Abstract
We consider the problem of the uniform (in \(L_\infty \)) recovery of ridge functions \(f(x)=\varphi (\langle a,x\rangle )\), \(x\in B_2^n\), using noisy evaluations \(y_1\approx f(x^1),\ldots ,y_N\approx f(x^N)\). It is known that for classes of functions \(\varphi \) of finite smoothness the problem suffers from the curse of dimensionality: in order to provide good accuracy for the recovery it is necessary to make exponential number of evaluations. We prove that if \(\varphi \) is analytic in a neighborhood of \([-1,1]\) and the noise is very small, \(\varepsilon \leqslant \exp (-c\log ^2n)\), then there is an efficient algorithm that recovers f with good accuracy using \(O(n\log ^2n)\) function evaluations.
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Acknowledgements
The authors wish to express their gratitude to S.V. Konyagin for his suggestion to consider the case of regular ridge functions and constant encouragement and to the anonymous referees for their careful work and precise comments.
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Communicated by Vladimir N. Temlyakov.
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The authors were supported by the Russian Federation Government Grant No. 14.W03.31.0031.
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Malykhin, Y., Ryutin, K. & Zaitseva, T. Recovery of Regular Ridge Functions on the Ball. Constr Approx 56, 687–708 (2022). https://doi.org/10.1007/s00365-022-09568-3
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DOI: https://doi.org/10.1007/s00365-022-09568-3