arXiv subjects

Thierry Klein

Publications and source records attributed to Thierry Klein.

At least 37 records · Page 2Linked to original sources

arXiv2020

Wide critical fluctuations of the field-induced phase transition in graphite

In the immediate vicinity of the critical temperature (T$_c$) of a phase transition, there are fluctuations of the order parameter, which reside beyond the mean-field approximation. Such critical fluctuations usually occur in a very narrow temperature window in contrast to Gaussian fluctuations. Here, we report on a study of specific heat in graphite subject to high magnetic field when all carriers are confined in the lowest Landau levels. The observation of a BCS-like specific heat jump in both temperature and field sweeps establishes that the phase transition discovered decades ago in graphite is of the second-order. The jump is preceded by a steady field-induced enhancement of the electronic specific heat. A modest (20 percent) reduction in the amplitude of the magnetic field (from 33 T to 27 T) leads to a threefold decrease of T$_c$ and a drastic widening of the specific heat anomaly, which acquires a tail spreading to two times T$_c$. We argue that the steady departure from the mean-field BCS behavior is the consequence of an exceptionally large Ginzburg number in this dilute metal, which grows steadily as the field lowers. Our fit of the critical fluctuations indicates that they belong to the $3DXY$ universality class, similar to the case of $^4$He superfluid transition.

cond-mat.str-el↗

arXiv2020

Large deviation results for triangular arrays of semiexponential random variables

Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated , in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when X 1 has a semiexponential distribution (see, [16, 17]). In the same setting, the authors of [4] derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition. In this paper, we exhibit the same asymptotic behaviour for triangular arrays of semiexponentially distributed random variables, no more supposed absolutely continuous.

math.PR↗

arXiv2020

Global sensitivity analysis and Wasserstein spaces

Sensitivity indices are commonly used to quantity the relative inuence of any specic group of input variables on the output of a computer code. In this paper, we focus both on computer codes the output of which is a cumulative distribution function and on stochastic computer codes. We propose a way to perform a global sensitivity analysis for these kinds of computer codes. In the rst setting, we dene two indices: the rst one is based on Wasserstein Fr{é}chet means while the second one is based on the Hoeding decomposition of the indicators of Wasserstein balls. Further, when dealing with the stochastic computer codes, we dene an ideal version of the stochastic computer code thats ts into the frame of the rst setting. Finally, we deduce a procedure to realize a second level global sensitivity analysis, namely when one is interested in the sensitivity related to the input distributions rather than in the sensitivity related to the inputs themselves. Several numerical studies are proposed as illustrations in the dierent settings.

math.ST↗

arXiv2020

High density of states in the pseudogap phase of the cuprate superconductor HgBa$_2$CuO$_{4 + δ}$

The specific heat $C$ of the single-layer cuprate superconductor HgBa$_2$CuO$_{4 + δ}$ was measured in an underdoped crystal with $T_{\rm c} = 72$ K at temperatures down to $2$ K in magnetic fields up to $35$ T, a field large enough to suppress superconductivity at that doping ($p \simeq 0.09$). In the normal state at $H = 35$ T, a residual linear term of magnitude $γ= 12 \pm 2$ mJ/K$^2$mol is observed in $C/T$ as $T \to 0$, a direct measure of the electronic density of states. This high value of $γ$ has two major implications. First, it is significantly larger than the value measured in overdoped cuprates outside the pseudogap phase ($p >p^\star$), such as La$_{2-x}$Sr$_x$CuO$_4$ and Tl$_2$Ba$_2$CuO$_{6 + δ}$ at $p \simeq 0.3$, where $γ\simeq 7$ mJ/K$^2$mol. Given that the pseudogap causes a loss of density of states, and assuming that HgBa$_2$CuO$_{4 + δ}$ has the same $γ$ value as other cuprates at $p \simeq 0.3$, this implies that $γ$ in HgBa$_2$CuO$_{4 + δ}$ must peak between $p \simeq 0.09$ and $p \simeq 0.3$, namely at (or near) the critical doping $p^\star$ where the pseudogap phase is expected to end ($p^\star\simeq 0.2$). Secondly, the high $γ$ value implies that the Fermi surface must consist of more than the single electron-like pocket detected by quantum oscillations in HgBa$_2$CuO$_{4 + δ}$ at $p \simeq 0.09$, whose effective mass $m^\star= 2.7\times m_0$ yields only $γ= 4.0$ mJ/K$^2$mol. This missing mass imposes a revision of the current scenario for how pseudogap and charge order respectively transform and reconstruct the Fermi surface of cuprates.

