Mean Dimension of Ridge Functions

07/01/2019

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by   Christopher R. Hoyt, et al.

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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→∞. 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 √(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→∞, then preintegration can reduce the mean dimension from O(√(d)) to O(1).