7,753 research outputs found
Sparsifying the Fisher Linear Discriminant by Rotation
Many high dimensional classification techniques have been proposed in the
literature based on sparse linear discriminant analysis (LDA). To efficiently
use them, sparsity of linear classifiers is a prerequisite. However, this might
not be readily available in many applications, and rotations of data are
required to create the needed sparsity. In this paper, we propose a family of
rotations to create the required sparsity. The basic idea is to use the
principal components of the sample covariance matrix of the pooled samples and
its variants to rotate the data first and to then apply an existing high
dimensional classifier. This rotate-and-solve procedure can be combined with
any existing classifiers, and is robust against the sparsity level of the true
model. We show that these rotations do create the sparsity needed for high
dimensional classifications and provide theoretical understanding why such a
rotation works empirically. The effectiveness of the proposed method is
demonstrated by a number of simulated and real data examples, and the
improvements of our method over some popular high dimensional classification
rules are clearly shown.Comment: 30 pages and 9 figures. This paper has been accepted by Journal of
the Royal Statistical Society: Series B (Statistical Methodology). The first
two versions of this paper were uploaded to Bin Dong's web site under the
title "A Rotate-and-Solve Procedure for Classification" in 2013 May and 2014
January. This version may be slightly different from the published versio
Global Structure of Exact Scalar Hairy Dynamical Black Holes
We study the global structure of some exact scalar hairy dynamical black
holes which were constructed in Einstein gravity either minimally or
non-minimally coupled to a scalar field. We find that both the apparent horizon
and the local event horizon (measured in luminosity coordinate) monotonically
increase with the advanced time as well as the Vaidya mass. At late advanced
times, the apparent horizon approaches the event horizon and gradually becomes
future outer. Correspondingly, the space-time arrives at stationary black hole
states with the relaxation time inversely proportional to the power
of the final black hole mass, where is the space-time dimension. These
results strongly support the solutions describing the formation of black holes
with scalar hair. We also obtain new charged dynamical solutions in the
non-minimal theory by introducing an Maxwell field which is non-minimally
coupled to the scalar.
The presence of the electric charge strongly modifies the dynamical evolution
of the space-time.Comment: 18 pages and 6 figures; minor corrections, publised versio
The Binary Space Partitioning-Tree Process
The Mondrian process represents an elegant and powerful approach for space
partition modelling. However, as it restricts the partitions to be
axis-aligned, its modelling flexibility is limited. In this work, we propose a
self-consistent Binary Space Partitioning (BSP)-Tree process to generalize the
Mondrian process. The BSP-Tree process is an almost surely right continuous
Markov jump process that allows uniformly distributed oblique cuts in a
two-dimensional convex polygon. The BSP-Tree process can also be extended using
a non-uniform probability measure to generate direction differentiated cuts.
The process is also self-consistent, maintaining distributional invariance
under a restricted subdomain. We use Conditional-Sequential Monte Carlo for
inference using the tree structure as the high-dimensional variable. The
BSP-Tree process's performance on synthetic data partitioning and relational
modelling demonstrates clear inferential improvements over the standard
Mondrian process and other related methods
Criticality in Einstein-Gauss-Bonnet Gravity: Gravity without Graviton
General Einstein-Gauss-Bonnet gravity with a cosmological constant allows two
(A)dS spacetimes as its vacuum solutions. We find a critical point in the
parameter space where the two (A)dS spacetimes coalesce into one and the
linearized perturbations lack any bilinear kinetic terms. The vacuum
perturbations hence loose their interpretation as linear graviton modes at the
critical point. Nevertheless, the critical theory admits black hole solutions
due to the nonlinear effect. We also consider Einstein gravity extended with
general quadratic curvature invariants and obtain critical points where the
theory has no bilinear kinetic terms for either the scalar trace mode or the
transverse modes. Such critical phenomena are expected to occur frequently in
general higher derivative gravities.Comment: 21 pages, no figures;refereces adde
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