Djalil Chafaï

Channel selection methods for Infrared Atmospheric Sounding Interferometer radiances

Advanced infrared sounders will provide thousands of radiance data at every observation location. The number of individual pieces of information is not usable in an operational numerical weather-prediction context, and we have investigated the possibilities of choosing an optimal subset of data. These issues have been addressed in the context of optimal linear estimation theory, using simulated Infrared Atmospheric Sounding Interferometer data. Several methods have been tried to select a set of the most useful channels for each individual atmospheric profile. These are two methods based on the data resolution matrix, one method based on the Jacobian matrix, and one iterative method selecting sequentially the channels with largest information content. The Jacobian method and the iterative method were found to be the most suitable for the problem. The iterative method was demonstrated to always produce the best results, but at a larger cost than the Jacobian method. To test the robustness of the iterative method, a variant has been tried. It consists in building a mean channel selection aimed at optimizing the results over the whole database, and then applying to each profile this constant selection. Results show that this constant iterative method is very promising, with results of intermediate quality between the ones obtained for the optimal iterative method and the Jacobian method. The practical advantage of this method for operational purposes is that the same set of channels can be used for various atmospheric profiles.

Around the circular law

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension

On gradient bounds for the heat kernel on the Heisenberg group

n

On the long time behavior of the TCP window size process

tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.

First-order global asymptotics for confined particles with singular pair repulsion

It is known that the couple formed by the two dimensional Brownian motion and its Levy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.

Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph.

The TCP window size process appears in the modeling of the famous transmission control protocol used for data transmission over the Internet. This continuous time Markov process takes its values in [0,[infinity]), and is ergodic and irreversible. It belongs to the additive increase-multiplicative decrease class of processes. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. Several aspects of this process have already been investigated in the literature. In the present paper, we mainly get quantitative estimates for the convergence to equilibrium, in terms of the W1 Wasserstein coupling distance, for the process and also for its embedded chain.

The Dirichlet Markov Ensemble

We study a physical system of N interacting particles in Rd, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as N tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension d>2, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as N tends to infinity. In the more specific case of Coulomb interaction in dimension d>2, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.

Confidence Regions for the Multinomial Parameter With Small Sample Size

We consider the random reversible Markov kernel K obtained by assigning i.i.d. non negative weights to the edges of the complete graph over n vertices, and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an alpha-stable law, with alpha in (0,2). When 1<= \alpha <2, we show that for a suitable regularly varying sequence kappa_n of index 1-1/alpha, the limiting spectral distribution mu_alpha of kappa_n K coincides with the one of the random symmetric matrix of the un-normalized weights (Levy matrix with i.i.d. entries). In contrast, when 0< alpha <1, we show that the empirical spectral distribution of K converges without rescaling to a non trivial law wmu_alpha supported on [-1,1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of mu_alpha and wmu_alpha. We also study the limiting behavior of the invariant probability measure of K.

Circular law for random matrices with exchangeable entries

We equip the polytope of nxn Markov matrices with the normalized trace of the Lebesgue measure of R^n^^^2. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,...,1/n). We show that if M is such a random matrix, then the empirical distribution built from the singular values of nM tends as n->~ to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of nM tends as n->~ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of M is of order 1-1/n when n is large.

Circular Law for Noncentral Random Matrices

Consider the observation of n iid realizations of an experiment with d≥2 possible outcomes, which corresponds to a single observation of a multinomial distribution ℳd(n, p) where p is an unknown discrete distribution on {1, …, d}. In many applications, the construction of a confidence region for p when n is small is crucial. This challenging concrete problem has a long history. It is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small. On the other hand, most available methods providing nonasymptotic regions with controlled coverage are limited to the binomial case d=2. Here we propose a new method valid for any d≥2 that provides confidence regions with controlled coverage and small volume. The method involves inversion of the “covering collection” associated with level sets of the likelihood. The behavior when d/n tends to infinity remains an interesting open problem beyond the scope of this work.