1,180 research outputs found
Statistical Mechanics of Learning: A Variational Approach for Real Data
Using a variational technique, we generalize the statistical physics approach
of learning from random examples to make it applicable to real data. We
demonstrate the validity and relevance of our method by computing approximate
estimators for generalization errors that are based on training data alone.Comment: 4 pages, 2 figure
Fast stable direct fitting and smoothness selection for Generalized Additive Models
Existing computationally efficient methods for penalized likelihood GAM
fitting employ iterative smoothness selection on working linear models (or
working mixed models). Such schemes fail to converge for a non-negligible
proportion of models, with failure being particularly frequent in the presence
of concurvity. If smoothness selection is performed by optimizing `whole model'
criteria these problems disappear, but until now attempts to do this have
employed finite difference based optimization schemes which are computationally
inefficient, and can suffer from false convergence. This paper develops the
first computationally efficient method for direct GAM smoothness selection. It
is highly stable, but by careful structuring achieves a computational
efficiency that leads, in simulations, to lower mean computation times than the
schemes based on working-model smoothness selection. The method also offers a
reliable way of fitting generalized additive mixed models
Study of fluid displacement in 3D porous media with an improved multi-component pseudopotential lattice Boltzmann method
We generalize to three dimensions (3D) a recently developed improved
multi-component pseudopotential lattice Boltzmann method and analyze its
applicability to simulate flows through realistic porous media. The model is
validated and characterized via benchmarks, and we investigate its performance
by simulating the displacement of immiscible fluids in 3D geometries. Two
samples are considered, namely, a pack of spheres obtained numerically, and a
Bentheimer sandstone rock sample obtained experimentally. We show that, with
this model it is possible to simulate realistic viscosity ratios, to tune
surface tension independently and, most importantly, to preserve the volume of
trapped fluid. We also evaluate the computational performance of the model on
the Graphical Processing Unit (GPU) and mention the implemented optimizations
to increase the computational speed and reduce the memory requirements.Comment: arXiv admin note: text overlap with arXiv:2111.0866
A Bayesian Approach to Inverse Quantum Statistics
A nonparametric Bayesian approach is developed to determine quantum
potentials from empirical data for quantum systems at finite temperature. The
approach combines the likelihood model of quantum mechanics with a priori
information over potentials implemented in form of stochastic processes. Its
specific advantages are the possibilities to deal with heterogeneous data and
to express a priori information explicitly, i.e., directly in terms of the
potential of interest. A numerical solution in maximum a posteriori
approximation was feasible for one--dimensional problems. Using correct a
priori information turned out to be essential.Comment: 4 pages, 6 figures, revte
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
Detecting Generalized Synchronization Between Chaotic Signals: A Kernel-based Approach
A unified framework for analyzing generalized synchronization in coupled
chaotic systems from data is proposed. The key of the proposed approach is the
use of the kernel methods recently developed in the field of machine learning.
Several successful applications are presented, which show the capability of the
kernel-based approach for detecting generalized synchronization. It is also
shown that the dynamical change of the coupling coefficient between two chaotic
systems can be captured by the proposed approach.Comment: 20 pages, 15 figures. massively revised as a full paper; issues on
the choice of parameters by cross validation, tests by surrogated data, etc.
are added as well as additional examples and figure
Estimation of Regularization Parameters in Multiple-Image Deblurring
We consider the estimation of the regularization parameter for the
simultaneous deblurring of multiple noisy images via Tikhonov regularization.
We approach the problem in three ways. We first reduce the problem to a
single-image deblurring for which the regularization parameter can be estimated
through a classic generalized cross-validation (GCV) method. A modification of
this function is used for correcting the undersmoothing typical of the original
technique. With a second method, we minimize an average least-squares fit to
the images and define a new GCV function. In the last approach, we use the
classical on a single higher-dimensional image obtained by concatanating
all the images into a single vector. With a reliable estimator of the
regularization parameter, one can fully exploit the excellent computational
characteristics typical of direct deblurring methods, which, especially for
large images, makes them competitive with the more flexible but much slower
iterative algorithms. The performance of the techniques is analyzed through
numerical experiments. We find that under the independent homoscedastic and
Gaussian assumptions made on the noise, the three approaches provide almost
identical results with the first single image providing the practical advantage
that no new software is required and the same image can be used with other
deblurring algorithms.Comment: To appear in Astronomy & Astrophysic
New Approaches To Photometric Redshift Prediction Via Gaussian Process Regression In The Sloan Digital Sky Survey
Expanding upon the work of Way and Srivastava 2006 we demonstrate how the use
of training sets of comparable size continue to make Gaussian process
regression (GPR) a competitive approach to that of neural networks and other
least-squares fitting methods. This is possible via new large size matrix
inversion techniques developed for Gaussian processes (GPs) that do not require
that the kernel matrix be sparse. This development, combined with a
neural-network kernel function appears to give superior results for this
problem. Our best fit results for the Sloan Digital Sky Survey (SDSS) Main
Galaxy Sample using u,g,r,i,z filters gives an rms error of 0.0201 while our
results for the same filters in the luminous red galaxy sample yield 0.0220. We
also demonstrate that there appears to be a minimum number of training-set
galaxies needed to obtain the optimal fit when using our GPR rank-reduction
methods. We find that morphological information included with many photometric
surveys appears, for the most part, to make the photometric redshift evaluation
slightly worse rather than better. This would indicate that most morphological
information simply adds noise from the GP point of view in the data used
herein. In addition, we show that cross-match catalog results involving
combinations of the Two Micron All Sky Survey, SDSS, and Galaxy Evolution
Explorer have to be evaluated in the context of the resulting cross-match
magnitude and redshift distribution. Otherwise one may be misled into overly
optimistic conclusions.Comment: 32 pages, ApJ in Press, 2 new figures, 1 new table of comparison
methods, updated discussion, references and typos to reflect version in Pres
Splines and Wavelets on Geophysically Relevant Manifolds
Analysis on the unit sphere found many applications in
seismology, weather prediction, astrophysics, signal analysis, crystallography,
computer vision, computerized tomography, neuroscience, and statistics.
In the last two decades, the importance of these and other applications
triggered the development of various tools such as splines and wavelet bases
suitable for the unit spheres , and the
rotation group . Present paper is a summary of some of results of the
author and his collaborators on generalized (average) variational splines and
localized frames (wavelets) on compact Riemannian manifolds. The results are
illustrated by applications to Radon-type transforms on and
.Comment: The final publication is available at http://www.springerlink.co
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