18,911 research outputs found
Polyharmonic approximation on the sphere
The purpose of this article is to provide new error estimates for a popular
type of SBF approximation on the sphere: approximating by linear combinations
of Green's functions of polyharmonic differential operators. We show that the
approximation order for this kind of approximation is for
functions having smoothness (for up to the order of the
underlying differential operator, just as in univariate spline theory). This is
an improvement over previous error estimates, which penalized the approximation
order when measuring error in , p>2 and held only in a restrictive setting
when measuring error in , p<2.Comment: 16 pages; revised version; to appear in Constr. Appro
Nonlinear Approximation Using Gaussian Kernels
It is well-known that non-linear approximation has an advantage over linear
schemes in the sense that it provides comparable approximation rates to those
of the linear schemes, but to a larger class of approximands. This was
established for spline approximations and for wavelet approximations, and more
recently by DeVore and Ron for homogeneous radial basis function (surface
spline) approximations. However, no such results are known for the Gaussian
function, the preferred kernel in machine learning and several engineering
problems. We introduce and analyze in this paper a new algorithm for
approximating functions using translates of Gaussian functions with varying
tension parameters. At heart it employs the strategy for nonlinear
approximation of DeVore and Ron, but it selects kernels by a method that is not
straightforward. The crux of the difficulty lies in the necessity to vary the
tension parameter in the Gaussian function spatially according to local
information about the approximand: error analysis of Gaussian approximation
schemes with varying tension are, by and large, an elusive target for
approximators. We show that our algorithm is suitably optimal in the sense that
it provides approximation rates similar to other established nonlinear
methodologies like spline and wavelet approximations. As expected and desired,
the approximation rates can be as high as needed and are essentially saturated
only by the smoothness of the approximand.Comment: 15 Pages; corrected typos; to appear in J. Funct. Ana
Penalized Likelihood and Bayesian Function Selection in Regression Models
Challenging research in various fields has driven a wide range of
methodological advances in variable selection for regression models with
high-dimensional predictors. In comparison, selection of nonlinear functions in
models with additive predictors has been considered only more recently. Several
competing suggestions have been developed at about the same time and often do
not refer to each other. This article provides a state-of-the-art review on
function selection, focusing on penalized likelihood and Bayesian concepts,
relating various approaches to each other in a unified framework. In an
empirical comparison, also including boosting, we evaluate several methods
through applications to simulated and real data, thereby providing some
guidance on their performance in practice
Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint
Inspired by ideas taken from the machine learning literature, new
regularization techniques have been recently introduced in linear system
identification. In particular, all the adopted estimators solve a regularized
least squares problem, differing in the nature of the penalty term assigned to
the impulse response. Popular choices include atomic and nuclear norms (applied
to Hankel matrices) as well as norms induced by the so called stable spline
kernels. In this paper, a comparative study of estimators based on these
different types of regularizers is reported. Our findings reveal that stable
spline kernels outperform approaches based on atomic and nuclear norms since
they suitably embed information on impulse response stability and smoothness.
This point is illustrated using the Bayesian interpretation of regularization.
We also design a new class of regularizers defined by "integral" versions of
stable spline/TC kernels. Under quite realistic experimental conditions, the
new estimators outperform classical prediction error methods also when the
latter are equipped with an oracle for model order selection
Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations
This paper is concerned with polynomial approximations of the spectral
abscissa function (the supremum of the real parts of the eigenvalues) of a
parameterized eigenvalue problem, which are closely related to polynomial chaos
approximations if the parameters correspond to realizations of random
variables.
Unlike in existing works, we highlight the major role of the smoothness
properties of the spectral abscissa function. Even if the matrices of the
eigenvalue problem are analytic functions of the parameters, the spectral
abscissa function may not be everywhere differentiable, even not everywhere
Lipschitz continuous, which is related to multiple rightmost eigenvalues or
rightmost eigenvalues with multiplicity higher than one.
The presented analysis demonstrates that the smoothness properties heavily
affect the approximation errors of the Galerkin and collocation-based
polynomial approximations, and the numerical errors of the evaluation of
coefficients with integration methods. A documentation of the experiments,
conducted on the benchmark problems through the software Chebfun, is publicly
available.Comment: This is a pre-print of an article published in Numerical Algorithms.
The final authenticated version is available online at:
https://doi.org/10.1007/s11075-018-00648-
On Homogeneous Decomposition Spaces and Associated Decompositions of Distribution Spaces
A new construction of decomposition smoothness spaces of homogeneous type is
considered. The smoothness spaces are based on structured and flexible
decompositions of the frequency space . We
construct simple adapted tight frames for that can be used
to fully characterise the smoothness norm in terms of a sparseness condition
imposed on the frame coefficients. Moreover, it is proved that the frames
provide a universal decomposition of tempered distributions with convergence in
the tempered distributions modulo polynomials. As an application of the general
theory, the notion of homogeneous -modulation spaces is introduced.Comment: 27 page
Surface Spline Approximation on SO(3)
The purpose of this article is to introduce a new class of kernels on SO(3)
for approximation and interpolation, and to estimate the approximation power of
the associated spaces. The kernels we consider arise as linear combinations of
Green's functions of certain differential operators on the rotation group. They
are conditionally positive definite and have a simple closed-form expression,
lending themselves to direct implementation via, e.g., interpolation, or
least-squares approximation. To gauge the approximation power of the underlying
spaces, we introduce an approximation scheme providing precise L_p error
estimates for linear schemes, namely with L_p approximation order conforming to
the L_p smoothness of the target function.Comment: 22 pages, to appear in Appl. Comput. Harmon. Ana
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