Approximation in Hilbert spaces of the Gaussian and other weighted power series kernels

This article considers linear approximation based on function evaluations in reproducing kernel Hilbert spaces of the Gaussian kernel and a more general class of weighted power series kernels on the interval $[-1, 1]$. We derive almost matching upper and lower bounds on the worst-case error, measured both in the uniform and $L^2([-1,1])$-norm, in these spaces. The results show that if the power series kernel expansion coefficients $\alpha_n^{-1}$ decay at least factorially, their rate of decay controls that of the worst-case error. Specifically, (i) the $n$th minimal error decays as $\alpha_n^{{ -1/2}}$ up to a sub-exponential factor and (ii) for any $n$ sampling points in $[-1, 1]$ there exists a linear algorithm whose error is $\alpha_n^{{ -1/2}}$ up to an exponential factor. For the Gaussian kernel the dominating factor in the bounds is $(n!)^{-1/2}$

Small Sample Spaces for Gaussian Processes

It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process X is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a "small" set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of X in the sigma-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of X and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen-Loeve theorem.Peer reviewe

被積分関数の高層偏微分のL1ノルムの増大度が高々指数的である場合の多次元数値積分の加速的な収束と計算容易性

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 坪井 俊, 東京大学教授 山本 昌宏, 東京大学教授 吉田 朋広, 東京大学准教授 長谷川 立, 東京大学准教授 齊藤 宣一, 東京大学名誉教授 楠岡 成雄, 広島大学教授 松本 眞University of Tokyo(東京大学