On the length of one-dimensional reactive paths

Motivated by some numerical observations on molecular dynamics simulations, we analyze metastable trajectories in a very simplecsetting, namely paths generated by a one-dimensional overdamped Langevin equation for a double well potential. More precisely, we are interested in so-called reactive paths, namely trajectories which leave definitely one well and reach the other one. The aim of this paper is to precisely analyze the distribution of the lengths of reactive paths in the limit of small temperature, and to compare the theoretical results to numerical results obtained by a Monte Carlo method, namely the multi-level splitting approach

Nearest neighbor classification in infinite dimension

Let $X$ be a random element in a metric space (\calF,d), and let $Y$ be a random variable with value $0$ or $1$. $Y$ is called the class, or the label, of $X$. Assume $n$ i.i.d. copies (X_i,Y_i)_1\leqi\leqn. The problem of classification is to predict the label of a new random element $X$. The $k$-nearest neighbor classifier consists in the simple following rule : look at the $k$ nearest neighbors of $X$ and choose $0$ or $1$ for its label according to the majority vote. If (\calF,d)=(R^d,||.||), Stone has proved in 1977 the universal consistency of this classifier : its probability of error converges to the Bayes error, whatever the distribution of $(X,Y)$. We show in this paper that this result is no more valid in general metric spaces. However, if (\calF,d) is separable and if a regularity condition is assumed, then the $k$-nearest neighbor classifier is weakly consistent

A multiple replica approach to simulate reactive trajectories

A method to generate reactive trajectories, namely equilibrium trajectories leaving a metastable state and ending in another one is proposed. The algorithm is based on simulating in parallel many copies of the system, and selecting the replicas which have reached the highest values along a chosen one-dimensional reaction coordinate. This reaction coordinate does not need to precisely describe all the metastabilities of the system for the method to give reliable results. An extension of the algorithm to compute transition times from one metastable state to another one is also presented. We demonstrate the interest of the method on two simple cases: a one-dimensional two-well potential and a two-dimensional potential exhibiting two channels to pass from one metastable state to another one

On the Rate of Convergence of the Functional kk-NN Estimates

Let $\mathcal F$ be a general separable metric space and denote by \mathcal D_n=\{(\bX_1,Y_1), \hdots, (\bX_n,Y_n)\} independent and identically distributed $\mathcal F\times \mathbb R$-valued random variables with the same distribution as a generic pair (\bX, Y). In the regression function estimation problem, the goal is to estimate, for fixed \bx \in \mathcal F, the regression function r(\bx)=\mathbb E[Y|\bX=\bx] using the data $\mathcal D_n$. Motivated by a broad range of potential applications, we propose, in the present contribution, to investigate the properties of the so-called $k_n$-nearest neighbor regression estimate. We present explicit general finite sample upper bounds, and particularize our results to important function spaces, such as reproducing kernel Hilbert spaces, Sobolev spaces or Besov spaces

New insights into Approximate Bayesian Computation

International audienceApproximate Bayesian Computation (ABC for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the present paper, we analyze the procedure from the point of view of k-nearest neighbor theory and explore the statistical properties of its outputs. We discuss in particular some asymptotic features of the genuine conditional density estimate associated with ABC, which is an interesting hybrid between a k-nearest neighbor and a kernel method

Sur la vitesse de convergence de l'estimateur du plus proche voisin baggé

International audienceOn s'intéresse dans cette communication à l'estimation de la fonction de

On the Hill relation and the mean reaction time for metastable processes

We illustrate how the Hill relation and the notion of quasi-stationary distribution can be used to analyse the error introduced by many algorithms that have been proposed in the literature, in particular in molecular dynamics, to compute mean reaction times between metastable states for Markov processes. The theoretical findings are illustrated on various examples demonstrating the sharpness of the error analysis as well as the applicability of our study to elliptic diffusions

Mathematical statistics.

Approximate Bayesian Computation (abc for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the present paper, we analyze the procedure from the point of view of k-nearest neighbor theory and explore the statistical properties of its outputs. We discuss in particular some asymptotic features of the genuine conditional density estimate associated with abc, which is an interesting hybrid between a k-nearest neighbor and a kernel method

Estimation de quantiles extrêmes et de probabilités d'événements rares

International audienceEtant donné une probabilité $\mu$ sur $\R^d$ ($d$ grand), on note $X$ un vecteur aléatoire générique de loi $\mu$ et $\Phi:\R^d\rightarrow\R$ une application ``boîte noire''. Un réel $q$ étant fixé, le but est de générer un échantillon i.i.d. $(X_1,\dots,X_N)$ tel que pour tout $i$ : $X_i\sim{\cal L}(X|\Phi(X)>q)$. Lorsque $q$ est grand comparé aux valeurs typiques de la variable $\Phi(X)$, la méthode Monte-Carlo classique devient trop coûteuse. Dans ce travail nous présentons et analysons une nouvelle approche pour ce problème. Celle-ci procède en plusieurs étapes, s'inspirant de l'algorithme de Metropolis-Hastings et des méthodes dites multi-niveaux en estimation d'événements rares. Deux problèmes peuvent être traités très facilement via cette nouvelle méthode : estimation de quantiles extrêmes et estimation d'événements rares. Les idées présentées seront illustrées sur un problème de tatouage numérique