Furthermore, we generalize these method to Gaussian processes, to include, for instance, the super-resolution case. In a third part, we present some applications of Rice formula to compute the cumulative distribution function of the maximum of a Gaussian process and derive corresponding numerical routines to investigate the efficiency of classical approximations. Finaly, we consider the asymp- totical comportement of the number of crossings of a differentiable Gaussian process for a given level u and time interval [0,T]

Dix ans d’interopérabilité, 1989 - 1999

La décennie 1989-1999 correspond à la période au cours de laquelle le réseau transeuropéen de trains à grande vitesse a pris forme et s’est développé. Les trains circulant sur ce réseau sont constitués de rames indéformables articulées ou non, composées d’un nombre d’éléments fixes. C’est dans ce contexte de réseau transeuropéen et de trains aptes à circuler sur l’ensemble de ce réseau que le concept d’interopérabilité a été élaboré et mis en œuvre progressivement en vue de surmonter les diff..

A Rice method proof of the Null-Space Property over the Grassmannian

18 Pages, some Figures.International audienceThe Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of $\ell_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP

El nuevo enfoque y la especificación técnica en el marco de la propuesta de directriz "productos para la construcción"

This article presents an interesting analysis of the EEC's proposed guidelines concerning "construction products". Two specific particularities of construction products with respect to other industrial products are pointed out, with the object of harmonizing the rules for free interchange: first, the necessity of evaluating their suitability for use not in an isolated fashion but as forming a part of the building as a whole; and second, the need to rely on the concept of contribution (utility) rather than base standardization on the traditional concept of the simple technical specification of characteristics.El presente artículo supone un interesante análisis sobre la propuesta de directriz de la CCE relativa a los "productos de construcción". En él se señalan como particularidades específicas de los productos para la construcción respecto a otros productos Industriales, a efectos de la armonización de las normas para el libre intercambio, la necesidad, por un lado, de evaluar su aptitud de empleo no de forma aislada sino formando parte del edificio como objeto final, y por otro, la necesidad de apoyarse en el concepto de prestación en lugar de basarse para su normalización en el concepto tradicional de simple especificación técnica de sus características

DNA demethylation agent 5azadC downregulates HPV16 E6 expression in cervical cancer cell lines independently of TBX2 expression

HPV16 is the most carcinogenic human papillomavirus and causes >50% of cervical cancers, the majority of anal cancers and 30% of oropharyngeal squamous cell carcinomas. HPV carcinogenesis relies on the continuous expression of the two main viral oncoproteins E6 and E7 that target >150 cellular proteins. Among them, epigenetic modifiers, including DNA Methyl Transferases (DNMT), are dysregulated, promoting an aberrant methylation pattern in HPV-positive cancer cells. It has been previously reported that the treatment of HPV-positive cervical cancer cells with DNMT inhibitor 5-aza-2'-deoxycytidine (5azadC) caused the downregulation of E6 expression due to mRNA destabilization that was mediated by miR-375. Recently, the T-box transcription factor 2 (TBX2) has been demonstrated to repress HPV LCR activity. In the current study, the role of TBX2 in E6 repression was investigated in HPV16 cervical cancer cell lines following 5azadC treatment. A decrease of E6 expression was accompanied by p53 and p21 restoration. While TBX2 mRNA was upregulated in 5azadC-treated SiHa and Ca Ski cells, TBX2 protein was not detectable. Furthermore, the overexpression of TBX2 protein in cervical cancer cells did not allow the repression of E6 expression. The TBX2 transcription factor is therefore unlikely to be associated with the repression of E6 following 5azadC treatment of SiHa and Ca Ski cells. © 2020 Spandidos Publications. All rights reserved

