Recovery of regular ridge functions on the ball

TL;DR

This paper demonstrates that ridge functions with analytic profiles can be efficiently recovered in high dimensions from noisy data using a number of evaluations that scales nearly linearly with the dimension, overcoming the curse of dimensionality.

Contribution

The authors show that for analytic ridge functions, an efficient recovery algorithm exists with near-linear complexity in the dimension, even under small noise conditions.

Findings

Recovery is feasible with $O(n ext{log}^2 n)$ evaluations for analytic $\

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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, $\varepsilon\le\exp(-c\log^2n)$, then there is an efficient algorithm that recovers $f$ with good accuracy using $O(n\log^2n)$ function evaluations.