869,314 research outputs found
Killing spinor initial data sets
A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is
made using the space spinor formalism. Conditions on initial data sets for the
Einstein vacuum equations are given so that their developments contain
solutions to the twistor and/or Killing equations. These lead to the notions of
twistor and Killing spinor initial data. These notions are used to obtain a
characterisation of initial data sets whose development are of Petrov type N or
D.Comment: 31 pages, submitted to J. Geom. Phy
Dimensionality reduction of clustered data sets
We present a novel probabilistic latent variable model to perform linear dimensionality reduction on data sets which contain clusters. We prove that the maximum likelihood solution of the model is an unsupervised generalisation of linear discriminant analysis. This provides a completely new approach to one of the most established and widely used classification algorithms. The performance of the model is then demonstrated on a number of real and artificial data sets
Decomposing data sets into skewness modes
We derive the nonlinear equations satisfied by the coefficients of linear
combinations that maximize their skewness when their variance is constrained to
take a specific value. In order to numerically solve these nonlinear equations
we develop a gradient-type flow that preserves the constraint. In combination
with the Karhunen-Lo\`eve decomposition this leads to a set of orthogonal modes
with maximal skewness. For illustration purposes we apply these techniques to
atmospheric data; in this case the maximal-skewness modes correspond to
strongly localized atmospheric flows. We show how these ideas can be extended,
for example to maximal-flatness modes.Comment: Submitted for publication, 12 pages, 4 figure
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