5,023 research outputs found
Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis
We study two classes of extension problems, and their interconnections: (i)
Extension of positive definite (p.d.) continuous functions defined on subsets
in locally compact groups ; (ii) In case of Lie groups, representations of
the associated Lie algebras by unbounded skew-Hermitian
operators acting in a reproducing kernel Hilbert space (RKHS)
.
Why extensions? In science, experimentalists frequently gather spectral data
in cases when the observed data is limited, for example limited by the
precision of instruments; or on account of a variety of other limiting external
factors. Given this fact of life, it is both an art and a science to still
produce solid conclusions from restricted or limited data. In a general sense,
our monograph deals with the mathematics of extending some such given partial
data-sets obtained from experiments. More specifically, we are concerned with
the problems of extending available partial information, obtained, for example,
from sampling. In our case, the limited information is a restriction, and the
extension in turn is the full positive definite function (in a dual variable);
so an extension if available will be an everywhere defined generating function
for the exact probability distribution which reflects the data; if it were
fully available. Such extensions of local information (in the form of positive
definite functions) will in turn furnish us with spectral information. In this
form, the problem becomes an operator extension problem, referring to operators
in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we
have stressed hands-on-examples. Extensions are almost never unique, and so we
deal with both the question of existence, and if there are extensions, how they
relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text
overlap with arXiv:1401.478
Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
Left-invariant PDE-evolutions on the roto-translation group (and
their resolvent equations) have been widely studied in the fields of cortical
modeling and image analysis. They include hypo-elliptic diffusion (for contour
enhancement) proposed by Citti & Sarti, and Petitot, and they include the
direction process (for contour completion) proposed by Mumford. This paper
presents a thorough study and comparison of the many numerical approaches,
which, remarkably, is missing in the literature. Existing numerical approaches
can be classified into 3 categories: Finite difference methods, Fourier based
methods (equivalent to -Fourier methods), and stochastic methods (Monte
Carlo simulations). There are also 3 types of exact solutions to the
PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in
previous works by Duits and van Almsick in 2005. Here we provide an overview of
these 3 types of exact solutions and explain how they relate to each of the 3
numerical approaches. We compute relative errors of all numerical approaches to
the exact solutions, and the Fourier based methods show us the best performance
with smallest relative errors. We also provide an improvement of Mathematica
algorithms for evaluating Mathieu-functions, crucial in implementations of the
exact solutions. Furthermore, we include an asymptotical analysis of the
singularities within the kernels and we propose a probabilistic extension of
underlying stochastic processes that overcomes the singular behavior in the
origin of time-integrated kernels. Finally, we show retinal imaging
applications of combining left-invariant PDE-evolutions with invertible
orientation scores.Comment: A final and corrected version of the manuscript is Published in
Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9),
p.1-50, 201
A Multivariate Fast Discrete Walsh Transform with an Application to Function Interpolation
For high dimensional problems, such as approximation and integration, one
cannot afford to sample on a grid because of the curse of dimensionality. An
attractive alternative is to sample on a low discrepancy set, such as an
integration lattice or a digital net. This article introduces a multivariate
fast discrete Walsh transform for data sampled on a digital net that requires
only operations, where is the number of data points. This
algorithm and its inverse are digital analogs of multivariate fast Fourier
transforms.
This fast discrete Walsh transform and its inverse may be used to approximate
the Walsh coefficients of a function and then construct a spline interpolant of
the function. This interpolant may then be used to estimate the function's
effective dimension, an important concept in the theory of numerical
multivariate integration. Numerical results for various functions are
presented
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}
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