We consider the mean dimension of some ridge functions of spherical Gaussian random vectors of dimension $d$. If the ridge function is Lipschitz continuous, then the mean dimension remains bounded as $d\to\infty$. If instead, the ridge function is discontinuous, then the mean dimension depends on a measure of the ridge function's sparsity, and absent sparsity the mean dimension can grow proportionally to $\sqrt{d}$. Preintegrating a ridge function yields a new, potentially much smoother ridge function. We include an example where, if one of the ridge coefficients is bounded away from zero as $d\to\infty$, then preintegration can reduce the mean dimension from $O(\sqrt{d})$ to $O(1)$.

中文翻译：

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岭函数的平均维数

我们考虑维数为 $d$ 的球面高斯随机向量的一些脊函数的平均维数。如果脊函数是 Lipschitz 连续的，则平均维度保持有界 $d\to\infty$。相反，如果脊函数是不连续的，则平均维度取决于脊函数稀疏度的度量，并且没有稀疏度，平均维度可以与 $\sqrt{d}$ 成比例地增长。对脊函数进行预积分会产生一个新的、可能更平滑的脊函数。我们包括一个例子，如果脊系数之一远离零为 $d\to\infty$，那么预积分可以将平均维数从 $O(\sqrt{d})$ 减少到 $O(1) $.