The Galactic Center stellar cluster: The central arcsecond

With 10 years of high-resolution imaging data now available on the stellar cluster in the Galactic Center, we analyze the dynamics of the stars at projected distances $\leq1.2''$ from the central black hole candidate Sagittarius A* (Sgr A*). We find evidence for radial anisotropy of the cluster of stars surrounding Sgr A*. We confirm/find accelerated motion for 6 stars, with 4 stars having passed the pericenter of their orbits during the observed time span. We calculated/constrained the orbital parameters of these stars. All orbits have moderate to high eccentricities. The center of acceleration coincides with the radio position of Sgr A*. From the orbit of the star S2, the currently most tightly constrained one, we determine the mass of Sgr A* to $3.3\pm0.7\times10^{6}$M$_{\odot}$ and its position to $2.0\pm2.4$ mas East and $2.7\pm4.5$ mas South of the nominal radio position. The data provide compelling evidence that Sgr A* is a single supermassive black hole.Comment: 7 pages, 3 Figures; to be published in Astron. Nachr., Vol. 324, No. S1 (2003), Special Supplement "The central 300 parsecs of the Milky Way", Eds. A. Cotera, H. Falcke, T. R. Geballe, S. Markof

Asymptotic Analysis of Inpainting via Universal Shearlet Systems

Recently introduced inpainting algorithms using a combination of applied harmonic analysis and compressed sensing have turned out to be very successful. One key ingredient is a carefully chosen representation system which provides (optimally) sparse approximations of the original image. Due to the common assumption that images are typically governed by anisotropic features, directional representation systems have often been utilized. One prominent example of this class are shearlets, which have the additional benefitallowing faithful implementations. Numerical results show that shearlets significantly outperform wavelets in inpainting tasks. One of those software packages, www.shearlab.org, even offers the flexibility of usingdifferent parameter for each scale, which is not yet covered by shearlet theory. In this paper, we first introduce universal shearlet systems which are associated with an arbitrary scaling sequence, thereby modeling the previously mentioned flexibility. In addition, this novel construction allows for a smooth transition between wavelets and shearlets and therefore enables us to analyze them in a uniform fashion. For a large class of such scaling sequences, we first prove that the associated universal shearlet systems form band-limited Parseval frames for $L^2(\mathbb{R}^2)$ consisting of Schwartz functions. Secondly, we analyze the performance for inpainting of this class of universal shearlet systems within a distributional model situation using an $\ell^1$-analysis minimization algorithm for reconstruction. Our main result in this part states that, provided the scaling sequence is comparable to the size of the (scale-dependent) gap, nearly-perfect inpainting is achieved at sufficiently fine scales

Robust 1-Bit Compressed Sensing via Hinge Loss Minimization

This work theoretically studies the problem of estimating a structured high-dimensional signal $x_0 \in \mathbb{R}^n$ from noisy $1$-bit Gaussian measurements. Our recovery approach is based on a simple convex program which uses the hinge loss function as data fidelity term. While such a risk minimization strategy is very natural to learn binary output models, such as in classification, its capacity to estimate a specific signal vector is largely unexplored. A major difficulty is that the hinge loss is just piecewise linear, so that its "curvature energy" is concentrated in a single point. This is substantially different from other popular loss functions considered in signal estimation, e.g., the square or logistic loss, which are at least locally strongly convex. It is therefore somewhat unexpected that we can still prove very similar types of recovery guarantees for the hinge loss estimator, even in the presence of strong noise. More specifically, our non-asymptotic error bounds show that stable and robust reconstruction of $x_0$ can be achieved with the optimal oversampling rate $O(m^{-1/2})$ in terms of the number of measurements $m$. Moreover, we permit a wide class of structural assumptions on the ground truth signal, in the sense that $x_0$ can belong to an arbitrary bounded convex set $K \subset \mathbb{R}^n$. The proofs of our main results rely on some recent advances in statistical learning theory due to Mendelson. In particular, we invoke an adapted version of Mendelson's small ball method that allows us to establish a quadratic lower bound on the error of the first order Taylor approximation of the empirical hinge loss function

The Galactic Center

In the past decade high resolution measurements in the infrared employing adaptive optics imaging on 10m telescopes have allowed determining the three dimensional orbits stars within ten light hours of the compact radio source at the center of the Milky Way. These observations show the presence of a three million solar mass black hole in Sagittarius A* beyond any reasonable doubt. The Galactic Center thus constitutes the best astrophysical evidence for the existence of black holes which have long been postulated, and is also an ideal `lab' for studying the physics in the vicinity of such an object. Remarkably, young massive stars are present there and probably have formed in the innermost stellar cusp. Variable infrared and X-ray emission from Sagittarius A* are a new probe of the physical processes and space-time curvature just outside the event horizon.Comment: Write up of the talk at IAU Symposium No. 238 (21-25 August 2006, Prague), to appear in Proceedings of "Black Holes: from Stars to Galaxies" (Cambridge University Press), p. 17

ℓ1\ell^1-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the $\ell^{1}$-analysis basis pursuit, enabling highly accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend on the Gram matrix of the analysis operator and therefore particularly account for its mutual coherence structure. Our findings defy conventional wisdom which promotes the sparsity of analysis coefficients as the crucial quantity to study. In fact, this common paradigm breaks down completely in many situations of practical interest, for instance, when applying a redundant (multilevel) frame as analysis prior. By extensive numerical experiments, we demonstrate that, in contrast, our theoretical sampling-rate bounds reliably capture the recovery capability of various examples, such as redundant Haar wavelets systems, total variation, or random frames. The proofs of our main results build upon recent achievements in the convex geometry of data mining problems. More precisely, we establish a sophisticated upper bound on the conic Gaussian mean width that is associated with the underlying $\ell^{1}$-analysis polytope. Due to a novel localization argument, it turns out that the presented framework naturally extends to stable recovery, allowing us to incorporate compressible coefficient sequences as well

Clockwise Stellar Disk and the Dark Mass in the Galactic Center

Two disks of young stars have recently been discovered in the Galactic Center. The disks are rotating in the gravitational field of the central black hole at radii r=0.1-0.3 pc and thus open a new opportunity to measure the central mass. We find that the observed motion of stars in the clockwise disk implies M=4.3+/-0.5 million solar masses for the fiducial distance to the Galactic Center R_0=8 kpc and derive the scaling of M with R_0. As a tool for our estimate we use orbital roulette, a recently developed method. The method reconstructs the three-dimensional orbits of the disk stars and checks the randomness of their orbital phases. We also estimate the three-dimensional positions and orbital eccentricities of the clockwise-disk stars.Comment: Comments: 16 pages, 5 figures, ApJ, in pres