We study the problem of approximating functions of $d$ variables in the average case setting for the $L_2$ space $L_{2,d}$ with the standard Gaussian weight equipped with a zero-mean Gaussian measure. The covariance kernel of this Gaussian measure takes the form of a Gaussian kernel with non-increasing positive shape parameters $γ_j^2$ for $j = 1, 2, \dots, d$. The error of approximation is defined in the norm of $L_{2,d}$. We study the average case error of algorithms that use at most $n$ arbitrary continuous linear functionals. The information complexity $n(\varepsilon, d)$ is defined as the minimal number of linear functionals which are needed to find an algorithm whose average case error is at most $\varepsilon$. We study different notions of tractability or exponentially-convergent tractability (EC-tractability) which the information complexity $n(\varepsilon, d)$ describe how behaves as a function of $d$ and $\varepsilon^{-1}$ or as one of $d$ and $(1+\ln\varepsilon^{-1})$.
We find necessary and sufficient conditions on various notions of tractability and EC-tractability in terms of shape parameters. In particular, for any positive $s>0$ and $t\in(0,1)$ we obtain that the sufficient and necessary condition on $γ^2_ j$ for which $$\lim_{d+\varepsilon^{-1}\to\infty}\frac{n(\varepsilon,d)}{\varepsilon^{-s}+d^t}=0$$ holds is $$ \lim_{j\to \infty}j^{1-t}γ_j^2\,\ln^+ γ_j^{-2}=0,$$where $\ln^+ x=\max(1,\ln x)$.
We investigate average case tractability of approximation of additive random fields with marginal random processes corresponding to the Korobov kernels for the non-homogeneous case. We use the absolute error criterion (ABS) or the normalized error criterion (NOR). We show that the problem is always polynomially tractable for ABS or NOR, and give sufficient and necessary conditions for strong polynomial tractability for ABS or NOR.
We study multivariate approximation in the average case setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{\rm all}$ for the absolute error criterion, and show that the power of $Λ^{\rm std}$ is the same as that of $Λ^{\rm all}$ for all notions of algebraic and exponential tr…
We study the worst case tractability of multivariate linear problems defined on separable Hilbert spaces. Information about a problem instance consists of noisy evaluations of arbitrary bounded linear functionals, where the noise is either deterministic or random. The cost of a single evaluation depends on its precision and is controlled by a cost function. We establish mutual interactions between tractability of a problem with noisy information, the cost function, and tractability of the same problem, but with exact information.
In this paper, we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences ${\bf a}=\{a_j\}_{j\geq1}$ and ${\bf b}=\{b_j\}_{j\geq1}$ of positive numbers. We obtain strong equivalences of the approximation numbers, and necessary and sufficient conditions on ${\bf a}$, ${\bf b}$ to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.
We study approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary linear functionals. We use the weighted least squares regression algorithm to obtain the upper estimates of the minimal randomized error using $Λ^{\rm std}$. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{…
Notes on $(s,t)$-weak tractability: A refined classification of problems with (sub)exponential information complexity
University of Warsaw, Faculty of Mathematics, Informatics and Mechanics（华沙大学数学、信息学与力学学院）；Philipps-University Marburg, Faculty of Mathematics and Computer Science（马尔堡菲利普斯大学数学与计算机科学系）
We show that Gaussian process regression (GPR) allows representing multivariate functions with low-dimensional terms via kernel design. When using a kernel built with HDMR (High-dimensional model representation), one obtains a similar type of representation as the previously proposed HDMR-GPR scheme while being faster and simpler to use. We tested the approach on cases where highly accurate machine learning is required from sparse data by fitting potential energy surfaces and kinetic energy densities.
In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between …
We consider tensor product random fields $Y_d$, $d\in\mathbb{N}$, whose covariance funtions are Gaussian kernels. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Namely, we find criteria of boundedness for $n^{Y_d}(\varepsilon)$ on $d$ and of tending $n^{Y_d}(\varepsilon)\to…
In recent years finite tensor products of reproducing kernel Hilbert spaces (RKHSs) of Gaussian kernels on the one hand and of Hermite spaces on the other hand have been considered in tractability analysis of multivariate problems. In the present paper we study countably infinite tensor products for both types of spaces. We show that the incomplete tensor product in the sense of von Neumann may be identified with an RKHS whose domain is a proper subset of the sequence space $\mathbb{R}^\mathbb{N}$. Moreover, we show that each tensor product of spaces of Gaussian kernels having square-summable …