Cadmium pigments in consumer products and their health risks

© 2018 Elsevier B.V. Cadmium is a toxic heavy metal that has been increasingly regulated over the past few decades. The main exposure routes for the general public are the consumption of certain foods and the inhalation of cigarette smoke. However, additional exposure may occur through the current and historical use of the metal in consumer products. In this paper, the uses of Cd in consumer goods are reviewed, with the focus on brightly-coloured Cd sulphide and sulphoselenide pigments, and measurements of Cd in historical and contemporary products ascertained by XRF are reported. Cadmium is encountered across a wide range of contemporary plastic products, mainly because of the unregulated recycling of electronic waste and polyvinyl chloride. However, concentrations are generally low (<100 μg g−1), conforming with current limits and posing minimal risk to consumers. Of greater concern is high concentrations of pigmented Cd (up to 2% by weight) in old products, and in particular children's toys that remain in circulation. Here, tests conducted suggest that Cd migration in some products exceeds the Toy Safety Directive limit of 17 μg g−1 by an order of magnitude. The principal current use of Cd pigments is in ceramic products where the metal is encapsulated and overglazed. Leaching tests on new and secondhand items of hollowware indicate compliance with respect to the current Cd limit of 300 μg L−1, but that non-compliance could occur for items of earthenware or damaged articles should a proposed limit of 5 μg L−1 be introduced. The greatest consumer risk identified is the use of Cd pigments in the enamels of decorated drinking glasses. Thus, while décor is restricted to the exterior, any enamel within the lip area is subject to ready attack from acidic beverages because the pigments are neither encapsulated nor overglazed. Glass bottles decorated with Cd-based enamel do not appear to represent a direct health hazard but have the propensity to contaminate recycled glass products. It is recommended that decorated glassware is better regulated and that old, brightly-coloured toys are treated cautiously

Méthodes gaussiennes et application à l'acquisition comprimée

Cette thèse s'inscrit dans le cadre de l'analyse statistique en grande dimension. Plus précisé- ment, l'objet de cette thèse est d'étudier la possible application d'outils issus de la théorie des processus Gaussiens afin de redémontrer certaines propriétés des matrices à entrées Gaussiennes et d'étendre certaines procédures de test du modèle linéaire Gaussien standard. Dans la première partie, nous nous concentrons sur les matrices Gaussiennes. Notre objectif est de démontrer, via des formules du type Kac-Rice, qu'une telle matrice satisfait, avec très grande probabilité, la Null Space Property (NSP) et la Propriété d'Isométrie Restreinte (RIP). De plus, nous déduisons des transitions de phases dépendant des paramètres classiques de la régression parcimonieuse, à savoir le nombre d'observations, le nombre de prédicteurs et le degré de sparsité. Dans la seconde partie, nous traitons le cas du test de nullité globale des paramètres pour le modèle linéaire Gaussien, afin de l'appliquer au cas de la sélection de modèle. Dans ces travaux, qui s'inscrivent dans la continuité de Taylor, Loftus et Tibshirani, nous proposons un test non conditionnel pour l'hypothèse de nullité globale dans le cadre du lasso et discutons autour de sa puissance. De plus, nous généralisons ces résultats aux processus Gaussiens, pour inclure, par exemple, la cas de la super-résolution. Dans une troisième partie, nous présentons quelques applications de la formule de Rice visantà calculer la fonction de répartition du maximum d'un processus Gaussien afin d'en déduire une version numériquement implémentable. Dans un deuxième temps, nous discutons de l'efficacité ou non de certaines approximations classiques pour la fonction de répartition du maximum. Enfin, nous étudions le comportement asymptotique du nombre de franchissements d'un niveau donné u sur un intervalle de temps [0,T] pour un processus Gaussien dérivable.This thesis fallin within the context of high-dimensional data analysis. More specificaly, the purpose is to study the possible application of some Gaussian tools to prove classical results on matrices with Gaussian entries and to extend existing test procedures for Gaussian linear models. In a first part, we focus on Gaussian matrices. Our aim is to prove, using a Kac-Rice formula on Gaussian processes, that such a matrice satisfies, with overwhelming probability, the Null Space Property (NSP) and the Restricted Isometry Property (RIP). Moreover, we derive phase transition graphs depending on the classical parameters of sparse regression, namely the number of observations, the number of predictors and the level of sparsity. In a second part, we deal with global null testing for Gaussian linear models, with application to Compressed Sensing. Following recent works of Taylor, Loftus and Tibshirani, we purpose a test for global null hypothesis in the lasso case and discuss about its power. Furthermore, we generalize these method to Gaussian processes, to include, for instance, the super-resolution case. In a third part, we present some applications of Rice formula to compute the cumulative distribution function of the maximum of a Gaussian process and derive corresponding numerical routines to investigate the efficiency of classical approximations. Finaly, we consider the asymp- totical comportement of the number of crossings of a differentiable Gaussian process for a given level u and time interval [0,T]