Relationships among Interpolation Bases of Wavelet Spaces and Approximation Spaces

A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis

Research in the Interpolation Representations of Stochastic Processes in the Two Types of Interpolation Knots

The article deals with some interpolation representations of stochastic processes with non-equidistance interpolation knots. Research is based on observations of the process and its derivatives of the first and second orders at some types of knots and observations of the process and its derivatives of the first orders at other types of knots. The necessary results from the theory of entire functions of complex variable are formulated. The function bounded on any bounded region of the complex plane is considered. The estimate of the residual of the interpolation series is obtained. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation is proved. Considering the separability of the process and the convergence of a row that the interpolation row converges to the stochastic process uniformly over in any bounded area of changing of parameter is obtained. The main purpose of this article is the obtained convergence with probability 1 of the corresponding interpolation series to a stochastic process in any bounded domain of changes of parameter. Obtained results may be applied in the modern theory of information transmission

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