11,203 research outputs found
A Convex Formulation for Spectral Shrunk Clustering
Spectral clustering is a fundamental technique in the field of data mining
and information processing. Most existing spectral clustering algorithms
integrate dimensionality reduction into the clustering process assisted by
manifold learning in the original space. However, the manifold in
reduced-dimensional subspace is likely to exhibit altered properties in
contrast with the original space. Thus, applying manifold information obtained
from the original space to the clustering process in a low-dimensional subspace
is prone to inferior performance. Aiming to address this issue, we propose a
novel convex algorithm that mines the manifold structure in the low-dimensional
subspace. In addition, our unified learning process makes the manifold learning
particularly tailored for the clustering. Compared with other related methods,
the proposed algorithm results in more structured clustering result. To
validate the efficacy of the proposed algorithm, we perform extensive
experiments on several benchmark datasets in comparison with some
state-of-the-art clustering approaches. The experimental results demonstrate
that the proposed algorithm has quite promising clustering performance.Comment: AAAI201
Constraining Primordial Magnetic Fields with Future Cosmic Shear Surveys
The origin of astrophysical magnetic fields observed in galaxies and clusters
of galaxies is still unclear. One possibility is that primordial magnetic
fields generated in the early Universe provide seeds that grow through
compression and turbulence during structure formation. A cosmological magnetic
field present prior to recombination would produce substantial matter
clustering at intermediate/small scales, on top of the standard inflationary
power spectrum. In this work we study the effect of this alteration on one
particular cosmological observable, cosmic shear. We adopt the semi-analytic
halo model in order to describe the non-linear clustering of matter, and feed
it with the altered mass variance induced by primordial magnetic fields. We
find that the convergence power spectrum is, as expected, substantially
enhanced at intermediate/small angular scales, with the exact amplitude of the
enhancement depending on the magnitude and power-law index of the magnetic
field power spectrum. We use the predicted statistical errors for a future
wide-field cosmic shear survey, on the model of the ESA Cosmic Vision mission
\emph{Euclid}, in order to forecast constraints on the amplitude of primordial
magnetic fields as a function of the spectral index. We find that the amplitude
will be constrained at the level of nG for , and at the
level of nG for . The latter is at the same level of
lower bounds coming from the secondary emission of gamma-ray sources, implying
that for high spectral indices \emph{Euclid} will certainly be able to detect
primordial magnetic fields, if they exist. The present study shows how
large-scale structure surveys can be used for both understanding the origins of
astrophysical magnetic fields and shedding new light on the physics of the
pre-recombination Universe. (abridged)Comment: 24 pages, 9 figures. To appear on JCA
Magnification bias as a novel probe for primordial magnetic fields
In this paper we investigate magnetic fields generated in the early Universe.
These fields are important candidates at explaining the origin of astrophysical
magnetism observed in galaxies and galaxy clusters, whose genesis is still by
and large unclear. Compared to the standard inflationary power spectrum,
intermediate to small scales would experience further substantial matter
clustering, were a cosmological magnetic field present prior to recombination.
As a consequence, the bias and redshift distribution of galaxies would also be
modified. Hitherto, primordial magnetic fields (PMFs) have been tested and
constrained with a number of cosmological observables, e.g. the cosmic
microwave background radiation, galaxy clustering and, more recently, weak
gravitational lensing. Here, we explore the constraining potential of the
density fluctuation bias induced by gravitational lensing magnification onto
the galaxy-galaxy angular power spectrum. Such an effect is known as
magnification bias. Compared to the usual galaxy clustering approach,
magnification bias helps in lifting the pathological degeneracy present amongst
power spectrum normalisation and galaxy bias. This is because magnification
bias cross-correlates galaxy number density fluctuations of nearby objects with
weak lensing distortions of high-redshift sources. Thus, it takes advantage of
the gravitational deflection of light, which is insensitive to galaxy bias but
powerful in constraining the density fluctuation amplitude. To scrutinise the
potentiality of this method, we adopt a deep and wide-field spectroscopic
galaxy survey. We show that magnification bias does contain important
information on primordial magnetism, which will be useful in combination with
galaxy clustering and shear. We find we shall be able to rule out at 95.4% CL
amplitudes of PMFs larger than 0.0005 nG for values of the PMF power spectral
index ~0.Comment: 21 pages, 9 figures; published on JCA
Simplified Energy Landscape for Modularity Using Total Variation
Networks capture pairwise interactions between entities and are frequently
used in applications such as social networks, food networks, and protein
interaction networks, to name a few. Communities, cohesive groups of nodes,
often form in these applications, and identifying them gives insight into the
overall organization of the network. One common quality function used to
identify community structure is modularity. In Hu et al. [SIAM J. App. Math.,
73(6), 2013], it was shown that modularity optimization is equivalent to
minimizing a particular nonconvex total variation (TV) based functional over a
discrete domain. They solve this problem, assuming the number of communities is
known, using a Merriman, Bence, Osher (MBO) scheme.
We show that modularity optimization is equivalent to minimizing a convex
TV-based functional over a discrete domain, again, assuming the number of
communities is known. Furthermore, we show that modularity has no convex
relaxation satisfying certain natural conditions. We therefore, find a
manageable non-convex approximation using a Ginzburg Landau functional, which
provably converges to the correct energy in the limit of a certain parameter.
We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et
al. and which is 7 times faster at solving the associated diffusion equation
due to the fact that the underlying discretization is unconditionally stable.
Our numerical tests include a hyperspectral video whose associated graph has
2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper
of Hu et al.Comment: 25 pages, 3 figures, 3 tables, submitted to SIAM J. App. Mat
Multiclass Semi-Supervised Learning on Graphs using Ginzburg-Landau Functional Minimization
We present a graph-based variational algorithm for classification of
high-dimensional data, generalizing the binary diffuse interface model to the
case of multiple classes. Motivated by total variation techniques, the method
involves minimizing an energy functional made up of three terms. The first two
terms promote a stepwise continuous classification function with sharp
transitions between classes, while preserving symmetry among the class labels.
The third term is a data fidelity term, allowing us to incorporate prior
information into the model in a semi-supervised framework. The performance of
the algorithm on synthetic data, as well as on the COIL and MNIST benchmark
datasets, is competitive with state-of-the-art graph-based multiclass
segmentation methods.Comment: 16 pages, to appear in Springer's Lecture Notes in Computer Science
volume "Pattern Recognition Applications and Methods 2013", part of series on
Advances in Intelligent and Soft Computin
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