Département de mathématiques et applications

Random forests are a scheme proposed by Leo Breiman in the 2000's for\nbuilding a predictor ensemble with a set of decision trees that grow in\nrandomly selected subspaces of data. Despite growing interest and practical\nuse, there has been little exploration of the statistical properties of random\nforests, and little is known about the mathematical forces driving the\nalgorithm. In this paper, we offer an in-depth analysis of a random forests\nmodel suggested by Breiman in \\cite{Bre04}, which is very close to the original\nalgorithm. We show in particular that the procedure is consistent and adapts to\nsparsity, in the sense that its rate of convergence depends only on the number\nof strong features and not on how many noise variables are present.\n

The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blow-up occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blows-up in finite time in the whole Euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with sub-critical mass, this allows us to give for large times an “intermediate asymptotics” description of the vanishing. In self-similar coordinates, we actually prove a convergence result to a limiting self-similar solution which is not a simple reflect of the diffusion.

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with nondegenerate critical points. Interestingly, the strong instability is due to viscosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.

Nous présentons de nouveaux M-estimateurs de la moyenne et de la variance d’une variable aléatoire réelle, fondés sur des bornes PAC-Bayésiennes. Nous analysons les propriétés minimax non-asymptotiques des déviations de ces estimateurs pour des distributions de l’échantillon soit de variance bornée, soit de variance et de kurtosis bornées. Sous ces hypothèses faibles, permettant des distributions à queue lourde, nous montrons que les déviations de la moyenne empirique sont dans le pire des cas sous-optimales. Nous prouvons en effet que pour tout niveau de confiance, il existe un M-estimateur dont les déviations sont du même ordre que les déviations de la moyenne empirique d’un échantillon Gaussien, même dans le cas où la véritable distribution de l’échantillon a une queue lourde. Le comportement expérimental de ces nouveaux estimateurs est du reste encore meilleur que ce que les bornes théoriques laissent prévoir, montrant que la fonction quantile des déviations est constamment en dessous de celle de la moyenne empirique pour des échantillons non Gaussiens aussi simples que des mélanges de deux distributions Gaussiennes.

The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass there is global existence of classical solutions and for large initial mass blow-up occurs. In this Note we complete this picture and give an explicit value for the critical mass when the system is set in the whole space.

This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.

International audience

The authors present a short review of the progress that has occurred during years 1955-91 in both the theoretical and practical characterization of frequency stability of precision frequency sources. The emphasis is on the evolution of ideas and concepts for the characterization of random noise processes in such standards in the time domain and the Fourier frequency domain, rather than a rigorous mathematical treatment of the problem. Numerous references to the mathematical treatments are made.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We consider a generalization of stochastic bandits where the set of arms, X, is allowed to be a generic measurable space and the mean-payoff function is Lipschitz with respect to a dissimilarity function that is known to the decision maker. Under this condition we construct an arm selection policy, called HOO (hierarchical optimistic optimization), with improved regret bounds compared to previous results for a large class of problems. In particular, our results imply that if X is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally continuous with a known smoothness degree, then the expected regret of HOO is bounded up to a logarithmic factor by √n, that is, the rate of growth of the regret is independent of the dimension of the space. We also prove the minimax optimality of our algorithm when the dissimilarity is a metric. Our basic strategy has quadratic computational complexity as a function of the number of time steps and does not rely on the doubling trick. We also introduce a modified strategy, which relies on the doubling trick but runs in linearithmic time. Both results are improvements with respect to previous approaches.

This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel system in the full space. We derive a critical mass threshold below which global existence is ensured. Carefully using energy methods and ad hoc functional inequalities, we improve and extend previous results in this direction. The given threshold is thought to be the optimal criterion, but this question is still open. This global existence result is accompanied by a detailed discussion on the duality between the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that underlie the following approach.

We derive new models for gravity driven shallow water flows in several space dimensions over a general topography. A first model is valid for small slope variation, i.e. small curvature, and a second model is valid for arbitrary topography. In both cases no particular assumption is made on the velocity profile in the material layer. The models are written for an arbitrary coordinate system, and several formulations are provided. A Coulomb friction term is derived within the same framework, relevant in particular for debris avalanches. All our models are invariant under rotation, admit a conservative energy equation, and preserve the steady state of a lake at rest.

In this paper, the stochastic characteristics of the electric consumption in France are analyzed. It is shown that the load time series exhibit lasting abrupt changes in the stochastic pattern, termed breaks. The goal is to propose an efficient and robust load forecasting method for prediction up to a day-ahead. To this end, two new robust procedures for outlier identification and suppression are developed. They are termed the multivariate ratio-of-medians-based estimator (RME) and the multivariate minimum-Hellinger-distance-based estimator (MHDE). The performance of the proposed methods has been evaluated on the French electric load time series in terms of execution times, ability to detect and suppress outliers, and forecasting accuracy. Their performances are compared with those of the robust methods proposed in the literature to estimate the parameters of SARIMA models and of the multiplicative double seasonal exponential smoothing. A new robust version of the latter is proposed as well. It is found that the RME approach outperforms all the other methods for “normal days” and presents several interesting properties such as good robustness, fast execution, simplicity, and easy online implementation. Finally, to deal with heteroscedasticity, we propose a simple novel multivariate modeling that improves the quality of the forecast.

