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Session III.2 - Approximation Theory

Wednesday, June 21, 16:30 ~ 17:00

Optimal sampling for $L^2$ approximation: from linear to nonlinear models

Anthony Nouy

Nantes Université - Centrale Nantes, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the approximation of functions in $L^2$ from point evaluations, using linear or nonlinear approximation tools. For linear approximation, recent results show that weighted least-squares projections allow to obtain quasi-optimal approximations with near to optimal sampling budget. This can be achieved by drawing i.i.d. samples from suitable distributions (depending on the linear approximation tool) and subsampling methods. In a first part of this talk, we review different strategies based on i.i.d. sampling and present alternative strategies based on repulsive point processes that allow to achieve the same task with a reduced sampling complexity. In a second part, we show how these methods can be used to approximate functions with nonlinear approximation tools, in an active learning setting, by coupling iterative algorithms and optimal sampling methods for the projection onto successive linear spaces. We particularly focus on the approximation using tree tensor networks, whose architectures allow for an efficient implementation of optimal sampling procedures within coordinate descent algorithms.

Joint work with Bertrand Michel (Nantes Université - Centrale Nantes, France), Charles Miranda (Nantes Université - Centrale Nantes, France, and WIAS Berlin, Germany) and Philipp Trunschke (Nantes Université - Centrale Nantes,, France).

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