Mean Dimension of Ridge Functions

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 $O(1)$.