The results in this paper concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is isomorphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I (GAFA, Geom. funct. anal. Vol. 9 (1999) 985-1033), others will be given in forthcoming papers. The results in this paper had been announced (with indications of proof) in a talk at the ICM 94 in Z{ü}rich. Up to typos, this is the revised version from 2003.

A general method for the derivation of asymptotic nonlinear models in shallow and deep water is presented. Starting from a general dimensionless version of the water wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet–Neumann operator and we show that many asymptotic models can be recovered by a simple analysis of this operator. Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to an uneven bottom the approach developed by Matsuno [Phys. Rev. E 47, 4593 (1993)] and Choi [J. Fluid Mech. 295, 381 (1995)]. This model is still valid in shallow water but with less precision than what can be achieved with the Green–Naghdi model when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equations with the fully nonlinear shallow water Green–Naghdi equations represents a relevant model for describing ocean wave propagation from deep to shallow waters.

The results in this paper concern computations of Floer cohomology using\ngenerating functions. The first part proves the isomorphism between Floer\ncohomology and Generating function cohomology introduced by Lisa Traynor. The\nsecond part proves that the Floer cohomology of the cotangent bundle (in the\nsense of Part I), is isomorphic to the cohomology of the loop space of the\nbase. This has many consequences, some of which were given in Part I (GAFA,\nGeom. funct. anal. Vol. 9 (1999) 985-1033), others will be given in forthcoming\npapers. The results in this paper had been announced (with indications of\nproof) in a talk at the ICM 94 in Z{\\"u}rich. Up to typos, this is the revised\nversion from 2003.\n

Comparing probability distributions is a fundamental problem in data\nsciences. Simple norms and divergences such as the total variation and the\nrelative entropy only compare densities in a point-wise manner and fail to\ncapture the geometric nature of the problem. In sharp contrast, Maximum Mean\nDiscrepancies (MMD) and Optimal Transport distances (OT) are two classes of\ndistances between measures that take into account the geometry of the\nunderlying space and metrize the convergence in law.\n This paper studies the Sinkhorn divergences, a family of geometric\ndivergences that interpolates between MMD and OT. Relying on a new notion of\ngeometric entropy, we provide theoretical guarantees for these divergences:\npositivity, convexity and metrization of the convergence in law. On the\npractical side, we detail a numerical scheme that enables the large scale\napplication of these divergences for machine learning: on the GPU, gradients of\nthe Sinkhorn loss can be computed for batches of a million samples.\n

Optimal transport (OT) and maximum mean discrepancies (MMD) are now routinely used in machine learning to compare probability measures. We focus in this paper on \emph{Sinkhorn divergences} (SDs), a regularized variant of OT distances which can interpolate, depending on the regularization strength $\varepsilon$, between OT ($\varepsilon=0$) and MMD ($\varepsilon=\infty$). Although the tradeoff induced by that regularization is now well understood computationally (OT, SDs and MMD require respectively $O(n^3\log n)$, $O(n^2)$ and $n^2$ operations given a sample size $n$), much less is known in terms of their \emph{sample complexity}, namely the gap between these quantities, when evaluated using finite samples \emph{vs.} their respective densities. Indeed, while the sample complexity of OT and MMD stand at two extremes, $1/n^{1/d}$ for OT in dimension $d$ and $1/\sqrt{n}$ for MMD, that for SDs has only been studied empirically. In this paper, we \emph{(i)} derive a bound on the approximation error made with SDs when approximating OT as a function of the regularizer $\varepsilon$, \emph{(ii)} prove that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and \emph{(iii)} provide the first sample complexity bound for SDs, obtained,by reformulating SDs as a maximization problem in a RKHS. We thus obtain a scaling in $1/\sqrt{n}$ (as in MMD), with a constant that depends however on $\varepsilon$, making the bridge between OT and MMD complete.

Mathematics cannot anymore be assimilated to a linguistic game, where formal proofs are strongly differentiated with conjectural thinking, without building any category of knowledge to understand the passage (Wittgenstein's gist). Nowadays, philosophy has to face with the growing, exponential ramified tree of speculative mathematical thinking. Our main (problematical) theses are: 1. In mathematics, there is no empirical automatism, and no separate, physical-like motricity. 2: The irreversible-synthetical must force to complexify the exegetical game of philosophy; numerical experiments in algebra and in number theory are a kind of letting blow up all possible problems; 4. The nature of mathematical questioning still remains in question.