Bernard Delyon’s research while affiliated with French National Centre for Scientific Research and other places

This paper addresses the problem of approximating an unknown probability distribution with density f -- which can only be evaluated up to an unknown scaling factor -- with the help of a sequential algorithm that produces at each iteration $n\geq 1$ an estimated density $q_n$.The proposed method optimizes the Kullback-Leibler divergence using a mirror descent (MD) algorithm directly on the space of density functions, while a stochastic approximation technique helps to manage between algorithm complexity and variability. One of the key innovations of this work is the theoretical guarantee that is provided for an algorithm with a fixed MD learning rate $\eta \in (0,1 )$. The main result is that the sequence $q_n$ converges almost surely to the target density f uniformly on compact sets. Through numerical experiments, we show that fixing the learning rate $\eta \in (0,1 )$ significantly improves the algorithm's performance, particularly in the context of multi-modal target distributions where a small value of $\eta$ allows to increase the chance of finding all modes. Additionally, we propose a particle subsampling method to enhance computational efficiency and compare our method against other approaches through numerical experiments.

In this paper, we prove a result of equivalence in law between a diffusion conditioned with respect to partial observations and an auxiliary process. By partial observations we mean coordinates (or linear transformation) of the process at a finite collection of deterministic times. Apart from the theoretical interest, this result allows to simulate the conditional diffusion through Monte Carlo methods, using the fact that the auxiliary process is easy to simulate.

The Kitanidis filter is a natural extension of the Kalman filter to systems subject to arbitrary unknown inputs or disturbances. Though the optimality of the Kitanidis filter was founded for general time varying systems more than 30 years ago, its boundedness and stability analysis is still limited to time invariant systems, up to the authors' knowledge. In the framework of general time varying systems, this paper establishes upper and lower bounds of the error covariance of the Kitanidis filter, as well as upper bounds of all the auxiliary variables involved in the filter. By preventing data overflow, upper bounds are crucial for all recursive algorithms in real time applications. The upper and lower bounds of the error covariance will also serve as the basis of the Kitanidis filter stability analysis, like in the case of time varying system Kalman filter.

As a natural extension of the Kalman filter to systems subject to arbitrary unknown inputs, the Kitanidis filter has been designed by one-step minimization of the trace of the state estimation error covariance matrix. In this technical communiqué, it is shown that the Kitanidis filter is also optimal for the whole gain sequence in the sense of matrix positive definiteness, which notably implies that the Kitanidis filter minimizes not only the trace criterion, but also the matrix spectral norm criterion.

Parametric motion models are commonly used in image sequence analysis for different tasks. A robust estimation framework is usually required to reliably compute the motion model over the estimation support in the presence of outliers, while the choice of the right motion model is also important to properly perform the task. However, dealing with model selection within a robust estimation setting remains an open question. We define two original propositions for robust motion-model selection. The first one is an extension of the Takeuchi information criterion. The second one is a new paradigm built from the Fisher statistic. We also derive an interpretation of the latter as a robust Mallows’ $C_P$ criterion. Both robust motion-model selection criteria are straightforward to compute. We have conducted a comparative objective evaluation on computer-generated image sequences with ground truth, along with experiments on real videos, for the parametric estimation of the 2D dominant motion in an image due to the camera motion. They demonstrate the interest and the efficiency of the proposed robust model-selection methods.

A key determinant of the success of Monte Carlo simulation is the sampling policy, the sequence of distribution used to generate the particles, and allowing the sampling policy to evolve adaptively during the algorithm provides considerable improvement in practice. The issues related to the adaptive choice of the sampling policy are addressed from a functional estimation point of view. %Uniform convergence of the sampling policy are established %The standard adaptive importance sampling approach is revisited The considered approach consists of modelling the sampling policy as a mixture distribution between a flexible kernel density estimate, based on the whole set of available particles, and a naive heavy tail density. When the share of samples generated according to the naive density goes to zero but not too quickly, two results are established. Uniform convergence rates are derived for the sampling policy estimate. A central limit theorem is obtained for the resulting integral estimates. The fact that the asymptotic variance is the same as the variance of an ``oracle'' procedure, in which the sampling policy is chosen as the optimal one, illustrates the benefits of the proposed approach.

Suppose that a mobile sensor describes a Markovian trajectory in the ambient space. At each time the sensor measures an attribute of interest, e.g., the temperature. Using only the location history of the sensor and the associated measurements, the aim is to estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g., Monte Carlo, the proposed approach does not require any knowledge on the distribution of the sensor trajectory. Probabilistic bounds on the convergence rates of the estimator are established. These rates are better than the traditional "root n"-rate, where n is the sample size, attached to other probabilistic integration methods. For finite sample sizes, the good behaviour of the procedure is demonstrated through simulations and an application to the evaluation of the average temperature of oceans is considered.

The \textit{sampling policy} of stage t, formally expressed as a probability density function $q_t$, stands for the distribution of the sample $(x_{t,1},\ldots, x_{t,n_t})$ generated at t. From the past samples, some information depending on some \textit{objective} is derived leading eventually to update the sampling policy to $q_{t+1}$. This generic approach characterizes \textit{adaptive importance sampling} (AIS) schemes. Each stage t is formed with two steps : (i) to explore the space with $n_t$ points according to $q_t$ and (ii) to exploit the current amount of information to update the sampling policy. The very fundamental question raised in the paper concerns the behavior of empirical sums based on AIS. Without making any assumption on the \textit{allocation policy} $n_t$, the theory developed involves no restriction on the split of computational resources between the explore (i) and the exploit (ii) step. It is shown that AIS is efficient : the asymptotic behavior of AIS is the same as some "oracle" strategy that knows the optimal sampling policy from the beginning. From a practical perspective, weighted AIS is introduced, a new method that allows to forget poor samples from early stages.

Adaptive Multilevel Splitting (AMS for short) is a generic Monte Carlo method for Markov processes that simulates rare events and estimates associated probabilities. Despite its practical efficiency, there are almost no theoretical results on the convergence of this algorithm. The purpose of this paper is to prove both consistency and asymptotic normality results in a general setting. This is done by associating to the original Markov process a level-indexed process, also called a stochastic wave, and by showing that AMS can then be seen as a Fleming-Viot type particle system. This being done, we can finally apply general results on Fleming-Viot particle systems that we have recently obtained.