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Mean Dimension of Ridge Functions

Abstract

We consider the mean dimension of some ridge functions of spherical Gaussian random vectors of dimension dd. If the ridge function is Lipschitz continuous, then the mean dimension remains bounded as dd\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 d\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 dd\to\infty, then preintegration can reduce the mean dimension from O(d)O(\sqrt{d}) to O(1)O(1).

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