MEAN DIMENSION OF RIDGE FUNCTIONS

摘要

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 -> infinity. 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 root 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 -> infinity, then preintegration can reduce the mean dimension from O(root d) to O(1).