A multivariate ridge function is a function of the form $f\left(x\right)=g\left(a^{T}x\right)$, where $g$ is univariate and $a\in {\mathbb{R}}^d$. We show that the recovery of an unknown ridge function defined on the hypercube ${[-1,1]}^d$ with Lipschitz-regular profile $g$ suffers from the curse of dimensionality when the recovery error is measured in the $L_{\infty }$-norm, even if we allow randomized algorithms. If a limited number of components of $a$ is substantially larger than the others, then the curse of dimensionality is not present and the problem is weakly tractable, provided the profile $g$ is sufficiently regular.

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超立方体上的岭函数的恢复遭受维数的诅咒

多元岭函数是形式的函数 $F（X）=G（{\mathrm{一种}}^{Ť}X）$，在哪里 $G$ 是单变量的 $\mathrm{一种}\in {\mathbb{[R}}^d$。我们显示了在超立方体上定义的未知岭函数的恢复${[-\mathrm{1个}，\mathrm{1个}]}^d$ 具有Lipschitz规则轮廓 $G$ 当测量恢复误差时，会遭受尺寸的诅咒。 ${\mathrm{大号}}_{\infty }$-范数，即使我们允许使用随机算法。如果部件数量有限$\mathrm{一种}$ 基本上大于其他维度，则不存在维数的诅咒，并且该问题很难解决。 $G$ 有足够的规律性。