Gradient density estimation in arbitrary finite dimensions using the method of stationary phase
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We prove that the density function of the gradient of a sufficiently smooth function S : Ω⊂R^d →R, obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of ϕ=(iS/τ) as the free parameter τ→ 0. The result is shown using the stationary phase approximation and standard integration techniques and requires proper ordering of limits. We highlight a relationship with the well-known characteristic function approach to density estimation, and detail why our result is distinct from this approach.
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