11,926 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
Approximation with Random Bases: Pro et Contra
In this work we discuss the problem of selecting suitable approximators from
families of parameterized elementary functions that are known to be dense in a
Hilbert space of functions. We consider and analyze published procedures, both
randomized and deterministic, for selecting elements from these families that
have been shown to ensure the rate of convergence in norm of order
, where is the number of elements. We show that both randomized and
deterministic procedures are successful if additional information about the
families of functions to be approximated is provided. In the absence of such
additional information one may observe exponential growth of the number of
terms needed to approximate the function and/or extreme sensitivity of the
outcome of the approximation to parameters. Implications of our analysis for
applications of neural networks in modeling and control are illustrated with
examples.Comment: arXiv admin note: text overlap with arXiv:0905.067
Exact heat kernel on a hypersphere and its applications in kernel SVM
Many contemporary statistical learning methods assume a Euclidean feature
space. This paper presents a method for defining similarity based on
hyperspherical geometry and shows that it often improves the performance of
support vector machine compared to other competing similarity measures.
Specifically, the idea of using heat diffusion on a hypersphere to measure
similarity has been previously proposed, demonstrating promising results based
on a heuristic heat kernel obtained from the zeroth order parametrix expansion;
however, how well this heuristic kernel agrees with the exact hyperspherical
heat kernel remains unknown. This paper presents a higher order parametrix
expansion of the heat kernel on a unit hypersphere and discusses several
problems associated with this expansion method. We then compare the heuristic
kernel with an exact form of the heat kernel expressed in terms of a uniformly
and absolutely convergent series in high-dimensional angular momentum
eigenmodes. Being a natural measure of similarity between sample points
dwelling on a hypersphere, the exact kernel often shows superior performance in
kernel SVM classifications applied to text mining, tumor somatic mutation
imputation, and stock market analysis
Neural Representations for Sensory-Motor Control I: Head-Centered 3-D Target Positions from Opponent Eye Commands
This article describes how corollary discharges from outflow eye movement commands can be transformed by two stages of opponent neural processing into a head-centered representation of 3-D target position. This representation implicitly defines a cyclopean coordinate system whose variables approximate the binocular vergence and spherical horizontal and vertical angles with respect to the observer's head. Various psychophysical data concerning binocular distance perception and reaching behavior are clarified by this representation. The representation provides a foundation for learning head-centered and body-centered invariant representations of both foveated and non-foveated 3-D target positions. It also enables a solution to be developed of the classical motor equivalence problem, whereby many different joint configurations of a redundant manipulator can all be used to realize a desired trajectory in 3-D space.Air Force Office of Scientific Research (URI 90-0175); Defense Advanced Research Projects Agency (AFOSR-90-0083); National Science Foundation (IRI-87-16960, IRI-90-24877
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