20,730 research outputs found
A tight lower bound instance for k-means++ in constant dimension
The k-means++ seeding algorithm is one of the most popular algorithms that is
used for finding the initial centers when using the k-means heuristic. The
algorithm is a simple sampling procedure and can be described as follows: Pick
the first center randomly from the given points. For , pick a point to
be the center with probability proportional to the square of the
Euclidean distance of this point to the closest previously chosen
centers.
The k-means++ seeding algorithm is not only simple and fast but also gives an
approximation in expectation as shown by Arthur and Vassilvitskii.
There are datasets on which this seeding algorithm gives an approximation
factor of in expectation. However, it is not clear from these
results if the algorithm achieves good approximation factor with reasonably
high probability (say ). Brunsch and R\"{o}glin gave a dataset where
the k-means++ seeding algorithm achieves an approximation ratio
with probability that is exponentially small in . However, this and all
other known lower-bound examples are high dimensional. So, an open problem was
to understand the behavior of the algorithm on low dimensional datasets. In
this work, we give a simple two dimensional dataset on which the seeding
algorithm achieves an approximation ratio with probability
exponentially small in . This solves open problems posed by Mahajan et al.
and by Brunsch and R\"{o}glin.Comment: To appear in TAMC 2014. arXiv admin note: text overlap with
arXiv:1306.420
Enhanced emission and light control with tapered plasmonic nanoantennas
We introduce a design of Yagi-Uda plasmonic nanoantennas for enhancing the
antenna gain and achieving control over the angular emission of light. We
demonstrate that tapering of antenna elements allows to decrease spacing
between the antenna elements tenfold also enhancing its emission directivity.
We find the optimal tapering angle that provides the maximum directivity
enhancement and the minimum end-fire beamwidth
Communications of the lunar planetary laboratory, volume 3, no. 40- the system of lunar craters, quadrant ii
Observable properties of craters in second lunar quadran
Measurement of the Spatial Cross-Correlation Function of Damped Lyman Alpha Systems and Lyman Break Galaxies
We present the first spectroscopic measurement of the spatial
cross-correlation function between damped Lyman alpha systems (DLAs) and Lyman
break galaxies (LBGs). We obtained deep u'BVRI images of nine QSO fields with
11 known z ~ 3 DLAs and spectroscopically confirmed 211 R < 25.5
photometrically selected z > 2 LBGs. We find strong evidence for an overdensity
of LBGs near DLAs versus random, the results of which are similar to that of
LBGs near other LBGs. A maximum likelihood cross-correlation analysis found the
best fit correlation length value of r_0 = 2.9^(+1.4)_(-1.5) h^(-1)Mpc using a
fixed value of gamma = 1.6. The implications of the DLA-LBG clustering
amplitude on the average dark matter halo mass of DLAs are discussed.Comment: 12 pages, 2 figures, accepted for publication in Astrophysical
Journal Letter
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