2,505 research outputs found
Learning Mixtures of Gaussians in High Dimensions
Efficiently learning mixture of Gaussians is a fundamental problem in
statistics and learning theory. Given samples coming from a random one out of k
Gaussian distributions in Rn, the learning problem asks to estimate the means
and the covariance matrices of these Gaussians. This learning problem arises in
many areas ranging from the natural sciences to the social sciences, and has
also found many machine learning applications. Unfortunately, learning mixture
of Gaussians is an information theoretically hard problem: in order to learn
the parameters up to a reasonable accuracy, the number of samples required is
exponential in the number of Gaussian components in the worst case. In this
work, we show that provided we are in high enough dimensions, the class of
Gaussian mixtures is learnable in its most general form under a smoothed
analysis framework, where the parameters are randomly perturbed from an
adversarial starting point. In particular, given samples from a mixture of
Gaussians with randomly perturbed parameters, when n > {\Omega}(k^2), we give
an algorithm that learns the parameters with polynomial running time and using
polynomial number of samples. The central algorithmic ideas consist of new ways
to decompose the moment tensor of the Gaussian mixture by exploiting its
structural properties. The symmetries of this tensor are derived from the
combinatorial structure of higher order moments of Gaussian distributions
(sometimes referred to as Isserlis' theorem or Wick's theorem). We also develop
new tools for bounding smallest singular values of structured random matrices,
which could be useful in other smoothed analysis settings
Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals
The equations for the three-dimensional incompressible flow of liquid
crystals are considered in a smooth bounded domain. The existence and
uniqueness of the global strong solution with small initial data are
established. It is also proved that when the strong solution exists, all the
global weak solutions constructed in [16] must be equal to the unique strong
solution
Orientability and energy minimization in liquid crystal models
Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory
through a unit vector field . This theory has the apparent drawback that it
does not respect the head-to-tail symmetry in which should be equivalent to
-. This symmetry is preserved in the constrained Landau-de Gennes theory
that works with the tensor .We study
the differences and the overlaps between the two theories. These depend on the
regularity class used as well as on the topology of the underlying domain. We
show that for simply-connected domains and in the natural energy class
the two theories coincide, but otherwise there can be differences
between the two theories, which we identify. In the case of planar domains we
completely characterise the instances in which the predictions of the
constrained Landau-de Gennes theory differ from those of the Oseen-Frank
theory
Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions
Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami
operator on a compact Riemannian manifold with boundary. We prove lower bounds
for the size of the nodal set {\phi=0}.Comment: 7 page
Testing Students with Special Educational Needs in Large-Scale Assessments – Psychometric Properties of Test Scores and Associations with Test Taking Behavior
Assessing competencies of students with special educational needs in learning (SEN-L) poses a challenge for large-scale assessments (LSAs). For students with SEN-L, the available competence tests may fail to yield test scores of high psychometric quality, which are—at the same time—measurement invariant to test scores of general education students. We investigated whether we can identify a subgroup of students with SEN-L, for which measurement invariant competence measures of adequate psychometric quality may be obtained with tests available in LSAs. We furthermore investigated whether differences in test-taking behavior may explain dissatisfying psychometric properties and measurement non-invariance of test scores within LSAs. We relied on person fit indices and mixture distribution models to identify students with SEN-L for whom test scores with satisfactory psychometric properties and measurement invariance may be obtained. We also captured differences in test-taking behavior related to guessing and missing responses. As a result we identified a subgroup of students with SEN-L for whom competence scores of adequate psychometric quality that are measurement invariant to those of general education students were obtained. Concerning test taking behavior, there was a small number of students who unsystematically picked response options. Removing these students from the sample slightly improved item fit. Furthermore, two different patterns of missing responses were identified that explain to some extent problems in the assessments of students with SEN-L.Peer Reviewe
Grid services for the MAGIC experiment
Exploring signals from the outer space has become an observational science
under fast expansion. On the basis of its advanced technology the MAGIC
telescope is the natural building block for the first large scale ground based
high energy gamma-ray observatory. The low energy threshold for gamma-rays
together with different background sources leads to a considerable amount of
data. The analysis will be done in different institutes spread over Europe.
Therefore MAGIC offers the opportunity to use the Grid technology to setup a
distributed computational and data intensive analysis system with the nowadays
available technology. Benefits of Grid computing for the MAGIC telescope are
presented.Comment: 5 pages, 1 figures, to be published in the Proceedings of the 6th
International Symposium ''Frontiers of Fundamental and Computational
Physics'' (FFP6), Udine (Italy), Sep. 26-29, 200
Nonlinear Hodge maps
We consider maps between Riemannian manifolds in which the map is a
stationary point of the nonlinear Hodge energy. The variational equations of
this functional form a quasilinear, nondiagonal, nonuniformly elliptic system
which models certain kinds of compressible flow. Conditions are found under
which singular sets of prescribed dimension cannot occur. Various degrees of
smoothness are proven for the sonic limit, high-dimensional flow, and flow
having nonzero vorticity. The gradient flow of solutions is estimated.
Implications for other quasilinear field theories are suggested.Comment: Slightly modified and updated version; tcilatex, 32 page
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