Linear and convex aggregation of density estimators

We study the problem of linear and convex aggregation of $M$ estimators of a density with respect to the mean squared risk. We provide procedures for linear and convex aggregation and we prove oracle inequalities for their risks. We also obtain lower bounds showing that these procedures are rate optimal in a minimax sense. As an example, we apply general results to aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense for a large class of kernels including Gaussian, Silverman's and Pinsker's ones. We prove that, for Pinsker's kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.Comment: 22 page

Exponential Screening and optimal rates of sparse estimation

In high-dimensional linear regression, the goal pursued here is to estimate an unknown regression function using linear combinations of a suitable set of covariates. One of the key assumptions for the success of any statistical procedure in this setup is to assume that the linear combination is sparse in some sense, for example, that it involves only few covariates. We consider a general, non necessarily linear, regression with Gaussian noise and study a related question that is to find a linear combination of approximating functions, which is at the same time sparse and has small mean squared error (MSE). We introduce a new estimation procedure, called Exponential Screening that shows remarkable adaptation properties. It adapts to the linear combination that optimally balances MSE and sparsity, whether the latter is measured in terms of the number of non-zero entries in the combination ($\ell_0$ norm) or in terms of the global weight of the combination ($\ell_1$ norm). The power of this adaptation result is illustrated by showing that Exponential Screening solves optimally and simultaneously all the problems of aggregation in Gaussian regression that have been discussed in the literature. Moreover, we show that the performance of the Exponential Screening estimator cannot be improved in a minimax sense, even if the optimal sparsity is known in advance. The theoretical and numerical superiority of Exponential Screening compared to state-of-the-art sparse procedures is also discussed

Isotropic Huygens dipoles and multipoles with colloidal particles

Huygens sources are elements that scatter light in the forward direction as used in the Huygens-Fresnel principle. They have remained fictitious until recently where experimental systems have been fabricated. In this letter, we propose isotropic meta-atoms that act as Huygens sources. Using clusters of plasmonic or dielectric colloidal particles, Huygens dipoles that resonate at visible frequencies can be achieved with scattering cross-sections as high as 5 times the geometric cross-section of the particle surpassing anything achievable with a hypothetical simple spherical particle. Examples are given that predict extremely broadband scattering in the forward direction over a 1000 nm wavelength range at optical frequencies. These systems are important to the fields of nanoantennas, metamaterials and wave physics in general as well as any application that requires local control over the radiation properties of a system as in solar cells or bio-sensing

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a non zero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term "quasi-satellite" more and more present in the literature. However, some authors rather use the term "retrograde satellite" when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.Comment: 30 pages, 13 figures, 1 tabl

Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems

We study qualitative properties of positive solutions of noncooperative, possibly nonvariational, elliptic systems. We obtain new classification and Liouville type theorems in the whole Euclidean space, as well as in half-spaces, and deduce a priori estimates and existence of positive solutions for related Dirichlet problems. We significantly improve the known results for a large class of systems involving a balance between repulsive and attractive terms. This class contains systems arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models of chemical reactions.Comment: 35 pages, to appear in Archive Rational Mech. Ana

Rigorous treatment of the averaging process for co-orbital motions in the planetary problem

We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamilto-nian, we estimate the size of the transformation that maps this Hamil-tonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale

The family of Quasi-satellite periodic orbits in the circular co-planar RTBP

In the circular case of the coplanar Restricted Three-body Problem, we studied how the family of quasi-satellite (QS) periodic orbits allows to define an associated libration center. Using the averaged problem, we highlighted a validity limit of this one: for QS orbits with low eccentricities, the averaged problem does not correspond to the real problem. We do the same procedure to L 3 , L 4 and L 5 emerging periodic orbits families and remarked that for very high eccentricities F L4 and F L5 merge with F L3 which bifurcates to a stable family

Spin-orbit coupling and chaotic rotation for coorbital bodies in quasi-circular orbits

Coorbital bodies are observed around the Sun sharing their orbits with the planets, but also in some pairs of satellites around Saturn. The existence of coorbital planets around other stars has also been proposed. For close-in planets and satellites, the rotation slowly evolves due to dissipative tidal effects until some kind of equilibrium is reached. When the orbits are nearly circular, the rotation period is believed to always end synchronous with the orbital period. Here we demonstrate that for coorbital bodies in quasi-circular orbits, stable non-synchronous rotation is possible for a wide range of mass ratios and body shapes. We show the existence of an entirely new family of spin-orbit resonances at the frequencies $n\pm k\nu/2$, where $n$ is the orbital mean motion, $\nu$ the orbital libration frequency, and $k$ an integer. In addition, when the natural rotational libration frequency due to the axial asymmetry, $\sigma$, has the same magnitude as $\nu$, the rotation becomes chaotic. Saturn coorbital satellites are synchronous since $\nu\ll\sigma$, but coorbital exoplanets may present non-synchronous or chaotic rotation. Our results prove that the spin dynamics of a body cannot be dissociated from its orbital environment. We further anticipate that a similar mechanism may affect the rotation of bodies in any mean-motion resonance.Comment: 6 pages. Astrophysical Journal (2013) 6p

Background subtraction based on Local Shape

We present a novel approach to background subtraction that is based on the local shape of small image regions. In our approach, an image region centered on a pixel is mod-eled using the local self-similarity descriptor. We aim at obtaining a reliable change detection based on local shape change in an image when foreground objects are moving. The method first builds a background model and compares the local self-similarities between the background model and the subsequent frames to distinguish background and foreground objects. Post-processing is then used to refine the boundaries of moving objects. Results show that this approach is promising as the foregrounds obtained are com-plete, although they often include shadows.Comment: 4 pages, 5 figures, 3 tabl