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

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&

Coarse wavelength division multiplexer on silicon-on-insulator for 100 GbE

A four-channel cascaded MZl based de-multiplexer at O-band with coarse channel spacing of 20 nm and band flatness of 13 nm is demonstrated on silicon-on-insulator. The device shows a mean crosstalk and insertion loss below -16 dB and 2.5 dB

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

Prospects for near-infrared characterisation of hot Jupiters with VSI

In this paper, we study the feasibility of obtaining near-infrared spectra of bright extrasolar planets with the 2nd generation VLTI Spectro-Imager instrument (VSI), which has the required angular resolution to resolve nearby hot Extrasolar Giant Planets (EGPs) from their host stars. Taking into account fundamental noises, we simulate closure phase measurements of several extrasolar systems using four 8-m telescopes at the VLT and a low spectral resolution (R = 100). Synthetic planetary spectra from T. Barman are used as an input. Standard chi2-fitting methods are then used to reconstruct planetary spectra from the simulated data. These simulations show that low-resolution spectra in the H and K bands can be retrieved with a good fidelity for half a dozen targets in a reasonable observing time (about 10 hours, spread over a few nights). Such observations would strongly constrain the planetary temperature and albedo, the energy redistribution mechanisms, as well as the chemical composition of their atmospheres. Systematic errors, not included in our simulations, could be a serious limitation to these performance estimations. The use of integrated optics is however expected to provide the required instrumental stability (around 10^-4 on the closure phase) to enable the first thorough characterisation of extrasolar planetary emission spectra in the near-infrared.Comment: 10 pages, 8 figures, Proc. SPIE conference 7013 "Optical and Infrared Interferometry" (Marseille 2008