Mean Dimension of Ridge Functions

Abstract
We consider the mean dimension of some ridge functions of spherical Gaussian random vectors of dimension . If the ridge function is Lipschitz continuous, then the mean dimension remains bounded as . 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 . 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 , then preintegration can reduce the mean dimension from to .
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