86 research outputs found
Next levels universal bounds for spherical codes: the Levenshtein framework lifted
We introduce a framework based on the Delsarte-Yudin linear programming
approach for improving some universal lower bounds for the minimum energy of
spherical codes of prescribed dimension and cardinality, and universal upper
bounds on the maximal cardinality of spherical codes of prescribed dimension
and minimum separation. Our results can be considered as next level universal
bounds as they have the same general nature and imply, as the first level
bounds do, necessary and sufficient conditions for their local and global
optimality. We explain in detail our approach for deriving second level bounds.
While there are numerous cases for which our method applies, we will emphasize
the model examples of points (-cell) and points (-cell) on
. In particular, we provide a new proof that the -cell is
universally optimal, and furthermore, we completely characterize the optimal
linear programing polynomials of degree at most by finding two new
polynomials, which together with the Cohn-Kumar's polynomial form the vertices
of the convex hull that consists of all optimal polynomials. Our framework also
provides a conceptual explanation of why polynomials of degree are needed
to handle the -cell via linear programming.Comment: 30 pages, 4 figures, 5 tables, submitte
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
In this paper we investigate the approximation properties of kernel
interpolants on manifolds. The kernels we consider will be obtained by the
restriction of positive definite kernels on , such as radial basis
functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d.
For restricted kernels having finite smoothness, we provide a complete
characterization of the native space on \M. After this and some preliminary
setup, we present Sobolev-type error estimates for the interpolation problem.
Numerical results verifying the theory are also presented for a one-dimensional
curve embedded in and a two-dimensional torus
Strictly and non-strictly positive definite functions on spheres
Isotropic positive definite functions on spheres play important roles in
spatial statistics, where they occur as the correlation functions of
homogeneous random fields and star-shaped random particles. In approximation
theory, strictly positive definite functions serve as radial basis functions
for interpolating scattered data on spherical domains. We review
characterizations of positive definite functions on spheres in terms of
Gegenbauer expansions and apply them to dimension walks, where monotonicity
properties of the Gegenbauer coefficients guarantee positive definiteness in
higher dimensions. Subject to a natural support condition, isotropic positive
definite functions on the Euclidean space , such as Askey's and
Wendland's functions, allow for the direct substitution of the Euclidean
distance by the great circle distance on a one-, two- or three-dimensional
sphere, as opposed to the traditional approach, where the distances are
transformed into each other. Completely monotone functions are positive
definite on spheres of any dimension and provide rich parametric classes of
such functions, including members of the powered exponential, Mat\'{e}rn,
generalized Cauchy and Dagum families. The sine power family permits a
continuous parameterization of the roughness of the sample paths of a Gaussian
process. A collection of research problems provides challenges for future work
in mathematical analysis, probability theory and spatial statistics.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP06 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Advances in radial and spherical basis function interpolation
The radial basis function method is a widely used technique for interpolation of scattered data. The method is meshfree, easy to implement independently of the number of dimensions, and for certain types of basis functions it provides spectral accuracy. All these properties also apply to the spherical basis function method, but the class of applicable basis functions, positive definite functions on the sphere, is not as well studied and understood as the radial basis functions for the Euclidean space. The aim of this thesis is mainly to introduce new techniques for construction of Euclidean basis functions and to establish new criteria for positive definiteness of functions on spheres.
We study multiply and completely monotone functions, which are important for radial basis function interpolation because their monotonicity properties are in some cases necessary and in some cases sufficient for the positive definiteness of a function. We enhance many results which were originally stated for completely monotone functions to the bigger class of multiply monotone functions and use those to derive new radial basis functions. Further, we study the connection of monotonicity properties and positive definiteness of spherical basis functions. In the processes several new sufficient and some new necessary conditions for positive definiteness of spherical radial functions are proven. We also describe different techniques of constructing new radial and spherical basis functions, for example shifts. For the shifted versions in the Euclidean space we prove conditions for positive definiteness, compute their Fourier transform and give integral representations. Furthermore, we prove that the cosine transforms of multiply monotone functions are positive definite under some mild extra conditions. Additionally, a new class of radial basis functions which is derived as the Fourier transforms of the generalised
Gaussian φ(t) = e−tβ is investigated.
We conclude with a comparison of the spherical basis functions, which we derived in this thesis and those spherical basis functions well known. For this numerical test a set of test functions as well as recordings of electroencephalographic data are used to evaluate the performance of the different basis functions
On polarization of spherical codes and designs
In this article we investigate the -point min-max and the max-min
polarization problems on the sphere for a large class of potentials in
. We derive universal lower and upper bounds on the polarization
of spherical designs of fixed dimension, strength, and cardinality. The bounds
are universal in the sense that they are a convex combination of potential
function evaluations with nodes and weights independent of the class of
potentials. As a consequence of our lower bounds, we obtain the
Fazekas-Levenshtein bounds on the covering radius of spherical designs.
Utilizing the existence of spherical designs, our polarization bounds are
extended to general configurations. As examples we completely solve the min-max
polarization problem for points on and show that the
-cell is universally optimal for that problem. We also provide alternative
methods for solving the max-min polarization problem when the number of points
does not exceed the dimension and when . We further show that
the cross-polytope has the best max-min polarization constant among all
spherical -designs of points for ; for , this
statement is conditional on a well-known conjecture that the cross-polytope has
the best covering radius. This max-min optimality is also established for all
so-called centered codes
Energy bounds for codes and designs in Hamming spaces
We obtain universal bounds on the energy of codes and for designs in Hamming
spaces. Our bounds hold for a large class of potential functions, allow unified
treatment, and can be viewed as a generalization of the Levenshtein bounds for
maximal codes.Comment: 25 page
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