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Session II.7 - Computational Harmonic Analysis and Data Science

Saturday, June 17, 16:30 ~ 17:00

Three Vignettes in Computational Optimal Recovery

Simon Foucart

Texas A&M University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The question addressed in this talk pertains to the utilization of observational data in order to optimally recover functions (or other objects) in a worst-case setting relative a model set based on approximation capabilities. The emphasis is put on the computational realization of the optimal recovery maps. In a first vignette, dealing with the space of continuous functions, I will showcase an algorithm to produce an optimal map---a linear one, to boot---for full recovery when the underlying approximation space is a Chebyshev space. In a second vignette, set in Hilbert spaces, I will indicate how to treat deterministically inaccurate data, especially given an $\ell_1$-bound, and again reveal the optimality of linear recovery maps. In a third vignette, focusing on the estimation of linear functionals but in arbitrary norm spaces, I will show that linear recovery maps are near optimal in the presence of stochastically inaccurate data when the noise distribution is log-concave.

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