789,305 research outputs found
Distributions on partitions, point processes, and the hypergeometric kernel
We study a 3-parametric family of stochastic point processes on the
one-dimensional lattice originated from a remarkable family of representations
of the infinite symmetric group. We prove that the correlation functions of the
processes are given by determinantal formulas with a certain kernel. The kernel
can be expressed through the Gauss hypergeometric function; we call it the
hypergeometric kernel.
In a scaling limit our processes approximate the processes describing the
decomposition of representations mentioned above into irreducibles. As we
showed before, see math.RT/9810015, the correlation functions of these limit
processes also have determinantal form with so-called Whittaker kernel. We show
that the scaling limit of the hypergeometric kernel is the Whittaker kernel.
The integral operator corresponding to the Whittaker kernel is an integrable
operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the
hypergeometric kernel can be considered as a kernel defining a `discrete
integrable operator'.
We also show that the hypergeometric kernel degenerates for certain values of
parameters to the Christoffel-Darboux kernel for Meixner orthogonal
polynomials. This fact is parallel to the degeneration of the Whittaker kernel
to the Christoffel-Darboux kernel for Laguerre polynomials.Comment: AMSTeX, 24 page
Operators for transforming kernels into quasi-local kernels that improve SVM accuracy
Motivated by the crucial role that locality plays in various learning approaches, we present, in the framework of kernel machines for classification, a novel family of operators on kernels able to integrate local information into any kernel obtaining quasi-local kernels. The quasi-local kernels maintain the possibly global properties of the input kernel and they increase the kernel value as the points get closer in the feature space of the input kernel, mixing the effect of the input kernel with a kernel which is local in the feature space of the input one. If applied on a local kernel the operators introduce an additional level of locality equivalent to use a local kernel with non-stationary kernel width. The operators accept two parameters that regulate the width of the exponential influence of points in the locality-dependent component and the balancing between the feature-space local component and the input kernel. We address the choice of these parameters with a data-dependent strategy. Experiments carried out with SVM applying the operators on traditional kernel functions on a total of 43 datasets with di®erent characteristics and application domains, achieve very good results supported by statistical significance
Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)
We introduce a new structured kernel interpolation (SKI) framework, which
generalises and unifies inducing point methods for scalable Gaussian processes
(GPs). SKI methods produce kernel approximations for fast computations through
kernel interpolation. The SKI framework clarifies how the quality of an
inducing point approach depends on the number of inducing (aka interpolation)
points, interpolation strategy, and GP covariance kernel. SKI also provides a
mechanism to create new scalable kernel methods, through choosing different
kernel interpolation strategies. Using SKI, with local cubic kernel
interpolation, we introduce KISS-GP, which is 1) more scalable than inducing
point alternatives, 2) naturally enables Kronecker and Toeplitz algebra for
substantial additional gains in scalability, without requiring any grid data,
and 3) can be used for fast and expressive kernel learning. KISS-GP costs O(n)
time and storage for GP inference. We evaluate KISS-GP for kernel matrix
approximation, kernel learning, and natural sound modelling.Comment: 19 pages, 4 figure
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