Recovery of regular ridge functions on the ball
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We consider the problem of the uniform (in L_∞) recovery of ridge functions f(x)=φ(⟨ a,x⟩), x∈ B_2^n, using noisy evaluations y_1≈ f(x^1),…,y_N≈ f(x^N). It is known that for classes of functions φ 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 φ is analytic in a neighborhood of [-1,1] and the noise is small, then there is an efficient algorithm that recovers f with good accuracy using ≍ nlog^2n function evaluations.
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