Abstract
We compare the average errors of n-term linear and nonlinear approximations assuming that the coefficients in an orthogonal expansion of the approximated element are scaled i.i.d. random variables. We show that generally the n-term nonlinear approximation can be even exponentially better than the n-term linear approximation. On the other hand, if the scaling parameters decay no faster than polynomially then the average errors of nonlinear approximations do not converge to zero faster than those of linear approximations, as nāāā+ā. The main motivation and application is the approximation of Gaussian processes. In this particular case, the nonlinear approximation is, roughly, no more than n times better than its linear counterpart.
Dedicated to Ian H. Sloan on the occasion of his 80th birthday.
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Notes
- 1.
The word āadaptiveā here means that the removed components depend on x.
- 2.
For two positive sequences, we write a nāāāb n iff there are 0ā<ācā<āCā<ā+ā and n 0 such that cāā¤āa nāb nāā¤āC holds for all nāā„ān 0. We write a nāāāb n iff limnā+āa nāb nā=ā1.
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Plaskota, L. (2018). On Linear Versus Nonlinear Approximation in the Average Case Setting. In: Dick, J., Kuo, F., WoÅŗniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_46
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