cond-mat.supr-con↗

arXiv2019

Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes

The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of non-Gaussian processes obtained by regular non-linear transformations of Gaussian processes. We provide the increasing-domain asymptotic properties of the (Gaussian) maximum likelihood and cross validation estimators of the covariance parameters of a non-Gaussian process of this class. We show that these estimators are consistent and asymptotically normal, although they are defined as if the process was Gaussian. They do not need to model or estimate the non-linear transformation. Our results can thus be interpreted as a robustness of (Gaussian) maximum likelihood and cross validation towards non-Gaussianity. Our proofs rely on two technical results that are of independent interest for the increasing-domain asymptotic literature of spatial processes. First, we show that, under mild assumptions, coefficients of inverses of large covariance matrices decay at an inverse polynomial rate as a function of the corresponding observation location distances. Second, we provide a general central limit theorem for quadratic forms obtained from transformed Gaussian processes. Finally, our asymptotic results are illustrated by numerical simulations.

math.ST↗

arXiv2018

A case study : Influence of Dimension Reduction on regression trees-based Algorithms -Predicting Aeronautics Loads of a Derivative Aircraft

In aircraft industry, market needs evolve quickly in a high competitiveness context. This requires adapting a given aircraft model in minimum time considering for example an increase of range or the number of passengers (cf A330 NEO family). The computation of loads and stress to resize the airframe is on the critical path of this aircraft variant definition: this is a consuming and costly process, one of the reason being the high dimen-sionality and the large amount of data. This is why Airbus has invested since a couple of years in Big Data approaches (statistic methods up to machine learning) to improve the speed, the data value extraction and the responsiveness of this process. This paper presents recent advances in this work made in cooperation between Airbus, ENAC and Institut de Math{é}-matiques de Toulouse in the framework of a proof of value sprint project. It compares the influence of three dimensional reduction techniques (PCA, polynomial fitting, combined) on the extrapolation capabilities of Regression Trees based algorithms for loads prediction. It shows that AdaBoost with Random Forest offers promising results in average in terms of accuracy and computational time to estimate loads on which a PCA is applied only on the outputs.

cs.LG↗

arXiv2018

Semi Parametric Estimations of rotating and scaling parameters for aeronautic loads

In this paper, we perform registration of noisy curves. We provide an appropriate model in estimating the rotation and scaling parameters to adjust a set of curves through a M-estimation procedure. We prove the consistency and the asymptotic normality of our estimators. Numerical simulation and a real life aeronautic example are given to illustrate our methodology.

stat.AP↗

arXiv2018

Prediction of Preliminary Maximum Wing Bending Moments under Discrete Gust

Many methodologies have been proposed to quickly identify among a very large number of flight conditions and maneuvers (i.e., steady, quasi-steady and unsteady loads cases) the ones which give the worst values for structural sizing (e.g., bending moments, shear forces, torques,...). All of these methods use both the simulation model of the aircraft under development and efficient algorithms to find out the critical points of the flight envelope. At the preliminary structural design phases detailed models are not available and airframe's loads are estimated by empirical relationships or engineering judgments. These approximations can induce load uncertainties and may lead to expensive redesign activities through the upcoming detailed sizing process. In the context of preliminary design phase for a weight aircraft variant without geometric change, to overcome this likely drawback, we propose a method based on the huge and reliable database of an initial aircraft from which the weight variant belongs. More precisely, from the load cases of this initial database, response surfaces are identified as functions of preliminary parameters (flight conditions and structural parameters). Then, these response surfaces are used to predict quickly the weight aircraft variant quantities of interest for preliminary structural design studies. Although the proposed method can be readily extended to any structural quantity of interest and to any flight conditions and maneuvers, it is presented here for the prediction of the bending moments due to discrete gust at different locations along a wing span.

cs.CE↗

arXiv2017

Sensitivity analysis based on Cram{é}r von Mises distance

In this paper, we first study a new sensitivity index that is based on higher moments and generalizes the so-called Sobol one. Further, following an idea of Borgonovo ([3]), we define and study a new sensitivity index based on the Cram{é}r von Mises distance. This new index appears to be more general than the Sobol one as it takes into account, not only the variance, but the whole distribution of the random variable. Furthermore, we study the statistical properties of a Monte Carlo estimate of this new index.

math.PR↗

arXiv2017

Characterization of barycenters in the Wasserstein space by averaging optimal transport maps

This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier (2011), and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.

math.ST↗

arXiv2017

A Central Limit Theorem for Wasserstein type distances between two different laws

This article is dedicated to the estimation of Wasserstein distances and Wasserstein costs between two distinct continuous distributions $F$ and $G$ on $\mathbb R$. The estimator is based on the order statistics of (possibly dependent) samples of $F$ resp. $G$. We prove the consistency and the asymptotic normality of our estimators. \begin{it}Keywords:\end{it} Central Limit Theorems- Generelized Wasserstein distances- Empirical processes- Strong approximation- Dependent samples.

math.ST↗

arXiv2017

Minimax convergence rate for estimating the Wasserstein barycenter of random measures on the real line

