Frédéric Cérou’s research while affiliated with French National Centre for Scientific Research and other places

This paper discusses a novel strategy for simulating rare events and an associated Monte Carlo estimation of tail probabilities. Our method uses a system of interacting particles and exploits a Feynman-Kac representation of that system to analyze their fluctuations. Our precise analysis of the variance of a standard multilevel splitting algorithm reveals an opportunity for improvement. This leads to a novel method that relies on adaptive levels and produces, in the limit of an idealized version of the algorithm, estimates with optimal variance. The motivation for this theoretical work comes from problems occurring in watermarking and fingerprinting of digital contents, which represents a new field of applications of rare event simulation techniques. Some numerical results show performance close to the idealized version of our technique for these practical applications. KeywordsRare event–Sequential importance sampling–Feynman-Kac formula–Metropolis-Hastings–Fingerprinting–Watermarking

We present an enhanced version of the splitting method, called the smoothed splitting method (SSM), for counting associated with complex sets, such as the set defined by the constraints of an integer program and in particular for counting the number of satisfiability assignments. Like the conventional splitting algorithms, ours uses a sequential sampling plan to decompose a “difficult” problem into a sequence of “easy” ones. The main difference between SSM and splitting is that it works with an auxiliary sequence of continuous sets instead of the original discrete ones. The rationale of doing so is that continuous sets are easier to handle. We show that while the proposed method and its standard splitting counterpart are similar in their CPU time and variability, the former is more robust and more flexible than the latter.

We present a nonasymptotic theorem for interacting particle approximations of unnormalized Feynman–Kac models. We provide an original stochastic analysis-based on Feynman–Kac semigroup techniques combined with recently developed coa-lescent tree-based functional representations of particle block distributions. We present some regularity conditions under which the L 2 -relative error of these weighted particle measures grows linearly with respect to the time horizon yielding what seems to be the first results of this type for this class of unnormalized models. We also illustrate these results in the context of particle absorption models, with a special interest in rare event analysis. Résumé. Nous présentons un théorème non asymptotique pour les approximation par systèmes de particules en interaction des modèles de Feynman–Kac non normalisés. Nous introduisons une analyse stochastique originale basée sur des techniques de semigroupes de Feynman–Kac, associées avec les représentation, récemment proposées, des distributions de blocks de particules, en terme de développement en arbre de coalescence. Nous présentons des conditions de régularité sous lesquelles l'erreur relative L 2 de ces mesures particulaires pondérées croît linéairement par rapport 'a l'horizon temporel, conduisant 'a ce qui semble être le premier résultat de ce type pour cette classe de modèles non normalisés. Nous illustrons ces résultats dans le contexte des mesures statiques de Boltzmann–Gibbs et des distributions restreintes, avec un intéret partuculier pour les événements rares.

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.

2 A simple mathematical model

Let F be a separable Banach space, and let (X, Y) be a random pair taking values in F × R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate r_n (x) of the regression function r(x) = E[Y|X = x], based on n independent copies of the pair (X, Y). Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference E[r_n(X) - r(X)]², and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces, and reproducing kernel Hilbert spaces.

Assessing that a probability of false alarm is below a given significance level is a crucial issue in watermarking. We propose an iterative and self-adapting algorithm which estimates very low probabilities of error. Some experimental investigations validates its performance for a rare detection scenario where there exists a close form formula of the probability of false alarm. Our algorithm appears to be much quicker and more accurate than a classical Monte Carlo estimator. It even allows the experimental measurement of error exponents.

This paper estimates the minimal length of a binary proba- bilistic traitor tracing code. We consider the code construction proposed by G. Tardos in 2003, with the symmetric accusation function as im- proved by B. Skoric et al. The length estimation is based on two pillars. First, we consider the Worst Case Attack that a group of c colluders can lead. This attack minimizes the mutual information between the code sequence of a colluder and the pirated sequence. Second, an algorithm pertaining to the field of rare event analysis is presented in order to es- timate the probabilities of error: the probability that an innocent user is framed, and the probabilities that all colluders are missed. Therefore, for a given collusion size, we are able to estimate the minimal length of the code satisfying some error probabilities constraints. This estimation is far lower than the known lower bounds.

This paper discusses the rare event simulation for a fixed probability law. The motivation comes from problems occurring in watermarking and fingerprinting of digital contents, which is a new application of rare event simulation techniques. We provide two versions of our algorithm, and discuss the convergence properties and implementation issues. A discussion on recent related works is also provided. Finally, we give some numerical results in watermarking context.

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.