Formation and evolution of planetary systems: the impact of high angular resolution optical techniques

The direct images of giant extrasolar planets recently obtained around several main sequence stars represent a major step in the study of planetary systems. These high-dynamic range images are among the most striking results obtained by the current generation of high angular resolution instruments, which will be superseded by a new generation of instruments in the coming years. It is therefore an appropriate time to review the contributions of high angular resolution visible/infrared techniques to the rapidly growing field of extrasolar planetary science. During the last 20 years, the advent of the Hubble Space Telescope, of adaptive optics on 4- to 10-m class ground-based telescopes, and of long-baseline infrared stellar interferometry has opened a new viewpoint on the formation and evolution of planetary systems. By spatially resolving the optically thick circumstellar discs of gas and dust where planets are forming, these instruments have considerably improved our models of early circumstellar environments and have thereby provided new constraints on planet formation theories. High angular resolution techniques are also directly tracing the mechanisms governing the early evolution of planetary embryos and the dispersal of optically thick material around young stars. Finally, mature planetary systems are being studied with an unprecedented accuracy thanks to single-pupil imaging and interferometry, precisely locating dust populations and putting into light a whole new family of long-period giant extrasolar planets.Comment: 71 pages, published in Astronomy and Astrophysics Review, online at http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s00159-009-0028-

A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds

The Newton iteration is a popular method for minimising a cost function on Euclidean space. Various generalisations to cost functions defined on manifolds appear in the literature. In each case, the convergence rate of the generalised Newton iteration needed establishing from first principles. The present paper presents a framework for generalising iterative methods from Euclidean space to manifolds that ensures local convergence rates are preserved. It applies to any (memoryless) iterative method computing a coordinate independent property of a function (such as a zero or a local minimum). All possible Newton methods on manifolds are believed to come under this framework. Changes of coordinates, and not any Riemannian structure, are shown to play a natural role in lifting the Newton method to a manifold. The framework also gives new insight into the design of Newton methods in general.Comment: 36 page

Searching for faint companions with VLTI/PIONIER. I. Method and first results

Context. A new four-telescope interferometric instrument called PIONIER has recently been installed at VLTI. It provides improved imaging capabilities together with high precision. Aims. We search for low-mass companions around a few bright stars using different strategies, and determine the dynamic range currently reachable with PIONIER. Methods. Our method is based on the closure phase, which is the most robust interferometric quantity when searching for faint companions. We computed the chi^2 goodness of fit for a series of binary star models at different positions and with various flux ratios. The resulting chi^2 cube was used to identify the best-fit binary model and evaluate its significance, or to determine upper limits on the companion flux in case of non detections. Results. No companion is found around Fomalhaut, tau Cet and Regulus. The median upper limits at 3 sigma on the companion flux ratio are respectively of 2.3e-3 (in 4 h), 3.5e-3 (in 3 h) and 5.4e-3 (in 1.5 h) on the search region extending from 5 to 100 mas. Our observations confirm that the previously detected near-infrared excess emissions around Fomalhaut and tau Cet are not related to a low-mass companion, and instead come from an extended source such as an exozodiacal disk. In the case of del Aqr, in 30 min of observation, we obtain the first direct detection of a previously known companion, at an angular distance of about 40 mas and with a flux ratio of 2.05e-2 \pm 0.16e-2. Due to the limited u,v plane coverage, its position can, however, not be unambiguously determined. Conclusions. After only a few months of operation, PIONIER has already achieved one of the best dynamic ranges world-wide for multi-aperture interferometers. A dynamic range up to about 1:500 is demonstrated, but significant improvements are still required to reach the ultimate goal of directly detecting hot giant extrasolar planets.Comment: 11 pages, 6 figures, accepted for publication in A&

Two Algorithms for Orthogonal Nonnegative Matrix Factorization with Application to Clustering

Approximate matrix factorization techniques with both nonnegativity and orthogonality constraints, referred to as orthogonal nonnegative matrix factorization (ONMF), have been recently introduced and shown to work remarkably well for clustering tasks such as document classification. In this paper, we introduce two new methods to solve ONMF. First, we show athematical equivalence between ONMF and a weighted variant of spherical k-means, from which we derive our first method, a simple EM-like algorithm. This also allows us to determine when ONMF should be preferred to k-means and spherical k-means. Our second method is based on an augmented Lagrangian approach. Standard ONMF algorithms typically enforce nonnegativity for their iterates while trying to achieve orthogonality at the limit (e.g., using a proper penalization term or a suitably chosen search direction). Our method works the opposite way: orthogonality is strictly imposed at each step while nonnegativity is asymptotically obtained, using a quadratic penalty. Finally, we show that the two proposed approaches compare favorably with standard ONMF algorithms on synthetic, text and image data sets.Comment: 17 pages, 8 figures. New numerical experiments (document and synthetic data sets

Cram\'er-Rao bounds for synchronization of rotations

Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise