73,676 research outputs found
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
The square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions
The momentum bands, energy dispersions, and velocities of the charge
fermions and spin-neutral two-spinon fermions of a square-lattice quantum
liquid referring to the Hubbard model on such a lattice of edge length in
the one- and two-electron subspace are studied. The model involves the
effective nearest-neighbor integral and on-site repulsion and can be
experimentally realized in systems of correlated ultra-cold fermionic atoms on
an optical lattice and thus our results are of interest for such systems. Our
investigations profit from a general rotated-electron description, which is
consistent with the model global symmetry. For
the model in the one- and two-electron subspace the discrete momentum values of
the and fermions are good quantum numbers so that in contrast to the
original strongly-correlated electronic problem their interactions are
residual. The use of our description renders an involved many-electron problem
into a quantum liquid with some similarities with a Fermi liquid.Comment: 61 pages, 1 figure, published in Nuclear Physics
Using state space differential geometry for nonlinear blind source separation
Given a time series of multicomponent measurements of an evolving stimulus,
nonlinear blind source separation (BSS) seeks to find a "source" time series,
comprised of statistically independent combinations of the measured components.
In this paper, we seek a source time series with local velocity cross
correlations that vanish everywhere in stimulus state space. However, in an
earlier paper the local velocity correlation matrix was shown to constitute a
metric on state space. Therefore, nonlinear BSS maps onto a problem of
differential geometry: given the metric observed in the measurement coordinate
system, find another coordinate system in which the metric is diagonal
everywhere. We show how to determine if the observed data are separable in this
way, and, if they are, we show how to construct the required transformation to
the source coordinate system, which is essentially unique except for an unknown
rotation that can be found by applying the methods of linear BSS. Thus, the
proposed technique solves nonlinear BSS in many situations or, at least,
reduces it to linear BSS, without the use of probabilistic, parametric, or
iterative procedures. This paper also describes a generalization of this
methodology that performs nonlinear independent subspace separation. In every
case, the resulting decomposition of the observed data is an intrinsic property
of the stimulus' evolution in the sense that it does not depend on the way the
observer chooses to view it (e.g., the choice of the observing machine's
sensors). In other words, the decomposition is a property of the evolution of
the "real" stimulus that is "out there" broadcasting energy to the observer.
The technique is illustrated with analytic and numerical examples.Comment: Contains 14 pages and 3 figures. For related papers, see
http://www.geocities.com/dlevin2001/ . New version is identical to original
version except for URL in the bylin
An analysis of a class of variational multiscale methods based on subspace decomposition
Numerical homogenization tries to approximate the solutions of elliptic
partial differential equations with strongly oscillating coefficients by
functions from modified finite element spaces. We present in this paper a class
of such methods that are very closely related to the method of M{\aa}lqvist and
Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and
Peterseim, these methods do not make explicit or implicit use of a scale
separation. Their compared to that in the work of M{\aa}lqvist and Peterseim
strongly simplified analysis is based on a reformulation of their method in
terms of variational multiscale methods and on the theory of iterative methods,
more precisely, of additive Schwarz or subspace decomposition methods.Comment: published electronically in Mathematics of Computation on January 19,
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