1,085,785 research outputs found
Contour projected dimension reduction
In regression analysis, we employ contour projection (CP) to develop a new
dimension reduction theory. Accordingly, we introduce the notions of the
central contour subspace and generalized contour subspace. We show that both of
their structural dimensions are no larger than that of the central subspace
Cook [Regression Graphics (1998b) Wiley]. Furthermore, we employ CP-sliced
inverse regression, CP-sliced average variance estimation and CP-directional
regression to estimate the generalized contour subspace, and we subsequently
obtain their theoretical properties. Monte Carlo studies demonstrate that the
three CP-based dimension reduction methods outperform their corresponding
non-CP approaches when the predictors have heavy-tailed elliptical
distributions. An empirical example is also presented to illustrate the
usefulness of the CP method.Comment: Published in at http://dx.doi.org/10.1214/08-AOS679 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Kernel dimension reduction in regression
We present a new methodology for sufficient dimension reduction (SDR). Our
methodology derives directly from the formulation of SDR in terms of the
conditional independence of the covariate from the response , given the
projection of on the central subspace [cf. J. Amer. Statist. Assoc. 86
(1991) 316--342 and Regression Graphics (1998) Wiley]. We show that this
conditional independence assertion can be characterized in terms of conditional
covariance operators on reproducing kernel Hilbert spaces and we show how this
characterization leads to an -estimator for the central subspace. The
resulting estimator is shown to be consistent under weak conditions; in
particular, we do not have to impose linearity or ellipticity conditions of the
kinds that are generally invoked for SDR methods. We also present empirical
results showing that the new methodology is competitive in practice.Comment: Published in at http://dx.doi.org/10.1214/08-AOS637 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Dimension Reduction via Colour Refinement
Colour refinement is a basic algorithmic routine for graph isomorphism
testing, appearing as a subroutine in almost all practical isomorphism solvers.
It partitions the vertices of a graph into "colour classes" in such a way that
all vertices in the same colour class have the same number of neighbours in
every colour class. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and
Ullman (Disc. Math., 1994) and Godsil (Lin. Alg. and its App., 1997)
established a tight correspondence between colour refinement and fractional
isomorphisms of graphs, which are solutions to the LP relaxation of a natural
ILP formulation of graph isomorphism.
We introduce a version of colour refinement for matrices and extend existing
quasilinear algorithms for computing the colour classes. Then we generalise the
correspondence between colour refinement and fractional automorphisms and
develop a theory of fractional automorphisms and isomorphisms of matrices.
We apply our results to reduce the dimensions of systems of linear equations
and linear programs. Specifically, we show that any given LP L can efficiently
be transformed into a (potentially) smaller LP L' whose number of variables and
constraints is the number of colour classes of the colour refinement algorithm,
applied to a matrix associated with the LP. The transformation is such that we
can easily (by a linear mapping) map both feasible and optimal solutions back
and forth between the two LPs. We demonstrate empirically that colour
refinement can indeed greatly reduce the cost of solving linear programs
Dimension reduction by random hyperplane tessellations
Given a subset K of the unit Euclidean sphere, we estimate the minimal number
m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense
that the fraction of the hyperplanes separating any pair x, y in K is nearly
proportional to the Euclidean distance between x and y. Random hyperplanes
prove to be almost ideal for this problem; they achieve the almost optimal
bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map
that sends x in K to the sign vector with respect to the hyperplanes, we
conclude that every bounded subset K of R^n embeds into the Hamming cube {-1,
1}^m with a small distortion in the Gromov-Haussdorf metric. Since for many
sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension
reduction for sets in Euclidean spaces.Comment: 17 pages, 3 figures, minor update
Nonlinear Dimension Reduction for Microarray Data
[Biomedical imaging
On dimension reduction in Gaussian filters
A priori dimension reduction is a widely adopted technique for reducing the
computational complexity of stationary inverse problems. In this setting, the
solution of an inverse problem is parameterized by a low-dimensional basis that
is often obtained from the truncated Karhunen-Loeve expansion of the prior
distribution. For high-dimensional inverse problems equipped with smoothing
priors, this technique can lead to drastic reductions in parameter dimension
and significant computational savings.
In this paper, we extend the concept of a priori dimension reduction to
non-stationary inverse problems, in which the goal is to sequentially infer the
state of a dynamical system. Our approach proceeds in an offline-online
fashion. We first identify a low-dimensional subspace in the state space before
solving the inverse problem (the offline phase), using either the method of
"snapshots" or regularized covariance estimation. Then this subspace is used to
reduce the computational complexity of various filtering algorithms - including
the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within
a novel subspace-constrained Bayesian prediction-and-update procedure (the
online phase). We demonstrate the performance of our new dimension reduction
approach on various numerical examples. In some test cases, our approach
reduces the dimensionality of the original problem by orders of magnitude and
yields up to two orders of magnitude in computational savings
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