Abstract : The recovery of high-dimensional functions from point evaluations
or more general linear measurements is a cornerstone of
approximation theory and numerical analysis. While both are
well-developed areas, the recent advances in learning theory
and high-dimensional statistics sparked several new relations and
tools for approximating functions. The research area hence has
seen a quite remarkable synthesis of old and new results,
especially in the context of nonlinear models such as neural
networks and randomized techniques.
In this minisymposium we aim at highlighting recent developments
of high-dimensional regression and sampling with modern
applications in machine learning and function approximation.
Jochen Garcke (Universität Bonn, and Fraunhofer SCAI)
Matthieu Dolbeault (Sorbonne University)
Takashi Goda (The University of Tokyo)
Stefan Heinrich (University of Kaiserslautern)
Frances Kuo (UNSW Sydney)
Joscha Prochno (University of Passau)
Anthony Nouy (Nantes Université)
Nikolaus Nüsken (King's College London)
Max Pfeffer (University of Chemnitz)
Nicolas Nagel (TU Chemnitz)
Frederiek Wesel (University of Delft)
Jan Vybiral (Czech Technical University)
Talks in Minisymposium :
[04266] Weighted least-squares approximation in expected $L^2$ norm
Author(s) :
Matthieu Dolbeault (Sorbonne Université)
Albert Cohen (Sorbonne Université)
Abdellah Chkifa (Mohammed VI Polytechnic University)
Abstract : We investigate the problem of approximating a function in $L^2$ with a linear space of functions of dimension $n$, using only evaluations at m chosen points. We improve on earlier results based on the solution to the Kadison-Singer problem, by using a randomized greedy strategy, which allows to reduce the oversampling ratio $m/n$ and provides an algorithm of polynomial complexity.
[04593] Polynomial tractability for integrating functions with slowly decaying Fourier series
Author(s) :
Takashi Goda (The University of Tokyo)
Abstract : This talk is concerned with high-dimensional numerical integration problem. In this context, polynomial tractability refers to the scenario where the minimum number of function evaluations required to make the worst-case error less than or equal to a tolerance $\varepsilon$ grows only polynomially with respect to $\varepsilon^{-1}$ and the dimension $d$. For function spaces that are \emph{unweighted}, meaning that all variables contribute equally to the norm of functions, there are many negative results known in the literature that exhibit the curse of dimensionality. The aim of this paper is to show a contrasting result by introducing a non-trivial unweighted function space with absolutely convergent Fourier series that exhibits polynomial tractability with an explicit quasi-Monte Carlo rule.