This paper is focused on the statistical analysis of probability measures $ν_{1},\ldots,ν_{n}$ on $\mathbb{R}$ that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures $ν_{i}$ are absolutely continuous with densities $f_{i}$ that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables $(X_{i,j})_{1 \leq i \leq n; \; 1 \leq j \leq p_{i} }$ organized in the form of $n$ experimental units, such that $X_{i,1},\ldots,X_{i,p_{i}}$ are iid observations sampled from a random measure $ν_{i}$ for each $1 \leq i \leq n$. In this setting, we focus on first-order statistics methods for estimating, from such data, a meaningful structural mean measure. For the purpose of taking into account phase and amplitude variations in the observations, we argue that the notion of Wasserstein barycenter is a relevant tool. The main contribution of this paper is to characterize the rate of convergence of a (possibly smoothed) empirical Wasserstein barycenter towards its population counterpart in the asymptotic setting where both $n$ and $\min_{1 \leq i \leq n} p_{i}$ may go to infinity. The optimality of this procedure is discussed from the minimax point of view with respect to the Wasserstein metric. We also highlight the connection between our approach and the curve registration problem in statistics. Some numerical experiments are used to illustrate the results of the paper on the convergence rate of empirical Wasserstein barycenters.

math.ST↗

arXiv2016

Dynamically current-driven de Gennes -St James states in Metal-Superconductor junctions

We investigate the conductance of a Normal-Normal'-Superconductor (NN'S) junction, in which current injection destroys superconductivity in a small region N' of the superconductor, with a size varying with the applied voltage V. Voltage-dependent de Gennes-Saint James (dGSJ) bound states appearing in the N' slab lead to two distinct sets of conductance oscillations. We show that this effect significantly alters the conductance of systems for which $κ^2v_F \sim 10^9$ m/s such as pnictides ($κ$ and $v_F$ being the Ginzburg number and the Fermi velocity, respectively), and we discuss their consequences on the identification of the bosonic modes of strongly coupled superconductors.

cond-mat.supr-con↗

arXiv2015

New Fréchet features for random distributions and associated sensitivity indices

In this article we define new FrÈchet features for random cumulative distribution functions using contrast. These contrasts allow to construct Wasserstein costs and our new features minimize the average costs as the FrÈchet mean minimizes the mean square Wasserstein$_2$ distance. An example of new features is the median, and more generally the quantiles. From these definitions, we are able to define sensitivity indices when the random distribution is the output of a stochastic code. Associated to the FrÈchet mean we extend the Sobol indices, and in general the indices associated to a contrast that we previously proposed.

math.ST↗

arXiv2015

A conditional Berry-Esseen bound and a conditional large deviation result without Laplace transform. Application to hashing with linear probing

\noindent We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. We prove a Berry-Esseen bound in a general setting and a large deviation result when the Laplace transform of the underlying distribution is not defined in a neighborhood of zero. Then we present several combinatorial applications. In particular, we prove a large deviation result for the model of hashing with linear probing.

math.PR↗

arXiv2014

Classification with the nearest neighbor rule in general finite dimensional spaces: necessary and sufficient conditions

Given an $n$-sample of random vectors $(X_i,Y_i)_{1 \leq i \leq n}$ whose joint law is unknown, the long-standing problem of supervised classification aims to \textit{optimally} predict the label $Y$ of a given a new observation $X$. In this context, the nearest neighbor rule is a popular flexible and intuitive method in non-parametric situations. Even if this algorithm is commonly used in the machine learning and statistics communities, less is known about its prediction ability in general finite dimensional spaces, especially when the support of the density of the observations is $\mathbb{R}^d$. This paper is devoted to the study of the statistical properties of the nearest neighbor rule in various situations. In particular, attention is paid to the marginal law of $X$, as well as the smoothness and margin properties of the \textit{regression function} $η(X) = \mathbb{E}[Y | X]$. We identify two necessary and sufficient conditions to obtain uniform consistency rates of classification and to derive sharp estimates in the case of the nearest neighbor rule. Some numerical experiments are proposed at the end of the paper to help illustrate the discussion.

math.ST↗

arXiv2014

Geodesic PCA in the Wasserstein space

We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable functions, with respect to an appropriate measure. Therefore, we consider the general problem of PCA in a closed convex subset of a separable Hilbert space, which serves as basis for the analysis of GPCA and also has interest in its own right. We provide illustrative examples on simple statistical models, to show the benefits of this approach for data analysis. The method is also applied to a real dataset of population pyramids.

stat.ME↗

arXiv2013

Sensitivity analysis for multidimensional and functional outputs

Let $X:=(X_1, \ldots, X_p)$ be random objects (the inputs), defined on some probability space $(Ω,{\mathcal{F}}, \mathbb P)$ and valued in some measurable space $E=E_1\times\ldots \times E_p$. Further, let $Y:=Y = f(X_1, \ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Y\in\mathbb R$ ), when the output belongs to $\mathbb{H}$. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme.

stat.AP↗