5,761 research outputs found
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 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
M and its position to mas East and
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 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
-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 from noisy -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 can be achieved with
the optimal oversampling rate in terms of the number of
measurements . Moreover, we permit a wide class of structural assumptions on
the ground truth signal, in the sense that can belong to an arbitrary
bounded convex set . 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
-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
-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 -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
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